How do you graph a transformed quadratic function?

Graph a Quadratic Function in the Form f(x)=a(x−h)2+k Using Properties Rewrite the function f(x)=a(x−h)2+k form. Determine whether the parabola opens upward, a>0, or downward, a<0. Find the axis of symmetry, x=h. Find the vertex, (h,k. Find the y-intercept. ... Find the x-intercepts. Graph the parabola. Dec 16, 2019

How do you graph a function using transformations?

5 Steps To Graph Function Transformations In Algebra Identify The Parent Function. ... Reflect Over X-Axis or Y-Axis. ... Shift (Translate) Vertically or Horizontally. ... Vertical and Horizontal Stretches/Compressions. ... Plug in a couple of your coordinates into the parent function to double check your work. More items... Jan 24, 2017

What is the rule for quadratic transformation?

The parent function of the quadratic family is f(x) = x2. A transformation of the graph of the parent function is represented by the function g(x) = a(x − h)2 + k, where a ≠ 0 .

What is a method used to transform quadratic functions?

Completing the square is a method that can be used to transform a quadratic equation in standard form to vertex form. Once in vertex form, a quadratic equation is easy to graph or solve. In other words, these two terms are equal to one another, so we are able to convert from one to the other.

How to tell how many solutions a quadratic graph has?

If b 2 - 4ac is positive (>0) then we have 2 solutions . If b 2 - 4ac is 0 then we have only one solution as the formula is reduced to x = [-b ± 0]/2a. So x = -b/2a, giving only one solution. Lastly, if b 2 - 4ac is less than 0 we have no solutions.

What is the formula to graph quadratic equations?

The U-shaped graph of any quadratic function defined by f(x)=ax2+bx+c , where a, b, and c are real numbers and a≠0. The point that defines the minimum or maximum of a parabola. The vertical line through the vertex, x=−b2a, about which the parabola is symmetric. A term used when referencing the line of symmetry.

How to solve quadratic functions?

The quadratic formula helps us solve any quadratic equation. First, we bring the equation to the form ax²+bx+c=0, where a, b, and c are coefficients. Then, we plug these coefficients in the formula: (-b±√(b²-4ac))/(2a) . See examples of using the formula to solve a variety of equations.

How to plot a parabola graph?

Given y = ax 2 + bx + c , we have to go through the following steps to find the points and shape of any parabola: Label a, b, and c. Decide the direction of the paraola: Find the x-intercepts: Find the y-intercept: Find the vertex (h,k): Plot the points and graph the parabola.

How do you graph translations?

To move a graph up, we add a positive value to the y-value. To move a graph down, we add a negative value to the y-value. To move a graph right, we add a negative value to the x-value. To move a graph left, we add a positive value to the x-value.

How is the graph of the parent quadratic function transformed to produce the graph?

Answer: The graph is reflected over the x-axis, compressed horizontally by a factor of 2, shifted left 6 units, and translated up 3 units .

9.8: Graph Quadratic Functions Using Transformations (2024)

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Learning Objectives

By the end of this section, you will be able to:

• Graph quadratic equations of the form $f (x) = x^{2} + k$
• Graph quadratic functions of the form $f (x) = {(x - h)}^{2}$
Graph quadratic functions of the form $f (x) = a x^{2}$
Graph quadratic functions using transformations
Find a quadratic function from its graph

Be Prepared 9.19

Before you get started, take this readiness quiz.

Graph the function $f (x) = x^{2}$ by plotting points.
If you missed this problem, review Example 3.54.

Be Prepared 9.20

Factor completely: $y^{2} - 14 y + 49 .$
If you missed this problem, review Example 6.24.

Be Prepared 9.21

Factor completely: $2 x^{2} - 16 x + 32 .$
If you missed this problem, review Example 6.26.

Graph Quadratic Functions of the form $f (x) = x^{2} + k$

In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of $f (x) = x^{2}$ and ‘moving’ it according to information given in the function equation. We call this graphing quadratic functions using transformations.

In the first example, we will graph the quadratic function $f (x) = x^{2}$ by plotting points. Then we will see what effect adding a constant, k, to the equation will have on the graph of the new function $f (x) = x^{2} + k .$

Example 9.53

Graph $f (x) = x^{2}, g (x) = x^{2} + 2,$ and $h (x) = x^{2} - 2$ on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Answer

Plotting points will help us see the effect of the constants on the basic $f (x) = x^{2}$ graph. We fill in the chart for all three functions.

9.8: Graph Quadratic Functions Using Transformations (2)

The g(x) values are two more than the f(x) values. Also, the h(x) values are two less than the f(x) values. Now we will graph all three functions on the same rectangular coordinate system.

9.8: Graph Quadratic Functions Using Transformations (3)

The graph of $g (x) = x^{2} + 2$ is the same as the graph of $f (x) = x^{2}$ but shifted up 2 units.

The graph of $h (x) = x^{2} - 2$ is the same as the graph of $f (x) = x^{2}$ but shifted down 2 units.

Try It 9.105

ⓐ Graph $f (x) = x^{2}, g (x) = x^{2} + 1,$ and $h (x) = x^{2} - 1$ on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

Try It 9.106

ⓐ Graph $f (x) = x^{2}, g (x) = x^{2} + 6,$ and $h (x) = x^{2} - 6$ on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

The last example shows us that to graph a quadratic function of the form $f (x) = x^{2} + k,$ we take the basic parabola graph of $f (x) = x^{2}$ and vertically shift it up $(k > 0)$ or shift it down $(k < 0)$ .

This transformation is called a vertical shift.

Graph a Quadratic Function of the form $f (x) = x^{2} + k$ Using a Vertical Shift

The graph of $f (x) = x^{2} + k$ shifts the graph of $f (x) = x^{2}$ vertically k units.

If k > 0, shift the parabola vertically up k units.
If k < 0, shift the parabola vertically down $| k |$ units.

Now that we have seen the effect of the constant, k, it is easy to graph functions of the form $f (x) = x^{2} + k .$ We just start with the basic parabola of $f (x) = x^{2}$ and then shift it up or down.

It may be helpful to practice sketching $f (x) = x^{2}$ quickly. We know the values and can sketch the graph from there.

Once we know this parabola, it will be easy to apply the transformations. The next example will require a vertical shift.

Example 9.54

Graph $f (x) = x^{2} - 3$ using a vertical shift.

Answer

We first draw the graph of $f (x) = x^{2}$ on the grid.
Determine $k$ .

Shift the graph $f (x) = x^{2}$ down 3.

Try It 9.107

Graph $f (x) = x^{2} - 5$ using a vertical shift.

Try It 9.108

Graph $f (x) = x^{2} + 7$ using a vertical shift.

Graph Quadratic Functions of the form $f (x) = {(x - h)}^{2}$

In the first example, we graphed the quadratic function $f (x) = x^{2}$ by plotting points and then saw the effect of adding a constant k to the function had on the resulting graph of the new function $f (x) = x^{2} + k .$

We will now explore the effect of subtracting a constant, h, from x has on the resulting graph of the new function $f (x) = {(x - h)}^{2} .$

Example 9.55

Graph $f (x) = x^{2}, g (x) = {(x - 1)}^{2},$ and $h (x) = {(x + 1)}^{2}$ on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Answer

Plotting points will help us see the effect of the constants on the basic $f (x) = x^{2}$ graph. We fill in the chart for all three functions.

9.8: Graph Quadratic Functions Using Transformations (9)

The g(x) values and the h(x) values share the common numbers 0, 1, 4, 9, and 16, but are shifted.

9.8: Graph Quadratic Functions Using Transformations (10)

9.8: Graph Quadratic Functions Using Transformations (11)

Try It 9.109

ⓐ Graph $f (x) = x^{2}, g (x) = {(x + 2)}^{2},$ and $h (x) = {(x - 2)}^{2}$ on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

Try It 9.110

ⓐ Graph $f (x) = x^{2}, g (x) = x^{2} + 5,$ and $h (x) = x^{2} - 5$ on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

The last example shows us that to graph a quadratic function of the form $f (x) = {(x - h)}^{2},$ we take the basic parabola graph of $f (x) = x^{2}$ and shift it left (h > 0) or shift it right (h < 0).

This transformation is called a horizontal shift.

Graph a Quadratic Function of the form $f (x) = {(x - h)}^{2}$ Using a Horizontal Shift

The graph of $f (x) = {(x - h)}^{2}$ shifts the graph of $f (x) = x^{2}$ horizontally $h$ units.

If h > 0, shift the parabola horizontally right h units.
If h < 0, shift the parabola horizontally left $| h |$ units.

Now that we have seen the effect of the constant, h, it is easy to graph functions of the form $f (x) = {(x - h)}^{2} .$ We just start with the basic parabola of $f (x) = x^{2}$ and then shift it left or right.

The next example will require a horizontal shift.

Example 9.56

Graph $f (x) = {(x - 6)}^{2}$ using a horizontal shift.

Answer

We first draw the graph of $f (x) = x^{2}$ on the grid.
Determine h.

Shift the graph $f (x) = x^{2}$ to the right 6 units.

Try It 9.111

Graph $f (x) = {(x - 4)}^{2}$ using a horizontal shift.

Try It 9.112

Graph $f (x) = {(x + 6)}^{2}$ using a horizontal shift.

Now that we know the effect of the constants h and k, we will graph a quadratic function of the form $f (x) = {(x - h)}^{2} + k$ by first drawing the basic parabola and then making a horizontal shift followed by a vertical shift. We could do the vertical shift followed by the horizontal shift, but most students prefer the horizontal shift followed by the vertical.

Example 9.57

Graph $f (x) = {(x + 1)}^{2} - 2$ using transformations.

Answer

This function will involve two transformations and we need a plan.

Let’s first identify the constants h, k.

9.8: Graph Quadratic Functions Using Transformations (16)

The h constant gives us a horizontal shift and the k gives us a vertical shift.

9.8: Graph Quadratic Functions Using Transformations (17)

We first draw the graph of $f (x) = x^{2}$ on the grid.

9.8: Graph Quadratic Functions Using Transformations (18)

9.8: Graph Quadratic Functions Using Transformations (19)

Try It 9.113

Graph $f (x) = {(x + 2)}^{2} - 3$ using transformations.

Try It 9.114

Graph $f (x) = {(x - 3)}^{2} + 1$ using transformations.

Graph Quadratic Functions of the Form $f (x) = a x^{2}$

So far we graphed the quadratic function $f (x) = x^{2}$ and then saw the effect of including a constant h or k in the equation had on the resulting graph of the new function. We will now explore the effect of the coefficient a on the resulting graph of the new function $f (x) = a x^{2} .$

If we graph these functions, we can see the effect of the constant a, assuming a > 0.

Graph of a Quadratic Function of the form $f (x) = a x^{2}$

The coefficient a in the function $f (x) = a x^{2}$ affects the graph of $f (x) = x^{2}$ by stretching or compressing it.

If $0 < | a | < 1,$ the graph of $f (x) = a x^{2}$ will be “wider” than the graph of $f (x) = x^{2} .$
If $| a | > 1$ , the graph of $f (x) = a x^{2}$ will be “skinnier” than the graph of $f (x) = x^{2} .$

Example 9.58

Graph $f (x) = 3 x^{2} .$

Answer: We will graph the functions $f (x) = x^{2}$ and $g (x) = 3 x^{2}$ on the same grid. We will choose a few points on $f (x) = x^{2}$ and then multiply the y-values by 3 to get the points for $g (x) = 3 x^{2} .$

Try It 9.115

Graph $f (x) = −3 x^{2} .$

Try It 9.116

Graph $f (x) = 2 x^{2} .$

Graph Quadratic Functions Using Transformations

We have learned how the constants a, h, and k in the functions, $f (x) = x^{2} + k, f (x) = {(x - h)}^{2},$ and $f (x) = a x^{2}$ affect their graphs. We can now put this together and graph quadratic functions $f (x) = a x^{2} + b x + c$ by first putting them into the form $f (x) = a {(x - h)}^{2} + k$ by completing the square. This form is sometimes known as the vertex form or standard form.

We must be careful to both add and subtract the number to the SAME side of the function to complete the square. We cannot add the number to both sides as we did when we completed the square with quadratic equations.

When we complete the square in a function with a coefficient of x² that is not one, we have to factor that coefficient from just the x-terms. We do not factor it from the constant term. It is often helpful to move the constant term a bit to the right to make it easier to focus only on the x-terms.

Once we get the constant we want to complete the square, we must remember to multiply it by that coefficient before we then subtract it.

Example 9.59

Rewrite $f (x) = −3 x^{2} - 6 x - 1$ in the $f (x) = a {(x - h)}^{2} + k$ form by completing the square.

Answer


Separate the x terms from the constant.
Factor the coefficient of $x^{2}$ , $−3$ .
Prepare to complete the square.
Take half of 2 and then square it to complete the square. ${(\frac{1}{2} \cdot 2)}^{2} = 1$
The constant 1 completes the square in the parentheses, but the parentheses is multiplied by $−3$ . So we are really adding $−3$ We must then add 3 to not change the value of the function.
Rewrite the trinomial as a square and subtract the constants.
The function is now in the $f (x) = a {(x - h)}^{2} + k$ form.

Try It 9.117

Rewrite $f (x) = −4 x^{2} - 8 x + 1$ in the $f (x) = a {(x - h)}^{2} + k$ form by completing the square.

Try It 9.118

Rewrite $f (x) = 2 x^{2} - 8 x + 3$ in the $f (x) = a {(x - h)}^{2} + k$ form by completing the square.

Once we put the function into the $f (x) = {(x - h)}^{2} + k$ form, we can then use the transformations as we did in the last few problems. The next example will show us how to do this.

Example 9.60

Graph $f (x) = x^{2} + 6 x + 5$ by using transformations.

Answer

Step 1. Rewrite the function in $f (x) = a {(x - h)}^{2} + k$ vertex form by completing the square.


Separate the x terms from the constant.
Take half of 6 and then square it to complete the square. ${(\frac{1}{2} \cdot 6)}^{2} = 9$
We both add 9 and subtract 9 to not change the value of the function.
Rewrite the trinomial as a square and subtract the constants.
The function is now in the $f (x) = {(x - h)}^{2} + k$ form.

Step 2: Graph the function using transformations.

Looking at the h, k values, we see the graph will take the graph of $f (x) = x^{2}$ and shift it to the left 3 units and down 4 units.

9.8: Graph Quadratic Functions Using Transformations (36)

We first draw the graph of $f (x) = x^{2}$ on the grid.

9.8: Graph Quadratic Functions Using Transformations (37)

9.8: Graph Quadratic Functions Using Transformations (38)

Try It 9.119

Graph $f (x) = x^{2} + 2 x - 3$ by using transformations.

Try It 9.120

Graph $f (x) = x^{2} - 8 x + 12$ by using transformations.

We list the steps to take to graph a quadratic function using transformations here.

How To

Graph a quadratic function using transformations.

Step 1. Rewrite the function in $f (x) = a {(x - h)}^{2} + k$ form by completing the square.
Step 2. Graph the function using transformations.

Example 9.61

Graph $f (x) = −2 x^{2} - 4 x + 2$ by using transformations.

Answer

Step 1. Rewrite the function in $f (x) = a {(x - h)}^{2} + k$ vertex form by completing the square.


Separate the x terms from the constant.
We need the coefficient of $x^{2}$ to be one. We factor $−2$ from the x-terms.
Take half of 2 and then square it to complete the square. ${(\frac{1}{2} \cdot 2)}^{2} = 1$
We add 1 to complete the square in the parentheses, but the parentheses is multiplied by $−2$ . Se we are really adding $−2$ . To not change the value of the function we add 2.
Rewrite the trinomial as a square and subtract the constants.
The function is now in the $f (x) = a {(x - h)}^{2} + k$ form.

Step 2. Graph the function using transformations.

9.8: Graph Quadratic Functions Using Transformations (45)

We first draw the graph of $f (x) = x^{2}$ on the grid.

9.8: Graph Quadratic Functions Using Transformations (46)

9.8: Graph Quadratic Functions Using Transformations (47)

Try It 9.121

Graph $f (x) = −3 x^{2} + 12 x - 4$ by using transformations.

Try It 9.122

Graph $f (x) = −2 x^{2} + 12 x - 9$ by using transformations.

Now that we have completed the square to put a quadratic function into $f (x) = a {(x - h)}^{2} + k$ form, we can also use this technique to graph the function using its properties as in the previous section.

If we look back at the last few examples, we see that the vertex is related to the constants h and k.

In each case, the vertex is (h, k). Also the axis of symmetry is the line x = h.

We rewrite our steps for graphing a quadratic function using properties for when the function is in $f (x) = a {(x - h)}^{2} + k$ form.

How To

Graph a quadratic function in the form $f (x) = a {(x - h)}^{2} + k$ using properties.

Step 1. Rewrite the function in $f (x) = a {(x - h)}^{2} + k$ form.
Step 2. Determine whether the parabola opens upward, a > 0, or downward, a < 0.
Step 3. Find the axis of symmetry, x = h.
Step 4. Find the vertex, (h, k).
Step 5. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
Step 6. Find the x-intercepts.
Step 7. Graph the parabola.

Example 9.62

ⓐ Rewrite $f (x) = 2 x^{2} + 4 x + 5$ in $f (x) = a {(x - h)}^{2} + k$ form and ⓑ graph the function using properties.

Answer

Rewrite the function in $f (x) = a {(x - h)}^{2} + k$ form by completing the square.	$f (x) = 2 x^{2} + 4 x + 5$
	$f (x) = 2 (x^{2} + 2 x) + 5$
	$f (x) = 2 (x^{2} + 2 x + 1) + 5 - 2$
	$f (x) = 2 {(x + 1)}^{2} + 3$
Identify the constants $a, h, k .$	$a = 2 h = −1 k = 3$
Since $a = 2$ , the parabola opens upward.
The axis of symmetry is $x = h$ .	The axis of symmetry is $x = −1$ .
The vertex is $(h, k)$ .	The vertex is $(−1, 3)$ .
Find the y-intercept by finding $f (0)$ .	$f (0) = 2 \cdot 0^{2} + 4 \cdot 0 + 5$
	$f (0) = 5$
	y-intercept $(0, 5)$
Find the point symmetric to $(0, 5)$ across the axis of symmetry.	$(- 2, 5)$
Find the x-intercepts.	The discriminant negative, so there are no x-intercepts. Graph the parabola.

Try It 9.123

ⓐ Rewrite $f (x) = 3 x^{2} - 6 x + 5$ in $f (x) = a {(x - h)}^{2} + k$ form and ⓑ graph the function using properties.

Try It 9.124

ⓐ Rewrite $f (x) = −2 x^{2} + 8 x - 7$ in $f (x) = a {(x - h)}^{2} + k$ form and ⓑ graph the function using properties.

Find a Quadratic Function from its Graph

So far we have started with a function and then found its graph.

Now we are going to reverse the process. Starting with the graph, we will find the function.

Example 9.63

Determine the quadratic function whose graph is shown.

Answer

Since it is quadratic, we start with the $f (x) = a {(x - h)}^{2} + k form.$
The vertex, $(h, k),$ is $(−2, −1)$ so $h = −2$ and $k = −1.$	$f (x) = a {(x - (−2))}^{2} - 1$
To find $a$ , we use the $y$ -intercept, $(0, 7)$ .
So $f (0) = 7$ .	$7 = a {(0 + 2)}^{2} - 1$
Solve for $a$ .	$7 = 4 a - 1$
	$8 = 4 a$
	$2 = a$
Write the function.	$f (x) = a {(x - h)}^{2} + k$
Substitute in $h = −2, k = −1$ and $a = 2$ .	$f (x) = 2 {(x + 2)}^{2} - 1$

Try It 9.125

Write the quadratic function in $f (x) = a {(x - h)}^{2} + k$ form whose graph is shown.

Try It 9.126

Determine the quadratic function whose graph is shown.

Media

Access these online resources for additional instruction and practice with graphing quadratic functions using transformations.

Section 9.7 Exercises

Practice Makes Perfect

Graph Quadratic Functions of the form $f (x) = x^{2} + k$

In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, k, to the function has on the basic parabola.

293.

$f (x) = x^{2}, g (x) = x^{2} + 4,$ and $h (x) = x^{2} - 4 .$

294.

$f (x) = x^{2}, g (x) = x^{2} + 7,$ and $h (x) = x^{2} - 7 .$

In the following exercises, graph each function using a vertical shift.

295.

$f (x) = x^{2} + 3$

296.

$f (x) = x^{2} - 7$

297.

$g (x) = x^{2} + 2$

298.

$g (x) = x^{2} + 5$

299.

$h (x) = x^{2} - 4$

300.

Writing Exercise

361.

Graph the quadratic function $f (x) = x^{2} + 4 x + 5$ first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?

362.

Graph the quadratic function $f (x) = 2 x^{2} - 4 x - 3$ first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

9.8: Graph Quadratic Functions Using Transformations (2024)

Learning Objectives

Be Prepared 9.19

Be Prepared 9.20

Be Prepared 9.21

Graph Quadratic Functions of the form f(x)=x2+kf(x)=x2+k

Example 9.53

Try It 9.105

Try It 9.106

Graph a Quadratic Function of the form f ( x ) = x 2 + k f ( x ) = x 2 + k Using a Vertical Shift

Example 9.54

Try It 9.107

Try It 9.108

Graph Quadratic Functions of the form f(x)=(x−h)2f(x)=(x−h)2

Example 9.55

Try It 9.109

Try It 9.110

Graph a Quadratic Function of the form f ( x ) = ( x − h ) 2 f ( x ) = ( x − h ) 2 Using a Horizontal Shift

Example 9.56

Try It 9.111

Try It 9.112

Example 9.57

Try It 9.113

Try It 9.114

Graph Quadratic Functions of the Form f(x)=ax2f(x)=ax2

Graph of a Quadratic Function of the form f ( x ) = a x 2 f ( x ) = a x 2

Example 9.58

Try It 9.115

Try It 9.116

Graph Quadratic Functions Using Transformations

Example 9.59

Try It 9.117

Try It 9.118

Example 9.60

Try It 9.119

Try It 9.120

How To

Graph a quadratic function using transformations.

Example 9.61

Try It 9.121

Try It 9.122

How To

Graph a quadratic function in the form f(x)=a(x−h)2+kf(x)=a(x−h)2+k using properties.

Example 9.62

Try It 9.123

Try It 9.124

Find a Quadratic Function from its Graph

Example 9.63

Try It 9.125

Try It 9.126

Media

Section 9.7 Exercises

Practice Makes Perfect

Writing Exercise

Self Check

FAQs

How do you graph a transformed quadratic function? ›

Graph Quadratic Functions of the form $f (x) = x^{2} + k$

Graph a Quadratic Function of the form $f (x) = x^{2} + k$ Using a Vertical Shift

Graph Quadratic Functions of the form $f (x) = {(x - h)}^{2}$

Graph a Quadratic Function of the form $f (x) = {(x - h)}^{2}$ Using a Horizontal Shift

Graph Quadratic Functions of the Form $f (x) = a x^{2}$

Graph of a Quadratic Function of the form $f (x) = a x^{2}$

Graph a quadratic function in the form $f (x) = a {(x - h)}^{2} + k$ using properties.