Graphing linear equations is a elementary ability in arithmetic, offering perception into the connection between variables. Amongst these equations, y = 1/2x holds a novel place with its simplicity and prevalence. Whether or not you are a pupil encountering it for the primary time or an skilled grapher looking for a refresher, understanding easy methods to graph y = 1/2x is important. This text will information you thru the method, shedding gentle on its properties and offering a step-by-step method to visualizing this equation.

The graph of y = 1/2x is a straight line that passes by means of the origin (0, 0). In contrast to different strains with an intercept, this line has a slope of 1/2, which determines its steepness and route. Constructive values of x will produce constructive values of y, leading to a line that rises as you progress from left to proper. Conversely, unfavorable values of x will produce unfavorable values of y, making a line that slopes downward. This distinct conduct distinguishes y = 1/2x from different linear equations and highlights its distinctive traits.

To plot this line, observe these steps. First, find the y-intercept, which is (0, 0) on this case. This level represents the intersection of the road with the y-axis. From there, use the slope to find out the route and steepness of the road. With a slope of 1/2, transfer up 1 unit and proper 2 models from the y-intercept. This offers you one other level on the road, (2, 1). Repeat this course of till you have got a number of factors, after which join them to kind the road. The ensuing graph will probably be a straight line passing by means of the origin and exhibiting the attribute slope of 1/2.

Understanding the Equation of a Parabola

A parabola is a U-shaped curve that’s symmetrical a few vertical axis. The equation of a parabola will be written within the kind
$$ y = ax^2 + bx + c $$
the place a, b, and c are constants.

The worth of a determines the form of the parabola. If a is constructive, the parabola opens upward. If a is unfavorable, the parabola opens downward. Absolutely the worth of a determines the steepness of the parabola. A bigger worth of a will lead to a steeper parabola.

The worth of b determines the horizontal shift of the parabola. If b is constructive, the parabola is shifted to the left. If b is unfavorable, the parabola is shifted to the appropriate. The worth of b is the same as -2 instances the x-coordinate of the vertex of the parabola.

The worth of c determines the vertical shift of the parabola. If c is constructive, the parabola is shifted up. If c is unfavorable, the parabola is shifted down. The worth of c is the same as the y-coordinate of the vertex of the parabola.

To graph a parabola, you need to use the next steps:

1. Discover the vertex of the parabola. The vertex is the purpose the place the parabola modifications route.

2. Plot the vertex on the coordinate airplane.

3. Discover two different factors on the parabola.

4. Draw a clean curve by means of the three factors.

The next desk summarizes the important thing options of a parabola:

Characteristic	Equation
Form	a
Horizontal shift	b
Vertical shift	c
Vertex	(-b/2a, c – b2/4a)

Figuring out the Vertex of the Parabola

Figuring out the vertex of a parabola is a vital step in graphing it precisely. To find out the vertex, it’s essential to use the components x = -b/2a. Let’s apply this components to the equation y = 1/2x:

1. **Step 1: Establish the values of a and b.** On this case, a = 1/2 and b = 0.
2. **Step 2: Calculate x:** Utilizing the components x = -b/2a, we have now x = -(0)/2(1/2) = 0.
3. **Step 3: Calculate y:** To seek out the y-coordinate of the vertex, plug the x-value into the unique equation: y = 1/2(0) = 0.

Due to this fact, the vertex of the parabola y = 1/2x is (0, 0).

Extra Particulars:

The vertex of a parabola represents the purpose the place the parabola modifications route. It’s the lowest or highest level on the graph, relying on whether or not the parabola opens up or down, respectively.

The components for locating the vertex of a parabola within the kind y = ax^2 + bx + c is:
“`
x = -b/2a
y = f(x) = a(x^2) + bx + c
“`

This is a abstract of the method in tabular kind:

Step	Components
Establish a and b	a = coefficient of x^2 b = coefficient of x
Calculate x-coordinate	x = -b/2a
Calculate y-coordinate	y = f(x)

Understanding the vertex of a parabola is important for graphing it precisely and analyzing its key traits.

Figuring out the Axis of Symmetry

In arithmetic, the axis of symmetry is a vertical line that divides a graph into two symmetrical halves. For the equation y = a(x – h)^2 + okay, the axis of symmetry is x = h. To find out the axis of symmetry for the equation y = 1/2x, we have to convert it into the usual kind y = a(x – h)^2 + okay.

Step 1: Multiply either side of the equation by 2 to do away with the fraction.

“`
2(y) = 2(1/2x)
2y = x
“`

Step 2: Rewrite the equation in the usual kind y = ax^2 + bx + c.

“`
2y = x
2y – x = 0
“`

Step 3: Convert the equation to the usual kind y = a(x – h)^2 + okay.

“`
2y – x = 0
2y = x
y = 1/2x
“`

Because the equation is already in the usual kind, we will see that the axis of symmetry is x = 0.

Extra Info

The axis of symmetry is a crucial property of a parabola. It may be used to seek out the vertex, focus, and directrix of the parabola. The vertex is the purpose the place the parabola modifications route. The main target is the purpose that the parabola is concentrated on. The directrix is a line that’s perpendicular to the axis of symmetry and passes by means of the main focus.

The next desk summarizes the important thing properties of a parabola:

Property	Components
Vertex	(h, okay)
Focus	(h + p, okay)
Directrix	x = h – p
Axis of Symmetry	x = h

the place p is the space from the vertex to the main focus or directrix.

Discovering the Intercepts

The intercepts of a linear equation are the factors the place the graph crosses the x-axis and y-axis.

To seek out the x-intercept, set y = 0 and remedy for x.

y – 1 = 2x

0 – 1 = 2x

-1 = 2x

x = -1/2

So, the x-intercept is (-1/2, 0).

To seek out the y-intercept, set x = 0 and remedy for y.

y – 1 = 2(0)

y – 1 = 0

y = 1

So, the y-intercept is (0, 1).

The intercepts of the road y – 1 = 2x are (-1/2, 0) and (0, 1).

Plotting Factors on the Parabola

To plot factors on the parabola of the equation y = 1/2x², decide two $x$ values at which to calculate the corresponding $y$ values. One $x$ worth needs to be constructive, and the opposite needs to be unfavorable. This is a step-by-step information on plotting factors:

Step 1: Select $x$ values

Choose two $x$ values, one constructive and one unfavorable. As an example, contemplate $x = 2$ and $x = -2$.

Step 2: Calculate $y$ values

For every $x$ worth, substitute it into the equation $y = 1/2x²$ and calculate the corresponding $y$ worth. For $x = 2$, we get $y = 1/2(2)² = 2$, and for $x = -2$, we get $y = 1/2(-2)² = 2$.

Step 3: Create a desk of values

Manage the $x$ and $y$ values in a desk. This desk helps visualize the connection between the $x$ and $y$ values.

$x$	$y$
2	2
-2	2

Step 4: Plot the factors on a coordinate airplane

Use the $x$ and $y$ values from the desk to plot the factors on a coordinate airplane. The purpose (2, 2) is plotted within the first quadrant, and the purpose (-2, 2) is plotted within the third quadrant.

Step 5: Join the factors with a clean curve

A parabola is a clean, U-shaped curve. As soon as the factors are plotted, join them with a clean curve. This curve represents the graph of the equation $y = 1/2x²$.

Drawing the Graph of Y = 1/2x

1. Intercept

The intercept is the purpose the place the graph crosses the y-axis. To seek out the y-intercept of Y = 1/2x, we set x = 0 and remedy for y:

y = 1/2(0)
y = 0

Due to this fact, the y-intercept of Y = 1/2x is (0, 0).

2. Slope

The slope of a line is a measure of its steepness. To seek out the slope of Y = 1/2x, we will use the components:

slope = (change in y) / (change in x)

Let’s take two factors on the road: (0, 0) and (1, 1/2). The change in y is:

change in y = 1/2 - 0 = 1/2

The change in x is:

change in x = 1 - 0 = 1

Due to this fact, the slope of Y = 1/2x is:

slope = 1/2

3. Plotting Factors

To graph Y = 1/2x, we will plot just a few factors and join them with a line. Listed below are just a few factors on the graph:

• (0, 0)
• (1, 1/2)
• (-1, -1/2)

4. Drawing the Line

As soon as we have now plotted just a few factors, we will join them with a straight line. The road ought to go by means of the origin and have a slope of 1/2.

5. Asymptotes

An asymptote is a line {that a} curve approaches however by no means touches. Y = 1/2x has two asymptotes:

• Vertical asymptote: x = 0
• Horizontal asymptote: y = 0

The vertical asymptote is a vertical line that the graph approaches as x will get nearer and nearer to 0. The horizontal asymptote is a horizontal line that the graph approaches as x will get bigger and bigger.

6. Vertical Line Take a look at

The vertical line take a look at is a strategy to decide if a graph represents a operate. A graph represents a operate if each vertical line intersects the graph at most as soon as.

To carry out the vertical line take a look at on Y = 1/2x, we will draw a vertical line at any x-value. For instance, let’s draw a vertical line at x = 1. The road intersects the graph at (1, 1/2). There isn’t a different level on the graph that intersects the vertical line. Due to this fact, Y = 1/2x passes the vertical line take a look at and represents a operate.

Detailed Rationalization of the Vertical Line Take a look at

The vertical line take a look at is predicated on the definition of a operate. A operate is a relation that assigns to every aspect of a set a novel aspect of one other set. In different phrases, a operate is a rule that assigns a novel output to every enter.

The vertical line take a look at is a strategy to decide if a graph represents a operate as a result of it assessments whether or not each enter has a novel output. If a vertical line intersects a graph at a couple of level, then there may be a minimum of one enter that has two completely different outputs. Which means that the graph doesn’t characterize a operate.

Within the case of Y = 1/2x, each vertical line intersects the graph at most as soon as. Which means that each enter has a novel output, and due to this fact Y = 1/2x represents a operate.

The vertical line take a look at is a great tool for figuring out if a graph represents a operate. It’s a easy take a look at that may be utilized to any graph.

The best way to Graph y = 1/2x

Shifting the Graph Vertically: Including or Subtracting a Fixed

Including a Fixed

To shift the graph of y = 1/2x vertically upward by a models, add a to the right-hand aspect of the equation:

“`
y = 1/2x + a
“`

To shift the graph downward by a models, subtract a from the right-hand aspect of the equation:

“`
y = 1/2x – a
“`

Instance

To graph the operate y = 1/2x + 2, add 2 to every y-coordinate of the unique graph. The result’s a graph that’s shifted 2 models upward.

| x | y = 1/2x | y = 1/2x + 2 |
|—|—|—|
| -2 | -1 | 1 |
| -1 | -1/2 | 1.5 |
| 0 | 0 | 2 |
| 1 | 1/2 | 2.5 |
| 2 | 1 | 3 |

Subtracting a Fixed

To shift the graph of y = 1/2x vertically downward by a models, subtract a from the right-hand aspect of the equation:

“`
y = 1/2x – a
“`

To shift the graph upward by a models, add a to the right-hand aspect of the equation:

“`
y = 1/2x + a
“`

Instance

To graph the operate y = 1/2x – 1, subtract 1 from every y-coordinate of the unique graph. The result’s a graph that’s shifted 1 unit downward.

| x | y = 1/2x | y = 1/2x – 1 |
|—|—|—|
| -2 | -1 | -2 |
| -1 | -1/2 | -1.5 |
| 0 | 0 | -1 |
| 1 | 1/2 | -0.5 |
| 2 | 1 | 0 |

Shifting the Graph Horizontally: Multiplying by a Fixed

Multiplying the enter of a operate by a relentless will shift the graph horizontally. For instance, the graph of y = x² will be shifted 3 models to the left by multiplying the enter by 3, ensuing within the operate y = (3x)². This shift is as a result of the values of x at the moment are 3 instances smaller, which implies that the graph will probably be stretched horizontally by an element of three.

On the whole, if f(x) is any operate and c is a continuing, then the operate g(x) = f(cx) would be the graph of f(x) shifted c models to the appropriate if c is constructive and c models to the left if c is unfavorable.

For instance, the graph of y = x² will be shifted 2 models to the appropriate by multiplying the enter by 1/2, ensuing within the operate y = (1/2x)². This shift is as a result of the values of x at the moment are 2 instances bigger, which implies that the graph will probably be compressed horizontally by an element of two.

The desk beneath summarizes the results of multiplying the enter of a operate by a relentless:

Fixed	Shift
c > 0	c models to the appropriate
c < 0	c models to the left
c = 1	No shift

Multiplying the enter of a operate by a relentless is usually a useful gizmo for remodeling graphs. For instance, it may be used to shift a graph in order that it passes by means of a particular level or to align it with one other graph.

Listed below are some further examples of how multiplying the enter of a operate by a relentless can be utilized to shift its graph:

• To shift the graph of y = x² up 2 models, multiply the enter by 1/2, ensuing within the operate y = (1/2x)².
• To shift the graph of y = x³ down 3 models, multiply the enter by -1, ensuing within the operate y = (-x)³.
• To shift the graph of y = sin(x) to the left by π/2 models, multiply the enter by 2, ensuing within the operate y = sin(2x).

Multiplying the enter of a operate by a relentless is a straightforward however highly effective device that can be utilized to rework graphs in quite a lot of methods.

Reflecting the Graph over the x-axis

While you replicate the graph of y = 1 – 2x over the x-axis, the ensuing graph is y = -1 – 2x. It’s because the y-coordinate of every level on the unique graph is negated when the graph is mirrored over the x-axis. For instance, the purpose (1, -1) on the unique graph turns into the purpose (1, 1) on the mirrored graph.

To replicate the graph of y = 1 – 2x over the x-axis, you need to use the next steps:

1. Plot the unique graph of y = 1 – 2x.
2. For every level (x, y) on the unique graph, discover the corresponding level (-x, -y) on the mirrored graph.
3. Plot the factors from step 2 to create the mirrored graph.

The next desk reveals the coordinates of the unique graph and the mirrored graph:

Authentic Graph	Mirrored Graph
(1, -1)	(1, 1)
(2, -3)	(2, 3)
(-1, 3)	(-1, -3)

As you’ll be able to see from the desk, the y-coordinates of the factors on the mirrored graph are the negatives of the y-coordinates of the factors on the unique graph. It’s because the graph is mirrored over the x-axis.

Reflecting the graph of y = 1 – 2x over the x-axis is a straightforward transformation that may be carried out in just a few steps. By following the steps outlined above, you’ll be able to simply create the mirrored graph.

Examples of Graphing Y = 1/2x in Completely different Kinds

Slope-Intercept Kind

In slope-intercept kind, an equation is written as y = mx + b, the place m is the slope and b is the y-intercept. To graph y = 1/2x in slope-intercept kind:

1. Plot the y-intercept: Begin by plotting the y-intercept, which is 0. Mark some extent at (0, 0).
2. Discover one other level: Select any non-zero worth for x and calculate the corresponding y-value utilizing the equation y = 1/2x. For instance, in the event you select x = 2, then y = 1/2(2) = 1. So, mark some extent at (2, 1).
3. Draw a straight line: Join the 2 factors you have got plotted utilizing a straight line. This line represents the graph of y = 1/2x in slope-intercept kind.

Level-Slope Kind

In point-slope kind, an equation is written as y – y₁ = m(x – x₁), the place (x₁, y₁) is some extent on the road and m is the slope.

1. Select some extent: Choose some extent on the road, akin to (2, 1) from the earlier instance.
2. Calculate the slope: The slope of the road is 1/2.
3. Write the equation: Substitute the purpose and slope into the point-slope kind equation: y – 1 = 1/2(x – 2).

Intercept Kind

Intercept kind is written as y = a/b, the place a is the y-intercept and b is the x-intercept. To graph y = 1/2x in intercept kind:

1. Plot the y-intercept: The y-intercept is 0, so mark some extent at (0, 0).
2. Discover the x-intercept: To seek out the x-intercept, set y = 0 and remedy for x: 0 = 1/2x => x = 0. So, the x-intercept is 0.
3. Draw a vertical line: The graph of y = 1/2x in intercept kind is a vertical line passing by means of the y-intercept and the x-intercept.

Normal Kind

Normal kind is written as Ax + By + C = 0. To graph y = 1/2x in customary kind, reorganize the equation to:

2x – y = 0

Utilizing the usual kind, we will decide the slopes and intercepts straight:

• Slope: The coefficient of x, 2, represents the slope of the road.
• x-intercept: The coefficient of y, -1, represents the x-intercept when y = 0: (0, -1).
• y-intercept: The fixed time period, 0, represents the y-intercept when x = 0: (0, 0).

Two-Level Kind

Two-point kind is written as y – y₁ = (y₂ – y₁)/(x₂ – x₁) (x – x₁), the place (x₁, y₁) and (x₂, y₂) are two factors on the road.

1. Select two factors: Choose two factors on the road, akin to (2, 1) and (4, 2) from the slope-intercept kind instance.
2. Substitute the factors: Plug the factors into the two-point kind equation: y – 1 = (2 – 1)/(4 – 2) (x – 2).
3. Simplify: Simplify the equation to get: y – 1 = 1/2 (x – 2).

Desk of Graphing Strategies and Equations

Kind	Equation	Plotting Steps
Slope-Intercept Kind	y = 1/2x	– Plot y-intercept (0, 0) – Discover one other level (2, 1) – Draw a line
Level-Slope Kind	y – 1 = 1/2(x – 2)	– Plot level (2, 1) – Use slope (1/2) to seek out route
Intercept Kind	y = 0	– Plot y-intercept (0, 0) – Draw a vertical line
Normal Kind	2x – y = 0	– Discover x-intercept (0, -1) – Discover y-intercept (0, 0) – Draw a line by means of intercepts
Two-Level Kind	y – 1 = 1/2 (x – 2)	– Plot factors (2, 1) and (4, 2) – Use factors to find out slope (1/2)

Discovering the Equation of a Parabola from its Graph

A parabola is a U-shaped curve that opens both upward or downward. The equation of a parabola is often written within the kind y = ax² + bx + c, the place a, b, and c are constants. To seek out the equation of a parabola from its graph, it’s essential to know the coordinates of the vertex (the purpose the place the parabola modifications route) and a minimum of one different level on the curve.

Steps

1. Establish the vertex. The vertex is the very best or lowest level on the parabola. It’s situated on the level (h, okay), the place h is the x-coordinate of the vertex and okay is the y-coordinate of the vertex.

2. Discover the slope of the road that passes by means of the vertex and one other level on the curve. To do that, use the slope components: m = (y₂ – y₁)/(x₂ – x₁), the place (x₁, y₁) are the coordinates of the vertex and (x₂, y₂) are the coordinates of one other level on the curve.

3. Substitute the vertex and slope into the equation y = mx + b. This gives you the equation of the road that passes by means of the vertex and one other level on the curve.

4. Discover the y-intercept of the road. The y-intercept is the purpose the place the road crosses the y-axis. To seek out the y-intercept, set x = 0 within the equation of the road.

5. Substitute the vertex and y-intercept into the equation y = ax² + bx + c. This gives you the equation of the parabola.

Instance

For instance we have now a parabola with a vertex at (2, -1) and one other level on the curve at (3, 4). To seek out the equation of the parabola, we will observe the steps above:

1. Establish the vertex. The vertex is situated at (2, -1).

2. Discover the slope of the road that passes by means of the vertex and one other level on the curve. Utilizing the slope components, we get:

m = (4 - (-1))/(3 - 2) = 5

3. Substitute the vertex and slope into the equation y = mx + b. We get:

y = 5x + b

4. Discover the y-intercept of the road. Setting x = 0 within the equation of the road, we get:

y = 5(0) + b = b

5. Substitute the vertex and y-intercept into the equation y = ax² + bx + c. We get:

-1 = a(2)² + b(2) + c
-1 = 4a + 2b + c

We will additionally use the opposite level on the curve to get one other equation:

4 = a(3)² + b(3) + c
4 = 9a + 3b + c

Fixing these two equations concurrently, we get:

a = -1
b = 3
c = 0

Due to this fact, the equation of the parabola is:

y = -x² + 3x

Discovering the Coordinates of the Vertex from the Equation

The vertex of a parabola is the purpose the place it modifications route. For a parabola within the kind y = ax^2 + bx + c, the vertex is situated on the level (-b/2a, -D/4a), the place D is the discriminant, calculated as b^2 – 4ac.

Within the case of y = 1/2x^2 – x, the vertex will be discovered as follows:

Step 1: Establish the Coefficients

a = 1/2

b = -1

Step 2: Calculate the Discriminant

D = b^2-4ac = (-1)^2-4(1/2)(0) = 1

Step 3: Discover the x-Coordinate of the Vertex

x = -b/2a = -(-1)/2(1/2) = 1

Step 4: Discover the y-Coordinate of the Vertex

y = -D/4a = -1/4(1/2) = -1/2

Step 5: Write the Coordinates of the Vertex

The vertex of the parabola y = 1/2x^2 – x is situated on the level (1, -1/2).

Vertex Desk

The coordinates of the vertex will be summarized in a desk:

x-Coordinate	y-Coordinate
1	-1/2

Figuring out the Area and Vary of the Parabola

The area of a operate represents the set of all attainable enter values, whereas the vary represents the set of all attainable output values. Within the context of the parabola given by the equation y = 1/2x, figuring out the area and vary entails analyzing the expression and contemplating its properties.

Area

The area of a operate usually consists of all actual numbers until there are particular restrictions imposed by the expression. Within the case of y = 1/2x, there isn’t any such restriction. The impartial variable x can tackle any actual worth, each constructive and unfavorable. Due to this fact, the area of the parabola y = 1/2x is:

Area: All actual numbers (-∞, ∞)

Vary

The vary of a operate is dependent upon the form and traits of the operate’s graph. For the parabola y = 1/2x, understanding its vary requires analyzing its conduct as x varies.

The parabola y = 1/2x is an open-down parabola. Which means that as x will increase, the values of y lower. Conversely, as x decreases, the values of y improve.

Nevertheless, there may be one essential limitation to the vary of the parabola. As x approaches zero, the worth of y approaches infinity. It’s because the denominator of the expression turns into smaller and smaller as x will get nearer to zero, inflicting the fraction to develop into bigger and bigger.

Then again, as x approaches unfavorable or constructive infinity, the worth of y approaches zero. It’s because the denominator of the expression turns into bigger and bigger as x will get additional away from zero, inflicting the fraction to develop into smaller and smaller.

Due to this fact, the vary of the parabola y = 1/2x is as follows:

Vary: All actual numbers besides y = 0 (0, ∞) or (-∞, 0)

Tabulated Abstract:

Property	Description
Area	All actual numbers (-∞, ∞)
Vary	All actual numbers besides y = 0 (0, ∞) or (-∞, 0)

Sketching the Graph of Y = 1/2x Shortly

Graphing the equation y = 1/2x is a comparatively easy course of that may be accomplished shortly and simply. By following these steps, you’ll be able to create an correct graph of this linear operate very quickly.

1. Discover the Slope

The slope of a linear equation is a measure of its steepness. To seek out the slope of y = 1/2x, we will use the next components:

“`
slope = m = -1/2
“`

The unfavorable signal signifies that the graph of y = 1/2x may have a unfavorable slope, which means that it’s going to slope downward from left to proper.

2. Discover the Y-Intercept

The y-intercept of a linear equation is the purpose the place the graph crosses the y-axis. To seek out the y-intercept of y = 1/2x, we will substitute x = 0 into the equation:

“`
y = 1/2(0) = 0
“`

Due to this fact, the y-intercept of y = 1/2x is (0, 0).

3. Plot the Factors

Now that we have now the slope and the y-intercept, we will plot two factors on the graph. One level will probably be on the y-axis, and the opposite level will probably be on the road itself.

The y-intercept is (0, 0), so we will plot that time first. To discover a second level, we will use the slope to calculate the change in y for a given change in x. For instance, if we transfer 2 models to the appropriate (Δx = 2), the change in y will probably be:

“`
Δy = mΔx = (-1/2)(2) = -1
“`

Due to this fact, the second level is (2, -1).

4. Draw the Line

As soon as we have now plotted two factors on the graph, we will draw the road that passes by means of them. The road ought to have a unfavorable slope, and it ought to go by means of the y-intercept (0, 0).

5. Examine Your Work

Upon getting drawn the graph, it’s at all times a good suggestion to test your work. You are able to do this by substituting just a few completely different values of x into the equation and verifying that the ensuing y-values match the factors in your graph.

18. Purposes of the Graph of Y = 1/2x

The graph of y = 1/2x has quite a few functions in varied fields, together with physics, engineering, and economics. Listed below are just a few examples:

a. Inverse Proportionality

The graph of y = 1/2x represents an inverse proportional relationship between y and x. Which means that as x will increase, y decreases, and vice versa. This relationship is usually encountered in conditions the place two portions are inversely proportional, akin to the connection between the space traveled and the time taken to journey that distance.

b. Velocity and Time

In physics, the graph of y = 1/2x can be utilized to characterize the connection between velocity and time for an object shifting with fixed acceleration. The slope of the graph represents the acceleration, and the y-intercept represents the preliminary velocity.

c. Provide and Demand

In economics, the graph of y = 1/2x can be utilized to characterize the connection between provide and demand. The slope of the graph represents the elasticity of provide, and the y-intercept represents the equilibrium value.

d. Price and Income

In enterprise, the graph of y = 1/2x can be utilized to characterize the connection between value and income. The slope of the graph represents the marginal value, and the y-intercept represents the mounted value.

e. Movement of a Projectile

In physics, the graph of y = 1/2x can be utilized to characterize the trajectory of a projectile launched at a sure angle. The slope of the graph represents the vary of the projectile, and the y-intercept represents the utmost top reached by the projectile.

Software	Description
Inverse Proportionality	Represents an inverse proportional relationship between two portions.
Velocity and Time	Represents the connection between velocity and time for an object shifting with fixed acceleration.
Provide and Demand	Represents the connection between provide and demand in economics.
Price and Income	Represents the connection between value and income in enterprise.
Movement of a Projectile	Represents the trajectory of a projectile launched at a sure angle.

Graphing Y = 1/2x Utilizing a Desk of Values

To graph the linear equation y = 1/2x utilizing a desk of values, observe these steps:

1. Select a set of x-values. Begin with just a few easy values of x, akin to -2, -1, 0, 1, and a pair of.
2. Calculate the corresponding y-values. For every x-value, plug it into the equation y = 1/2x to seek out the corresponding y-value. For instance, when x = -2, y = 1/2(-2) = -1.
3. Create a desk of values. Manage the x- and y-values in a desk, as proven beneath:

x	y
-2	-1
-1	-0.5
0	0
1	0.5
2	1

4. Plot the factors. Mark every level on the coordinate airplane utilizing a small circle or dot.

5. Draw a line. Join the factors with a straight line. This line represents the graph of the equation y = 1/2x.

Part 20: Detailed Rationalization of the Worth 20

Within the desk of values, we selected the worth 20 for x as an instance easy methods to discover the corresponding y-value.

1. Plug 20 into the equation y = 1/2x.
   y = 1/2(20)
   y = 10

2. Affirm the end result.
   We will test our reply by substituting x = 20 and y = 10 again into the equation:
   10 = 1/2(20)
   10 = 10

Because the equation holds true, we all know that the purpose (20, 10) lies on the graph of y = 1/2x.

3. Plot the purpose.
   Mark the purpose (20, 10) on the coordinate airplane, and join it with a line to the opposite factors to finish the graph.

Using Intercepts to Plot the Graph Precisely

Intercepts are essential factors the place the graph of a operate crosses the coordinate axes. Figuring out the intercepts of the road y = 1/2x permits us to pinpoint two distinct factors on the graph, offering a basis for correct plotting.

To seek out the x-intercept, we set y = 0 within the equation and remedy for x:

Step	Equation
1	0 = 1/2x
2	x = 0

Due to this fact, the x-intercept is (0, 0).

To seek out the y-intercept, we set x = 0 within the equation and remedy for y:

Step	Equation
1	y = 1/2(0)
2	y = 0

Thus, the y-intercept is (0, 0).

With each intercepts recognized, we will now plot the graph of y = 1/2x:

1. Plot the x-intercept (0, 0) on the coordinate airplane.
2. Plot the y-intercept (0, 0) on the coordinate airplane.
3. Draw a straight line connecting the 2 intercepts. Because the slope of the road is 1/2, the road will rise 1 unit for each 2 models it strikes to the appropriate.

By using the intercepts, we have now successfully plotted the graph of y = 1/2x, making certain its accuracy and offering a transparent visible illustration of the operate.

Understanding the Asymptotes of the Parabola

Asymptotes are essential strains that present beneficial details about the conduct of a parabola. They will help decide the form and limits of the parabola, permitting us to raised perceive its total traits. Within the case of the parabola outlined by the equation y = 1/2x, we have now two forms of asymptotes: vertical and horizontal.

Vertical Asymptote

The vertical asymptote is a vertical line that the parabola approaches however by no means intersects. For the equation y = 1/2x, the vertical asymptote is at x = 0. It’s because as x approaches 0, the worth of y approaches infinity or unfavorable infinity, relying on whether or not x is constructive or unfavorable. The parabola will get nearer and nearer to the vertical asymptote, however it’s going to by no means really contact it.

Horizontal Asymptote

The horizontal asymptote is a horizontal line that the parabola approaches as x approaches constructive or unfavorable infinity. For the equation y = 1/2x, the horizontal asymptote is at y = 0. It’s because as x turns into very massive, both positively or negatively, the worth of y approaches 0. The parabola will get nearer and nearer to the horizontal asymptote, however it’s going to by no means really intersect it.

Significance of Asymptotes

Asymptotes have a number of essential implications for the graph of y = 1/2x:

* They divide the coordinate airplane into areas the place the parabola behaves otherwise.
* They assist decide the area and vary of the parabola.
* They can be utilized to sketch the approximate form of the parabola with out plotting each level.
* They supply insights into the boundaries and conduct of the operate as x approaches sure values.

Understanding the asymptotes of a parabola is important for absolutely comprehending its graph and conduct. By figuring out the vertical and horizontal asymptotes for the equation y = 1/2x, we achieve beneficial details about the form, limits, and tendencies of this explicit parabola.

Deciphering the Slope of the Parabola

The slope of a parabola is a vital aspect in understanding the parabola’s form, route, and charge of change. The slope of a parabola is represented by the numerical coefficient (a) that accompanies the x-term within the quadratic equation. On this case, for the parabola y = 1/2x^2, the slope is 24, which performs a big position in figuring out the parabola’s traits.

Affect of Slope on Parabola Form

The slope of a parabola primarily impacts its concavity. A constructive slope, like on this case, signifies that the parabola opens upward, resembling a U-shaped curve. Conversely, a unfavorable slope would lead to a parabola opening downward, forming an inverted U-shape.

Figuring out the Path of Opening

The slope of a parabola additionally gives beneficial details about its route of opening. A constructive slope signifies that the parabola opens upward, with the vertex pointing towards the constructive y-axis. In distinction, a unfavorable slope signifies that the parabola opens downward, with the vertex pointing towards the unfavorable y-axis.

Measuring the Steepness

The magnitude of the slope governs the steepness of the parabola’s curvature. A bigger slope, akin to 24 on this occasion, signifies that the parabola is extra sharply curved, leading to a narrower, extra compact form. A smaller slope would yield a parabola with a gentler curvature and a wider, extra elongated form.

Calculating the Charge of Change

The slope of a parabola is intimately related to the speed of change of the parabola. The slope represents the vertical change within the y-coordinate relative to the horizontal change within the x-coordinate. On this case, for the parabola y = 1/2x^2, the slope of 24 implies that when x will increase by 1 unit, the corresponding y-coordinate will increase by 24 models. This worth represents the speed of change or the gradient of the parabola.

Instance

Take into account the parabola y = 1/2x^2, which has a slope of 24. Which means that the parabola opens upward and is extra sharply curved than a parabola with a smaller slope. The speed of change for this parabola is 24, indicating that for every unit improve in x, the y-coordinate will increase by 24 models.

Significance of the Slope

The slope of a parabola is a elementary attribute that influences the parabola’s total look, conduct, and charge of change. Understanding the slope allows you to interpret the parabola’s form, route of opening, curvature, and charge of change precisely.

Slope	Opening	Charge of Change
Constructive	Upward	Constructive
Unfavourable	Downward	Unfavourable
Bigger Constructive	Sharply Curved	Steeper
Smaller Constructive	Gently Curved	Much less Steep
Bigger Unfavourable	Sharply Curved Downward	Steeper Downward
Smaller Unfavourable	Gently Curved Downward	Much less Steep Downward

Graphing Y = 1/2x by Finishing the Sq.

To graph the equation y = 1/2x utilizing the tactic of finishing the sq., observe these steps:

1. Rewrite the equation: Multiply either side of the equation by 2 to do away with the fraction:

2y = 1x

2. Full the sq.: To finish the sq., add and subtract the sq. of half the coefficient of x, which is (1/2)^2 = 1/4:

2y + 1/4 – 1/4 = 1x

3. Issue the right sq. trinomial: The left-hand aspect of the equation will be factored as an ideal sq. trinomial:

2y + 1/4 = (√2y)^2 – (1/2)^2

4. Write the equation in vertex kind: The vertex type of a parabola is:

y = a(x – h)^2 + okay

the place (h, okay) is the vertex of the parabola.

Substituting the values from the earlier step, we get:

2y = (√2y – 1/2)^2

Dividing either side by 2, we get:

y = (1/2)(√2y – 1/2)^2

5. Establish the vertex: The vertex of the parabola is (1/2, 0).
6. Plot the vertex: Plot the purpose (1/2, 0) on the coordinate airplane.
7. Discover further factors: To seek out further factors on the parabola, you need to use the equation:

y = (1/2)(√2y – 1/2)^2

For instance, you’ll find the x-intercepts by setting y = 0 and fixing for x:

0 = (1/2)(√2(0) – 1/2)^2

0 = (-1/4)^2

x = ±1/4

So the x-intercepts are (-1/4, 0) and (1/4, 0).

8. Sketch the parabola: Use the vertex, the x-intercepts, and another factors you have got discovered to sketch the parabola.

Here’s a desk summarizing the steps required to graph y = 1/2x utilizing the tactic of finishing the sq.:

Step	Description
1	Multiply either side of the equation by 2 to do away with the fraction
2	Add and subtract the sq. of half the coefficient of x
3	Issue the right sq. trinomial
4	Write the equation in vertex kind
5	Establish the vertex
6	Plot the vertex
7	Discover further factors
8	Sketch the parabola

Figuring out the Symmetry Level of the Parabola

The symmetry level of a parabola, also referred to as the vertex, is the purpose the place the parabola modifications route. It’s the lowest level on a parabola that opens upward or the very best level on a parabola that opens downward. The symmetry level is a key characteristic of a parabola and can be utilized to find out different essential traits of the graph.

Discovering the Symmetry Level

To seek out the symmetry level of a parabola, it’s essential to first decide the equation of the parabola. The equation of a parabola is often written within the kind y = ax^2 + bx + c, the place a, b, and c are constants. Upon getting the equation of the parabola, you need to use the next steps to seek out the symmetry level:

1. Set the spinoff of the parabola equal to zero. The spinoff of a parabola is the same as 2ax + b. To seek out the symmetry level, it’s essential to set the spinoff equal to zero and remedy for x.
2. Resolve for x. Upon getting set the spinoff equal to zero, you’ll be able to remedy for x by dividing either side of the equation by 2a.
3. Plug x again into the equation of the parabola. Upon getting solved for x, you’ll be able to plug the worth of x again into the equation of the parabola to seek out the y-coordinate of the symmetry level.

Instance

Let’s discover the symmetry level of the parabola y = x^2 – 4x + 3.

1. Set the spinoff of the parabola equal to zero. The spinoff of this parabola is y’ = 2x – 4. To set the spinoff equal to zero, we will write: 2x – 4 = 0.

2. Resolve for x. Fixing for x, we get: x = 2.

3. Plug x again into the equation of the parabola. Plugging x = 2 again into the equation of the parabola, we get: y = 2^2 – 4(2) + 3 = -1.

Due to this fact, the symmetry level of the parabola y = x^2 – 4x + 3 is (2, -1).

Properties of the Symmetry Level

The symmetry level of a parabola has a number of essential properties:

• The symmetry level is the turning level of the parabola. Which means that the parabola modifications route on the symmetry level.
• The symmetry level is the midpoint of the road that connects the x-intercepts of the parabola.
• The symmetry level is the vertex of the parabola. Which means that the parabola is symmetric concerning the symmetry level.

Abstract

The symmetry level of a parabola is a key characteristic of the graph. It’s the level the place the parabola modifications route and is the turning level of the parabola. The symmetry level will be discovered by setting the spinoff of the parabola equal to zero and fixing for x. The symmetry level has a number of essential properties, together with being the midpoint of the road that connects the x-intercepts of the parabola and being the vertex of the parabola.

Property	Description
Turning level	The purpose the place the parabola modifications route.
Midpoint of x-intercepts	The purpose that’s midway between the 2 x-intercepts of the parabola.
Vertex	The very best or lowest level on the parabola.

Discovering the Focus and Directrix of the Parabola

To find out the main focus and directrix of the parabola represented by the equation y = 1/2x^2, we have to first establish the vertex (h, okay) of the parabola. Because the equation is within the kind y = a(x – h)^2 + okay, we will straight learn off the vertex as (0, 0).

Subsequent, we have to decide the worth of “a” from the equation, which is 1/2. The worth of “a” determines the vertical stretch or compression of the parabola.

Focus

The main target of a parabola is some extent (h + p, okay) which is p models from the vertex alongside the axis of the parabola. Because the parabola opens upwards (i.e., the coefficient of x^2 is constructive), the axis of the parabola is the y-axis. Due to this fact, the main focus will probably be situated at a distance of p models above the vertex.

To find out the worth of “p,” we use the components p = 1/4a. Substituting the worth of “a” (1/2) into this components, we get:

“`
p = 1/4 * 1/2 = 1/8
“`

Due to this fact, the main focus of the parabola y = 1/2x^2 is situated on the level (0, 1/8).

Directrix

The directrix of a parabola is a horizontal line situated at a distance of p models beneath the vertex. On this case, the directrix will probably be situated at a distance of 1/8 models beneath the vertex (0, 0).

The equation of the directrix is y = okay – p, the place (h, okay) is the vertex. Substituting the values of h and okay (each equal to 0) and p (1/8), we get:

“`
y = 0 – 1/8
“`

Due to this fact, the equation of the directrix is y = -1/8.

Desk summarizing the main focus and directrix:

Focus:	(0, 1/8)
Directrix:	y = -1/8

Graphing Y = 1/2x Utilizing Transformations

Step 1: Graph Y = 1/x

The guardian operate for Y = 1/2x is Y = 1/x, which is a hyperbola. To graph it:

• Plot the purpose (1, 1) on the coordinate airplane.
• From this level, transfer 1 unit to the appropriate and 1 unit as much as plot (2, 1/2).
• Repeat this course of till you have got a number of factors on the higher and decrease branches of the hyperbola.
• Join the factors with a clean curve to finish the graph.

Step 2: Shrink the Graph Vertically

To remodel Y = 1/x to Y = 1/2x, we have to shrink the graph vertically by an element of two.

• For every level (x, y) on the Y = 1/x graph, the corresponding level on the Y = 1/2x graph will probably be (x, y/2).

• For instance, the purpose (2, 1/2) on the Y = 1/x graph turns into (2, 1/4) on the Y = 1/2x graph.

• Plot the brand new factors and join them with a clean curve to finish the graph of Y = 1/2x.

Extra Notes on Vertical Shrinkage

• Vertical shrinkage doesn’t have an effect on the form of the graph.
• It solely scales the peak of the graph relative to the y-axis.
• If the vertical shrinkage issue is larger than 1, the graph will develop into wider than the unique.
• If the vertical shrinkage issue is lower than 1, the graph will develop into narrower than the unique.

• The overall equation for vertical shrinkage is:
  $$
  y = frac{1}{a} f(x)
  $$
  the place:
• $$a$$ is the vertical shrinkage issue.

Scaling the Graph of Y = 1/2x

31. The Impact of Multiplying the Unbiased Variable by a Fixed: Transformations with Respect to the x-axis

When the impartial variable (x) is multiplied by a relentless (a), the graph of the operate (y = f(ax)) undergoes the next transformations:

Scaling within the x-direction: The graph is stretched or compressed horizontally by an element of (1/a).

Reflection over the x-axis: If (a) is unfavorable, the graph is mirrored over the x-axis.

For instance this, let’s contemplate the operate (y = 1/2x) for instance.

a) Scaling within the x-direction:

• If (a > 1) (e.g., (a = 2)), the graph of (y = 1/2(2x)) is stretched horizontally by an element of (1/2). Which means that the x-coordinates of the factors on the graph are halved.

• If (0 < a < 1) (e.g., (a = 0.5)), the graph of (y = 1/2(0.5x)) is compressed horizontally by an element of (1/0.5 = 2). Which means that the x-coordinates of the factors on the graph are doubled.

b) Reflection over the x-axis:

• If (a) is unfavorable (e.g., (a = -2)), the graph of (y = 1/2(-2x)) is mirrored over the x-axis. Which means that the factors on the graph are mirrored throughout the x-axis.

To summarize these transformations, the next desk reveals the results of multiplying the impartial variable (x) by completely different constants:

Fixed (a), (worth of x-scaling issue, 1/a)	Transformation
(a > 1)	Stretched horizontally by an element of (1/a)
(0 < a < 1)	Compressed horizontally by an element of (1/a)
(a < 0)	Mirrored over the x-axis and stretched horizontally by an element of (1/a)

Stretching or Compressing the Graph of Y = 1/2x

The graph of Y = 1/2x is a hyperbola that opens to the left and proper. We will stretch or compress this graph by multiplying the x-coordinate of every level by a relentless. If the fixed is larger than 1, the graph will probably be stretched. If the fixed is between 0 and 1, the graph will probably be compressed.

For instance, let’s graph the operate Y = 1/2x.

x	y
-2	-1
-1	-2
0	0
1	2
2	1

Now let’s graph the operate Y = 1/x.

x	y
-2	-1/2
-1	-1
0	undefined
1	1
2	1/2

As you’ll be able to see, the graph of Y = 1/x is stretched within the x-direction in comparison with the graph of Y = 1/2x. It’s because the fixed 1 is larger than 1.

We will additionally compress the graph of Y = 1/2x by multiplying the x-coordinate of every level by a relentless between 0 and 1. For instance, let’s graph the operate Y = 1/4x.

x	y
-2	-1/4
-1	-1/2
0	0
1	1/4
2	1/2

As you’ll be able to see, the graph of Y = 1/4x is compressed within the x-direction in comparison with the graph of Y = 1/2x. It’s because the fixed 1/4 is between 0 and 1.

Rotating the Graph of Y = 1/2x

To rotate the graph of Y = 1/2x, we have to apply a metamorphosis to the equation. The transformation will contain rotating the graph across the origin by a sure angle. The angle of rotation is often measured in levels or radians.

The components for rotating some extent (x, y) across the origin by an angle θ is as follows:

“`
x’ = x cos(θ) – y sin(θ)
y’ = x sin(θ) + y cos(θ)
“`

The place (x’, y’) are the coordinates of the rotated level.

To rotate the graph of Y = 1/2x by an angle of 45 levels, we will apply the next transformation:

“`
x’ = x cos(45°) – y sin(45°)
y’ = x sin(45°) + y cos(45°)
“`

Substituting the equation of the road into the transformation equations, we get the next:

“`
x’ = x cos(45°) – (1/2x) sin(45°)
y’ = x sin(45°) + (1/2x) cos(45°)
“`

Simplifying the equations, we get the next:

“`
x’ = (√2/2) x – (√2/4) y
y’ = (√2/4) x + (√2/2) y
“`

That is the equation of the rotated graph. The graph has been rotated by 45 levels across the origin.

Steps to Rotate the Graph of Y = 1/2x by 45 Levels

1. Substitute the equation of the road into the transformation equations.
2. Simplify the equations to get the equation of the rotated graph.
3. Graph the rotated graph.

Instance

To graph the graph of Y = 1/2x rotated by 45 levels across the origin, observe these steps:

1. Substitute the equation of the road into the transformation equations.

x’ = x cos(45°) – (1/2x) sin(45°)

y’ = x sin(45°) + (1/2x) cos(45°)

2. Simplify the equations to get the equation of the rotated graph.

x’ = (√2/2) x – (√2/4) y

y’ = (√2/4) x + (√2/2) y

3. Graph the rotated graph.

The graph of the rotated graph is proven beneath.

Figuring out the Vertex of the Inverse Perform

Changing the Equation of the Hyperbola to the Equation of the Inverse Perform

To establish the vertex of the inverse operate, we first have to convert the equation of the hyperbola to the equation of the inverse operate. The overall equation of a hyperbola is:

$$y^2 – {b^2}{x^2} = {a^2}$$

Given the equation $y = 1 + 2x$, we will establish $a$ and $b$ as follows:

$$a = 1, b = 2$$

The equation of the inverse operate is:

$$x = 1 + 2y$$

Discovering the Vertex of the Inverse Perform

The vertex of a hyperbola within the kind $x^2 – {b^2}{y^2} = {a^2}$ is given by the purpose $(0, pm a)$.

In our case, the inverse operate is within the kind $y^2 – {2^2}{x^2} = {1^2}$, so the vertex of the inverse operate is:

$$(0, pm 1)$$

Due to this fact, the vertex of the inverse operate $x = 1 + 2y$ is on the level (0, 1).

Discovering the Intercepts of the Inverse Perform

Discovering the x-intercept of f^-1(x)

To seek out the x-intercept of the inverse operate f^-1(x), we set y = 0 and remedy for x:

“`
0 = 1/2x
x = 0
“`

Due to this fact, the x-intercept is (0, 0).

Discovering the y-intercept of f^-1(x)

To seek out the y-intercept of the inverse operate f^-1(x), we set x = 0 and remedy for y:

“`
y = 1/2(0)
y = 0
“`

Due to this fact, the y-intercept can also be (0, 0).

Abstract of the intercepts of f^-1(x)

The intercepts of the inverse operate f^-1(x) are summarized within the following desk:

Intercept	Coordinates
x-intercept	(0, 0)
y-intercept	(0, 0)

Graphical interpretation of the intercepts of f^-1(x)

The intercepts of the inverse operate f^-1(x) present essential details about its graph:

* The x-intercept represents the purpose the place the graph of f^-1(x) intersects the x-axis.
* The y-intercept represents the purpose the place the graph of f^-1(x) intersects the y-axis.
* The truth that each intercepts are on the origin signifies that the graph of f^-1(x) is symmetric with respect to the road y = x.

Plotting Factors on the Inverse Perform

To graph the inverse operate, we have to discover the inverse of the unique operate, which is $y = 1/2x$. To do that, we swap the roles of $x$ and $y$ and remedy for $y$.

$$x = 1/2y$$
$$2x = y$$
$$y = 2x$$

So the inverse operate is $y = 2x$. Now we will plot factors on the inverse operate identical to we did with the unique operate. Let’s begin by discovering the $x$- and $y$-intercepts.

1. Discover the $x$-intercept:
Set $y = 0$ and remedy for $x$.

$$0 = 2x$$
$$x = 0$$

So the $x$-intercept is $(0, 0)$.

2. Discover the $y$-intercept:
Set $x = 0$ and remedy for $y$.

$$y = 2(0)$$
$$y = 0$$

So the $y$-intercept is $(0, 0)$.

3. Plot the factors:
We will now plot the factors $(0, 0)$ and another factors that we need to discover. For instance, we may plot the purpose $(1, 2)$ by substituting $x = 1$ into the equation $y = 2x$.

$$y = 2(1)$$
$$y = 2$$

So the purpose $(1, 2)$ is on the graph of the inverse operate.

4. Draw the road:
As soon as we have now plotted just a few factors, we will draw the road that passes by means of them. The road needs to be a straight line with a slope of two.

The graph of the inverse operate is proven beneath.

Abstract of steps to plot factors on the inverse operate:

• Discover the inverse operate.
• Discover the $x$- and $y$-intercepts.
• Plot the intercepts and another factors that you simply need to discover.
• Draw the road that passes by means of the factors.

Figuring out the Slope and y-Intercept of Y = 2x

To graph any linear equation, we have to decide its slope and y-intercept. For the equation Y = 2x, the slope is 2 and the y-intercept is 0. This info will assist us plot the graph precisely.

Slope: 2

The slope represents the speed of change within the y-coordinate relative to the change within the x-coordinate. In different phrases, it tells us how a lot the y-coordinate will increase (or decreases) for each unit improve (or lower) within the x-coordinate. In our case, the slope is 2, which implies that for each unit improve in x, the y-coordinate will increase by 2 models.

y-Intercept: 0

The y-intercept is the purpose the place the graph crosses the y-axis. It represents the worth of y when x is the same as 0. In our equation, the y-intercept is 0, which implies that the graph intersects the y-axis on the level (0, 0).

Plotting the Graph

Now that we all know the slope and y-intercept, we will begin plotting the graph. We are going to use a coordinate airplane with x and y axes.

Step 1: Plot the y-Intercept

Begin by plotting the y-intercept, which is (0, 0). This level represents the purpose the place the graph crosses the y-axis.

Step 2: Decide the Slope

The slope of the equation is 2, which implies that for each unit improve in x, the y-coordinate will increase by 2 models.

Step 3: Use the Slope to Discover Extra Factors

From the y-intercept (0, 0), we will use the slope to find out further factors on the graph.

• Improve x by 1 unit and improve y by 2 models. This offers us the purpose (1, 2).
• Improve x by one other 1 unit and improve y by 2 models once more. This offers us the purpose (2, 4).

Step 4: Draw the Line

Plot the extra factors (1, 2) and (2, 4) on the coordinate airplane. Join these factors with a straight line. This line represents the graph of Y = 2x.

Step 5: Examine for Accuracy

Be sure that the road passes by means of the y-intercept (0, 0) and has the right slope of two. Additionally, test that the road passes by means of the extra factors (1, 2) and (2, 4). This verification ensures the accuracy of the graph.

Shifting the Graph of the Inverse Perform Horizontally

Step 41: Shifting the Graph of f-1(x) to the Left by h Items

To shift the graph of f-1(x) h models to the left, we have to exchange x within the equation of f-1(x) with (x + h). It’s because shifting the graph to the left implies that for any given worth of y, the corresponding worth of x will probably be h models lower than it might be on the unique graph.

For instance, if we need to shift the graph of f-1(x) = 2x + 1 to the left by 3 models, we’d exchange x with (x + 3) within the equation, giving us the equation f-1(x) = 2(x + 3) + 1 = 2x + 7.

The next desk summarizes the steps concerned in shifting the graph of f-1(x) h models to the left:

Step	Equation
Exchange x with (x + h) within the equation of f-1(x).	f-1(x + h) = 2x + 1
Simplify the equation.	f-1(x + h) = 2x + 7

Step 42: Shifting the Graph of f-1(x) to the Proper by h Items

To shift the graph of f-1(x) h models to the appropriate, we have to exchange x within the equation of f-1(x) with (x – h). It’s because shifting the graph to the appropriate implies that for any given worth of y, the corresponding worth of x will probably be h models larger than it might be on the unique graph.

For instance, if we need to shift the graph of f-1(x) = 2x + 1 to the appropriate by 3 models, we’d exchange x with (x – 3) within the equation, giving us the equation f-1(x) = 2(x – 3) + 1 = 2x – 5.

The next desk summarizes the steps concerned in shifting the graph of f-1(x) h models to the appropriate:

Step	Equation
Exchange x with (x – h) within the equation of f-1(x).	f-1(x – h) = 2x + 1
Simplify the equation.	f-1(x – h) = 2x – 5

Reflecting the Graph of the Inverse Perform over the y-axis

When reflecting a graph over the y-axis, we basically flip the graph across the vertical axis. This operation basically replaces x with -x, as each level’s distance from the y-axis modifications its signal.

To replicate the graph of the inverse operate over the y-axis, we observe these steps:

1. Exchange x with -x within the operate’s equation:

y = 1 / (2x) turns into y = 1 / (2(-x)) = 1 / (-2x)

2. Simplify the equation:

y = 1 / (-2x) = -1 / (2x)

The ensuing equation, y = -1 / (2x), represents the inverse operate mirrored over the y-axis.

Analyzing the Mirrored Graph

The mirrored graph of the inverse operate is a vertical stretch of the unique inverse operate by an element of 1. Which means that the graph is stretched away from the x-axis by an element of 1.

The mirrored graph additionally has a mirrored image symmetry concerning the y-axis. Which means that for any level (x, y) on the graph, the purpose (-x, y) can also be on the graph.

Key Factors

• Reflecting a graph over the y-axis flips the graph across the vertical axis.
• To replicate the graph of an inverse operate over the y-axis, exchange x with -x within the operate’s equation.
• The mirrored graph is a vertical stretch of the unique inverse operate by an element of 1 and has a mirrored image symmetry concerning the y-axis.

Authentic Perform	Inverse Perform	Mirrored Inverse Perform
y = 2x	y = 1 / (2x)	y = -1 / (2x)

Combining Transformations for the Inverse Perform

Up to now, we have now solely labored with particular person transformations of the operate $y = f(x)$. On this part, we’ll discover what occurs once we mix two or extra transformations. For instance, we could have a operate that’s vertically stretched after which shifted to the left. To graph this operate, we’d carry out the transformations within the order by which they’re given. First, we’d stretch the operate vertically, after which we’d shift it to the left.

Combining transformations is usually a bit difficult, however with observe, it turns into simpler. The bottom line is to do not forget that the transformations are utilized within the order by which they’re written. For instance, the operate $y = 2f(x – 3)$ is first shifted 3 models to the appropriate after which vertically stretched by an element of two.

Horizontal vs. Vertical Transformations

When combining transformations, you will need to distinguish between horizontal and vertical transformations. Horizontal transformations have an effect on the x-coordinates of the graph, whereas vertical transformations have an effect on the y-coordinates of the graph. For instance, the operate $y = f(x + 3)$ is a horizontal translation, whereas the operate $y = 2f(x)$ is a vertical stretch.

When combining horizontal and vertical transformations, the order by which the transformations are utilized issues. For instance, the operate $y = 2f(x – 3)$ is first shifted 3 models to the appropriate after which vertically stretched by an element of two. Nevertheless, the operate $y = f(x – 3) + 2$ is first shifted 3 models to the appropriate after which vertically translated 2 models up.

Reflections

Reflections are one other kind of transformation that may be utilized to features. A mirrored image over the x-axis modifications the signal of the y-coordinate of each level on the graph, whereas a mirrored image over the y-axis modifications the signal of the x-coordinate of each level on the graph.

Reflections will be mixed with different transformations to create much more complicated graphs. For instance, the operate $y = -f(x – 3)$ is a mirrored image over the x-axis and a horizontal translation 3 models to the appropriate.

Instance

Graph the operate $y = |x – 2| + 1$.

First, we’ll graph the operate $y = |x|$. This operate is a V-shaped graph with vertex on the origin. Subsequent, we’ll translate the graph 2 models to the appropriate. It will give us the graph of the operate $y = |x – 2|$. Lastly, we’ll translate the graph 1 unit up. It will give us the graph of the operate $y = |x – 2| + 1$.

The next desk summarizes the transformations that had been utilized to the operate $y = |x|$ to create the graph of the operate $y = |x – 2| + 1$.

Transformation	Impact on the Graph
Translate 2 models to the appropriate	Shifts the graph 2 models to the appropriate
Translate 1 unit up	Shifts the graph 1 unit up

Purposes of the Graph of Y = 2x

The graph of the operate y = 2x is a straight line that passes by means of the origin and has a slope of two. This graph can be utilized to resolve quite a lot of issues, together with:

• Discovering the equation of a line.

• Graphing a linear equation.

• Fixing techniques of linear equations.

• Discovering the slope of a line.

• Figuring out the y-intercept of a line.

• Calculating the world of triangles and parallelograms.

• Fixing phrase issues involving linear equations.

46. Discovering the x-intercept of a line

The x-intercept of a line is the purpose the place the road crosses the x-axis. To seek out the x-intercept of the road y = 2x, set y = 0 and remedy for x:

0 = 2x
x = 0

Due to this fact, the x-intercept of the road y = 2x is (0, 0).

Here’s a desk summarizing the important thing factors concerning the graph of y = 2x:

Level	y-value
(0, 0)	0
(1, 2)	2
(2, 4)	4
(-1, -2)	-2
(-2, -4)	-4

The graph of y = 2x is a straight line that passes by means of the origin and has a slope of two. This graph can be utilized to resolve quite a lot of issues, together with these listed above.

Discovering the Coordinates of the Vertex of the Inverse Perform from the Equation

Step 1: Resolve the equation for x when it comes to y

Rewrite the given equation, y = 1/2x, as:

x = 2y

Step 2: Swap x and y

To seek out the inverse operate, swap the roles of x and y:

y = 2x

Step 3: Discover the equation of the vertex

The vertex of a parabola is at all times on the level (h, okay) the place h is the x-coordinate and okay is the y-coordinate. For the equation y = 2x, the vertex is on the level:

(0, 0)

Step 4: Decide the form of the inverse operate

Because the coefficient of x within the equation y = 2x is constructive, the parabola opens upward. Which means that the inverse operate will probably be a parabola that opens downward.

Step 5: Decide the route of the opening

Because the coefficient of x within the equation y = 2x is constructive, the parabola opens upward. Which means that the inverse operate will open downward.

Step 6: Decide the coordinates of the vertex of the inverse operate

Because the inverse operate is a parabola that opens downward with a vertex at (0, 0), the inverse operate has a vertex at:

(0, 0)

Abstract Desk:

Authentic Perform	Inverse Perform
y = 1/2x	y = 2x
Vertex: (0, 0)	Vertex: (0, 0)

Figuring out the Area and Vary of the Inverse Perform

The inverse of a operate is a brand new operate that "undoes" the unique operate. To find out the area and vary of the inverse operate, we have to change the roles of the enter (x) and output (y) variables.

Step 1: Resolve for y

Start by isolating the output variable (y) within the authentic operate:

y = 1/2x

2y = x

Step 2: Swap x and y

Interchange the roles of x and y to acquire the inverse operate:

x = 2y

Step 3: Decide the Area of the Inverse Perform

The area of the inverse operate contains all attainable values of x within the authentic operate. From the unique operate, we observe that x can tackle any actual quantity besides zero:

x ≠ 0

Due to this fact, the area of the inverse operate is:

x ∈ (-∞, 0) ∪ (0, ∞)

Step 4: Decide the Vary of the Inverse Perform

The vary of the inverse operate contains all attainable values of y within the authentic operate. From the unique operate, we observe that y can tackle any actual quantity:

y ∈ (-∞, ∞)

Due to this fact, the vary of the inverse operate is:

y ∈ (-∞, ∞)

Step 5: Properties of the Inverse Perform

The inverse operate shares a number of properties with the unique operate:

• Symmetry concerning the line y = x: The graph of the inverse operate is a mirrored image of the unique operate over the road y = x.

• Linear Perform: Each the unique operate and its inverse are linear features with a slope of 1/2.

• Reciprocal Relationship: The inverse operate is the reciprocal of the unique operate:

f(x) = 1/2x
g(x) = 2x

the place g(x) is the inverse operate of f(x).

The best way to Graph y = 1/2x

To graph the equation y = 1/2x, observe these steps:

1. Plot the intercepts. The y-intercept is discovered by setting x = 0. y = 1/2(0) = 0, so the y-intercept is (0, 0). The x-intercept is discovered by setting y = 0. 0 = 1/2x, so x = 0. Due to this fact, the x-intercept is (0, 0).

2. Draw a line by means of the intercepts. The road passes by means of the factors (0, 0).

3. Examine your graph. You’ll be able to test your graph by plugging in just a few factors to verify they fulfill the equation. For instance, when x = 1, y = 1/2(1) = 1/2. So the purpose (1, 1/2) is on the road.

Individuals Additionally Ask About 115 How To Graph Y 1 2x

What’s the slope of the road y = 1/2x?

The slope of the road y = 1/2x is 1/2.

What’s the y-intercept of the road y = 1/2x?

The y-intercept of the road y = 1/2x is 0.

How do you graph the road y = 1/2x?

To graph the road y = 1/2x, observe these steps:

1. Plot the y-intercept at (0, 0).
2. Use the slope to seek out one other level on the road. For instance, you’ll be able to go up 1 unit and over 2 models to get to the purpose (2, 1).
3. Draw a line by means of the 2 factors.