Learning Objectives
- Graph quadratic functions of the form
- Graph quadratic functions of the form
- Graph quadratic functions of the form
- Graph quadratic functions using transformations
- Find a quadratic function from its graph
Before you get started, take this readiness quiz.
- Graph the function by plotting points.
If you missed this problem, review Example 3.54. - Factor completely:
If you missed this problem, review Example 6.24. - Factor completely:
If you missed this problem, review Example 6.26.
Graph Quadratic Functions of the form
In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of 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 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
Example 9.53
Graph and on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.
ⓐ Graph and on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.
ⓐ Graph and 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 we take the basic parabola graph of and vertically shift it up or shift it down .
This transformation is called a vertical shift.
Graph a Quadratic Function of the form Using a Vertical Shift
The graph of shifts the graph of vertically k units.
- If k > 0, shift the parabola vertically up k units.
- If k < 0, shift the parabola vertically down units.
Now that we have seen the effect of the constant, k, it is easy to graph functions of the form We just start with the basic parabola of and then shift it up or down.
It may be helpful to practice sketching 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 using a vertical shift.
Graph using a vertical shift.
Graph using a vertical shift.
Graph Quadratic Functions of the form
In the first example, we graphed the quadratic function 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
We will now explore the effect of subtracting a constant, h, from x has on the resulting graph of the new function
Example 9.55
Graph and on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.
ⓐ Graph and on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.
ⓐ Graph and 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 we take the basic parabola graph of 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 Using a Horizontal Shift
The graph of shifts the graph of horizontally units.
- If h > 0, shift the parabola horizontally left h units.
- If h < 0, shift the parabola horizontally right units.
Now that we have seen the effect of the constant, h, it is easy to graph functions of the form We just start with the basic parabola of and then shift it left or right.
The next example will require a horizontal shift.
Example 9.56
Graph using a horizontal shift.
Graph using a horizontal shift.
Graph 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 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 using transformations.
Graph using transformations.
Graph using transformations.
Graph Quadratic Functions of the Form
So far we graphed the quadratic function 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
If we graph these functions, we can see the effect of the constant a, assuming a > 0.
To graph a function with constant a it is easiest to choose a few points on and multiply the y-values by a.
Graph of a Quadratic Function of the form
The coefficient a in the function affects the graph of by stretching or compressing it.
- If the graph of will be “wider” than the graph of
- If , the graph of will be “skinnier” than the graph of
Example 9.58
Graph
Graph
Graph
Graph Quadratic Functions Using Transformations
We have learned how the constants a, h, and k in the functions, and affect their graphs. We can now put this together and graph quadratic functions by first putting them into the form 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 in the form by completing the square.
Rewrite in the form by completing the square.
Rewrite in the form by completing the square.
Once we put the function into the 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 by using transformations.
Graph by using transformations.
Graph 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 form by completing the square.
- Step 2. Graph the function using transformations.
Example 9.61
Graph by using transformations.
Graph by using transformations.
Graph by using transformations.
Now that we have completed the square to put a quadratic function into 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 form.
How To
Graph a quadratic function in the form using properties.
- Step 1. Rewrite the function in 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 in form and ⓑ graph the function using properties.
ⓐ Rewrite in form and ⓑ graph the function using properties.
ⓐ Rewrite in 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.
Write the quadratic function in form whose graph is shown.
Determine the quadratic function whose graph is shown.
Media Access Additional Online Resources
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
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.
and
In the following exercises, graph each function using a vertical shift.
Graph Quadratic Functions of the form
In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, , inside the parentheses has
and
In the following exercises, graph each function using a horizontal shift.
In the following exercises, graph each function using transformations.
Graph Quadratic Functions of the form
In the following exercises, graph each function.
Graph Quadratic Functions Using Transformations
In the following exercises, rewrite each function in the form by completing the square.
In the following exercises, ⓐ rewrite each function in form and ⓑ graph it by using transformations.
In the following exercises, ⓐ rewrite each function in form and ⓑ graph it using properties.
Matching
In the following exercises, match the graphs to one of the following functions: ⓐ ⓑ ⓒ ⓓ ⓔ ⓕ ⓖ ⓗ
Find a Quadratic Function from its Graph
In the following exercises, write the quadratic function in form whose graph is shown.
Writing Exercise
Graph the quadratic function first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?
Graph the quadratic function 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?