Quadratic Equations and Functions

### Learning Objectives

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

- 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 (Figure).

- Factor completely:
If you missed this problem, review (Figure).

- Factor completely:
If you missed this problem, review (Figure).

### 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

Graph and on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Plotting points will help us see the effect of the constants on the basic graph. We fill in the chart for all three functions.

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.

The graph of is the same as the graph of but shifted up 2 units.

The graph of is the same as the graph of but shifted down 2 units.

ⓐ Graph and on the same rectangular coordinate system.

ⓑ Describe what effect adding a constant to the function has on the basic parabola.

ⓐ

ⓑ The graph of is the same as the graph of but shifted up 1 unit. The graph of is the same as the graph of but shifted down 1 unit.

ⓐ Graph and on the same rectangular coordinate system.

ⓑ Describe what effect adding a constant to the function has on the basic parabola.

ⓐ

ⓑ The graph of is the same as the graph of but shifted up 6 units. The graph of is the same as the graph of but shifted down 6 units.

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.

Graph using a vertical shift.

We first draw the graph of on the grid. | |

Determine . | |

Shift the graph down 3. |

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

Graph and on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Plotting points will help us see the effect of the constants on the basic graph. We fill in the chart for all three functions.

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

ⓐ Graph and on the same rectangular coordinate system.

ⓑ Describe what effect adding a constant to the function has on the basic parabola.

ⓐ

ⓑ The graph of is the same as the graph of but shifted left 2 units. The graph of is the same as the graph of but shift right 2 units.

ⓐ Graph and on the same rectangular coordinate system.

ⓑ Describe what effect adding a constant to the function has on the basic parabola.

ⓐ

ⓑ The graph of is the same as the graph of but shifted left 5 units. The graph of is the same as the graph of but shifted right 5 units.

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.

Graph using a horizontal shift.

We first draw the graph of on the grid. | |

Determine h. | |

Shift the graph to the right 6 units. |

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.

Graph using transformations.

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

Let’s first identify the constants *h*, *k*.

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

We first draw the graph of on the grid.

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

Graph

We will graph the functions and on the same grid. We will choose a few points on and then multiply the *y*-values by 3 to get the points for

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 *x*^{2} 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.

Rewrite in the form by completing the square.

Separate the x terms from the constant. | |

Factor the coefficient of , . | |

Prepare to complete the square. | |

Take half of 2 and then square it to complete the square. | |

The constant 1 completes the square in the parentheses, but the parentheses is multiplied by . So we are really adding 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 form. |

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.

Graph by using transformations.

**Step 1.** Rewrite the function in 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. | |

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 form. |

**Step 2:** Graph the function using transformations.

Looking at the *h*, *k* values, we see the graph will take the graph of and shift it to the left 3 units and down 4 units.

We first draw the graph of on the grid.

Graph by using transformations.

Graph by using transformations.

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

Graph a quadratic function using transformations.

- Rewrite the function in form by completing the square.
- Graph the function using transformations.

Graph by using transformations.

**Step 1.** Rewrite the function in vertex form by completing the square.

Separate the x terms from the constant. | |

We need the coefficient of to be one. We factor from the | |

Take half of 2 and then square it to complete the square. | |

We add 1 to complete the square in the parentheses, but the parentheses is multiplied by . Se we are really adding . 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 form. |

**Step 2.** Graph the function using transformations.

We first draw the graph of on the grid.

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.

Graph a quadratic function in the form using properties.

- Rewrite the function in 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 point symmetric to the*y*-intercept across the axis of symmetry. - Find the
*x*-intercepts. - Graph the parabola.

ⓐ Rewrite in form and ⓑ graph the function using properties.

Rewrite the function in form by completing the square. | |

Identify the constants | |

Since , the parabola opens upward. | |

The axis of symmetry is . | The axis of symmetry is . |

The vertex is . | The vertex is . |

Find the y-intercept by finding . | |

y-intercept | |

Find the point symmetric to across the axis of symmetry. | |

Find the x-intercepts. | The discriminant negative, so there are no |

ⓐ 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.

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.

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

### Key Concepts

- 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.

- If

- The graph of shifts the graph of vertically k units.
- Graph a Quadratic Function of the form Using a Horizontal Shift
- The graph of shifts the graph of horizontally h units.
- If
*h*> 0, shift the parabola horizontally left*h*units. - If
*h*< 0, shift the parabola horizontally right units.

- If

- The graph of shifts the graph of horizontally h units.
- Graph of a Quadratic Function of the form
- The coefficient
*a*in the function affects the graph of by stretching or compressing it.If then the graph of will be “wider” than the graph of

If then the graph of will be “skinnier” than the graph of

- The coefficient
- How to graph a quadratic function using transformations
- Rewrite the function in form by completing the square.
- Graph the function using transformations.

- Graph a quadratic function in the vertex form using properties
- Rewrite the function in 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 point symmetric to the*y*-intercept across the axis of symmetry. - Find the
*x*-intercepts, if possible. - Graph the parabola.

#### 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

ⓐ

ⓑ The graph of is the same as the graph of but shifted up 4 units. The graph of is the same as the graph of but shift down 4 units.

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

ⓐ

ⓑ The graph of is the same as the graph of but shifted right 3 units. The graph of is the same as the graph of but shifted left 3 units.

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?

Answers will vary.

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?