Quadratic Equations
85 Graphing Quadratic Equations
Learning Objectives
By the end of this section, you will be able to:
 Recognize the graph of a quadratic equation in two variables
 Find the axis of symmetry and vertex of a parabola
 Find the intercepts of a parabola
 Graph quadratic equations in two variables
 Solve maximum and minimum applications
Before you get started, take this readiness quiz.
Recognize the Graph of a Quadratic Equation in Two Variables
We have graphed equations of the form . We called equations like this linear equations because their graphs are straight lines.
Now, we will graph equations of the form . We call this kind of equation a quadratic equation in two variables.
A quadratic equation in two variables, where are real numbers and , is an equation of the form
Just like we started graphing linear equations by plotting points, we will do the same for quadratic equations.
Let’s look first at graphing the quadratic equation . We will choose integer values of between and 2 and find their values. See (Figure).
0  0 
1  1 
1  
2  4 
4 
Notice when we let and , we got the same value for .
The same thing happened when we let and .
Now, we will plot the points to show the graph of . See (Figure).
The graph is not a line. This figure is called a parabola. Every quadratic equation has a graph that looks like this.
In (Figure) you will practice graphing a parabola by plotting a few points.
Graph .
We will graph the equation by plotting points.
Choose integers values for x, substitute them into the equation and solve for y.  
Record the values of the ordered pairs in the chart.  
Plot the points, and then connect them with a smooth curve. The result will be the graph of the equation. 
Graph .
Graph .
How do the equations and differ? What is the difference between their graphs? How are their graphs the same?
All parabolas of the form open upwards or downwards. See (Figure).
Notice that the only difference in the two equations is the negative sign before the in the equation of the second graph in (Figure). When the term is positive, the parabola opens upward, and when the term is negative, the parabola opens downward.
For the quadratic equation , if:
Determine whether each parabola opens upward or downward:
ⓐⓑ
ⓐ Find the value of “a“. 
Since the “a” is negative, the parabola will open downward. 

ⓑ Find the value of “a“. 
Since the “a” is positive, the parabola will open upward. 
Determine whether each parabola opens upward or downward:
ⓐⓑ
ⓐ up ⓑ down
Determine whether each parabola opens upward or downward:
ⓐⓑ
ⓐ down ⓑ up
Find the Axis of Symmetry and Vertex of a Parabola
Look again at (Figure). Do you see that we could fold each parabola in half and that one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry of the parabola.
We show the same two graphs again with the axis of symmetry in red. See (Figure).
The equation of the axis of symmetry can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of is
So, to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula .
Look back at (Figure). Are these the equations of the dashed red lines?
The point on the parabola that is on the axis of symmetry is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards. This point is called the vertex of the parabola.
We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means its xcoordinate is . To find the ycoordinate of the vertex, we substitute the value of the xcoordinate into the quadratic equation.
For a parabola with equation :
 The axis of symmetry of a parabola is the line .
 The vertex is on the axis of symmetry, so its xcoordinate is .
To find the ycoordinate of the vertex, we substitute into the quadratic equation.
For the parabola find: ⓐ the axis of symmetry and ⓑ the vertex.
ⓐ  
The axis of symmetry is the line .  
Substitute the values of a, b into the equation.  
Simplify.  
The axis of symmetry is the line .  
ⓑ  
The vertex is on the line of symmetry, so its xcoordinate will be .  
Substitute into the equation and solve for y.  
Simplify.  
This is the ycoordinate.  The vertex is 
For the parabola find: ⓐ the axis of symmetry and ⓑ the vertex.
ⓐⓑ
For the parabola find: ⓐ the axis of symmetry and ⓑ the vertex.
ⓐⓑ
Find the Intercepts of a Parabola
When we graphed linear equations, we often used the x– and yintercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.
Remember, at the yintercept the value of is zero. So, to find the yintercept, we substitute into the equation.
Let’s find the yintercepts of the two parabolas shown in the figure below.
At an xintercept, the value of is zero. To find an xintercept, we substitute into the equation. In other words, we will need to solve the equation for .
But solving quadratic equations like this is exactly what we have done earlier in this chapter.
We can now find the xintercepts of the two parabolas shown in (Figure).
First, we will find the xintercepts of a parabola with equation .
Let .  
Factor.  
Use the zero product property.  
Solve.  
The x intercepts are and 
Now, we will find the xintercepts of the parabola with equation .
Let .  
This quadratic does not factor, so we use the Quadratic Formula.  
, ,  
Simplify.  
The x intercepts are and . 
We will use the decimal approximations of the xintercepts, so that we can locate these points on the graph.
Do these results agree with our graphs? See (Figure).
To find the intercepts of a parabola with equation :
Find the intercepts of the parabola .
To find the yintercept, let and solve for y.  
When , then . The yintercept is the point . 

To find the xintercept, let and solve for x.  
Solve by factoring.  
When , then . The xintercepts are the points and .
Find the intercepts of the parabola
Find the intercepts of the parabola
In this chapter, we have been solving quadratic equations of the form . We solved for and the results were the solutions to the equation.
We are now looking at quadratic equations in two variables of the form . The graphs of these equations are parabolas. The xintercepts of the parabolas occur where .
For example:
The solutions of the quadratic equation are the values of the xintercepts.
Earlier, we saw that quadratic equations have 2, 1, or 0 solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the equations give the xintercepts of the graphs, the number of xintercepts is the same as the number of solutions.
Previously, we used the discriminant to determine the number of solutions of a quadratic equation of the form . Now, we can use the discriminant to tell us how many xintercepts there are on the graph.
Before you start solving the quadratic equation to find the values of the xintercepts, you may want to evaluate the discriminant so you know how many solutions to expect.
Find the intercepts of the parabola .
To find the yintercept, let and solve for y.  When , then . The yintercept is the point . 

To find the xintercept, let and solve for x.  
Find the value of the discriminant to predict the number of solutions and so xintercepts.  
Since the value of the discriminant is negative, there is no real solution to the equation.  There are no xintercepts. 
Find the intercepts of the parabola
Find the intercepts of the parabola
Find the intercepts of the parabola .
To find the yintercept, let and solve for y.  
When , then . The yintercept is the point . 

To find the xintercept, let and solve for x.  
Find the value of the discriminant to predict the number of solutions and so xintercepts.  
Since the value of the discriminant is 0, there is no real solution to the equation. So there is one xintercept.  
Solve the equation by factoring the perfect square trinomial.  
Use the Zero Product Property.  
Solve for x.  
When , then  
The xintercept is the point 
Find the intercepts of the parabola
Find the intercepts of the parabola
Graph Quadratic Equations in Two Variables
Now, we have all the pieces we need in order to graph a quadratic equation in two variables. We just need to put them together. In the next example, we will see how to do this.
Graph .
Graph the parabola
; ;
axis: ; vertex: ;
Graph the parabola
axis: ;
 Write the quadratic equation with on one side.
 Determine whether the parabola opens upward or downward.
 Find the axis of symmetry.
 Find the vertex.
 Find the yintercept. Find the point symmetric to the yintercept across the axis of symmetry.
 Find the xintercepts.
 Graph the parabola.
We were able to find the xintercepts in the last example by factoring. We find the xintercepts in the next example by factoring, too.
Graph .
The equation y has on one side.  
Since a is , the parabola opens downward. To find the axis of symmetry, find . 
The axis of symmetry is The vertex is on the line 

Find y when  The vertex is 

The yintercept occurs when Substitute Simplify. The point is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is Point symmetric to the yintercept is 
The yintercept is 

The xintercept occurs when  
Substitute  
Factor the GCF.  
Factor the trinomial.  
Solve for x.  
Connect the points to graph the parabola. 
Graph the parabola
axis: ;
Graph the parabola
axis: ;
For the graph of , the vertex and the xintercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation is 0, so there is only one solution. That means there is only one xintercept, and it is the vertex of the parabola.
How many xintercepts would you expect to see on the graph of ?
Graph .
The equation has y on one side.  
Since a is 1, the parabola opens upward.  
To find the axis of symmetry, find  The axis of symmetry is 

The vertex is on the line  
Find y when  The vertex is 

The yintercept occurs when Substitute Simplify. The point is two units to the right of the line of symmetry. The point two units to the left of the line of symmetry is 
The yintercept is Point symmetric to the y intercept is . 

The x– intercept occurs when  
Substitute Test the discriminant. 

Since the value of the discriminant is negative, there is no solution and so no x intercept. Connect the points to graph the parabola. You may want to choose two more points for greater accuracy. 
Graph the parabola
axis: ;
Graph the parabola
axis: ;
Finding the yintercept by substituting into the equation is easy, isn’t it? But we needed to use the Quadratic Formula to find the xintercepts in (Figure). We will use the Quadratic Formula again in the next example.
Graph .
The equation y has one side. Since a is 2, the parabola opens upward. 

To find the axis of symmetry, find .  The axis of symmetry is . 

The vertex on the line  
Find y when .  The vertex is . 

The yintercept occurs when  
Substitute  
Simplify.  The yintercept is . 

The point is one unit to the left of the line of symmetry. The point one unit to the right of the line of symmetry is 
Point symmetric to the yintercept is  
The xintercept occurs when .  
Substitute .  
Use the Quadratic Formula.  
Substitute in the values of a, b, c.  
Simplify.  
Simplify inside the radical.  
Simplify the radical.  
Factor the GCF.  
Remove common factors.  
Write as two equations.  
Approximate the values.  
The approximate values of the xintercepts are and .  
Graph the parabola using the points found. 
Graph the parabola
axis: ;
Graph the parabola
axis: ;
Solve Maximum and Minimum Applications
Knowing that the vertex of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic equation. The ycoordinate of the vertex is the minimum yvalue of a parabola that opens upward. It is the maximum yvalue of a parabola that opens downward. See (Figure).
The ycoordinate of the vertex of the graph of a quadratic equation is the
 minimum value of the quadratic equation if the parabola opens upward.
 maximum value of the quadratic equation if the parabola opens downward.
Find the minimum value of the quadratic equation .
Since a is positive, the parabola opens upward.  
The quadratic equation has a minimum.  
Find the axis of symmetry.  The axis of symmetry is . 

The vertex is on the line  
Find y when  The vertex is . 

Since the parabola has a minimum, the ycoordinate of the vertex is the minimum yvalue of the quadratic equation.  
The minimum value of the quadratic is and it occurs when .  
Show the graph to verify the result. 
Find the maximum or minimum value of the quadratic equation .
The minimum value is when .
Find the maximum or minimum value of the quadratic equation .
The maximum value is 5 when .
We have used the formula
to calculate the height in feet, , of an object shot upwards into the air with initial velocity, , after seconds.
This formula is a quadratic equation in the variable , so its graph is a parabola. By solving for the coordinates of the vertex, we can find how long it will take the object to reach its maximum height. Then, we can calculate the maximum height.
The quadratic equation models the height of a volleyball hit straight upwards with velocity 176 feet per second from a height of 4 feet.
 ⓐ How many seconds will it take the volleyball to reach its maximum height?
 ⓑ Find the maximum height of the volleyball.
Since a is negative, the parabola opens downward.
The quadratic equation has a maximum.
 ⓐ
 ⓑ
Find h when . Use a calculator to simplify. The vertex is . Since the parabola has a maximum, the hcoordinate of the vertex is the maximum yvalue of the quadratic equation. The maximum value of the quadratic is 488 feet and it occurs when seconds.
The quadratic equation is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height? Round answers to the nearest tenth.
It will take 4 seconds to reach the maximum height of 288 feet.
A toy rocket shot upward from the ground at a rate of 208 ft/sec has the quadratic equation of . When will the rocket reach its maximum height? What will be the maximum height? Round answers to the nearest tenth.
It will take 6.5 seconds to reach the maximum height of 676 feet.
Access these online resources for additional instruction and practice graphing quadratic equations:
Key Concepts
 The graph of every quadratic equation is a parabola.
 Parabola Orientation For the quadratic equation , if
 , the parabola opens upward.
 , the parabola opens downward.
 Axis of Symmetry and Vertex of a Parabola For a parabola with equation :
 The axis of symmetry of a parabola is the line .
 The vertex is on the axis of symmetry, so its xcoordinate is .
 To find the ycoordinate of the vertex we substitute into the quadratic equation.
 Find the Intercepts of a Parabola To find the intercepts of a parabola with equation :
 To Graph a Quadratic Equation in Two Variables
 Write the quadratic equation with on one side.
 Determine whether the parabola opens upward or downward.
 Find the axis of symmetry.
 Find the vertex.
 Find the yintercept. Find the point symmetric to the yintercept across the axis of symmetry.
 Find the xintercepts.
 Graph the parabola.
 Minimum or Maximum Values of a Quadratic Equation
 The y–coordinate of the vertex of the graph of a quadratic equation is the
 minimum value of the quadratic equation if the parabola opens upward.
 maximum value of the quadratic equation if the parabola opens downward.
Section Exercises
Practice Makes Perfect
Recognize the Graph of a Quadratic Equation in Two Variables
In the following exercises, graph:
In the following exercises, determine if the parabola opens up or down.
down
up
Find the Axis of Symmetry and Vertex of a Parabola
In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex.
ⓐⓑ
ⓐⓑ
Find the Intercepts of a Parabola
In the following exercises, find the x– and yintercepts.
Graph Quadratic Equations in Two Variables
In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.
axis:
axis:
axis:
axis:
axis:
axis:
axis:
axis:
axis:
Solve Maximum and Minimum Applications
In the following exercises, find the maximum or minimum value.
The minimum value is when .
The minimum value is 6 when .
The maximum value is 16 when .
In the following exercises, solve. Round answers to the nearest tenth.
An arrow is shot vertically upward from a platform 45 feet high at a rate of 168 ft/sec. Use the quadratic equation to find how long it will take the arrow to reach its maximum height, and then find the maximum height.
In 5.3 sec the arrow will reach maximum height of 486 ft.
A stone is thrown vertically upward from a platform that is 20 feet high at a rate of 160 ft/sec. Use the quadratic equation to find how long it will take the stone to reach its maximum height, and then find the maximum height.
A computer store owner estimates that by charging dollars each for a certain computer, he can sell computers each week. The quadratic equation is used to find the revenue, , received when the selling price of a computer is . Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.
20 computers will give the maximum of ?400 in receipts.
A retailer who sells backpacks estimates that, by selling them for dollars each, he will be able to sell backpacks a month. The quadratic equation is used to find the received when the selling price of a backpack is . Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.
A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using 240 feet of fencing. The quadratic equation gives the area of the corral, , for the length, , of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the corral.
The length of the side along the river of the corral is 120 feet and the maximum area is 7,200 sq ft.
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation gives the area, , of the dog run for the length, , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Everyday Math
In the previous set of exercises, you worked with the quadratic equation that modeled the revenue received from selling computers at a price of dollars. You found the selling price that would give the maximum revenue and calculated the maximum revenue. Now you will look at more characteristics of this model.
ⓐ Graph the equation . ⓑ Find the values of the xintercepts.
 ⓐ
 ⓑ
In the previous set of exercises, you worked with the quadratic equation that modeled the revenue received from selling backpacks at a price of dollars. You found the selling price that would give the maximum revenue and calculated the maximum revenue. Now you will look at more characteristics of this model.
ⓐ Graph the equation . ⓑ Find the values of the xintercepts.
Writing Exercises
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
Chapter 10 Review Exercises
10.1 Solve Quadratic Equations Using the Square Root Property
In the following exercises, solve using the Square Root Property.
no solution
In the following exercises, solve using the Square Root Property.
no solution
10.2 Solve Quadratic Equations Using Completing the Square
In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
In the following exercises, solve by completing the square.
no solution
10.3 Solve Quadratic Equations Using the Quadratic Formula
In the following exercises, solve by using the Quadratic Formula.
no real solution
In the following exercises, determine the number of solutions to each quadratic equation.
 ⓐ
 ⓑ
 ⓒ
 ⓓ
ⓐ 1 ⓑ 2 ⓒ 2 ⓓ none
 ⓐ
 ⓑ
 ⓒ
 ⓓ
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation.
 ⓐ
 ⓑ
 ⓒ
ⓐ factor ⓑ Quadratic Formula ⓒ square root
 ⓐ
 ⓑ
 ⓒ
10.4 Solve Applications Modeled by Quadratic Equations
In the following exercises, solve by using methods of factoring, the square root principle, or the quadratic formula.
Find two consecutive odd numbers whose product is 323.
Two consecutive odd numbers whose product is 323 are 17 and 19, and and
Find two consecutive even numbers whose product is 624.
A triangular banner has an area of 351 square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base.
The height of the banner is 13 cm and the length of the side is 54 cm.
Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.
A tile mosaic in the shape of a right triangle is used as the corner of a rectangular pathway. The hypotenuse of the mosaic is 5 feet. One side of the mosaic is twice as long as the other side. What are the lengths of the sides? Round to the nearest tenth.
The lengths of the sides of the mosaic are 2.2 and 4.4 feet.
A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth.
The front walk from the street to Pam’s house has an area of 250 square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth.
The width of the front walk is 8.1 feet and its length is 30.8 feet.
For Sophia’s graduation party, several tables of the same width will be arranged end to end to give a serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth. Round answer to the nearest tenth.
A ball is thrown vertically in the air with a velocity of 160 ft/sec. Use the formula to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.
The ball will reach 384 feet on its way up in 4 seconds and on the way down in 6 seconds.
A bullet is fired straight up from the ground at a velocity of 320 ft/sec. Use the formula to determine when the bullet will reach 800 feet. Round to the nearest tenth.
10.5 Graphing Quadratic Equations in Two Variables
In the following exercises, graph by plotting point.
Graph
Graph
In the following exercises, determine if the following parabolas open up or down.
down
up
In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex.
ⓐⓑ
In the following exercises, find the x– and yintercepts.
In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.
axis:
axis:
axis:
axis:
In the following exercises, find the minimum or maximum value.
The minimum value is when .
In the following exercises, solve. Rounding answers to the nearest tenth.
A ball is thrown upward from the ground with an initial velocity of 112 ft/sec. Use the quadratic equation to find how long it will take the ball to reach maximum height, and then find the maximum height.
In 3.5 seconds the ball is at its maximum height of 196 feet.
A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation gives the area, , of the yard for the length, , of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.
Practice Test
Use the Square Root Property to solve the quadratic equation: .
Use Completing the Square to solve the quadratic equation: .
Use the Quadratic Formula to solve the quadratic equation: .
Solve the following quadratic equations. Use any method.
no real solution
Use the discriminant to determine the number of solutions of each quadratic equation.
2
Solve by factoring, the Square Root Property, or the Quadratic Formula.
Find two consecutive even numbers whose product is 360.
Two consecutive even number are and and 18 and 20.
The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)
For each parabola, find ⓐ which ways it opens, ⓑ the axis of symmetry, ⓒ the vertex, ⓓ the x– and yintercepts, and ⓔ the maximum or minimum value.
ⓐ up ⓑ ⓒ ⓓ ⓔ minimum value of 5 when
ⓐ up ⓑ ⓒ ⓓ ⓔ minimum value of when
ⓐ down ⓑ
ⓒ ⓓ
ⓔ maximum value of when
Graph the following parabolas by using intercepts, the vertex, and the axis of symmetry.
axis:
Solve.
A water balloon is launched upward at the rate of 86 ft/sec. Using the formula , find how long it will take the balloon to reach the maximum height and then find the maximum height. Round to the nearest tenth.
Glossary
 axis of symmetry
 The axis of symmetry is the vertical line passing through the middle of the parabola
 parabola
 The graph of a quadratic equation in two variables is a parabola.
 quadratic equation in two variables
 A quadratic equation in two variables, where are real numbers and is an equation of the form
 vertex
 The point on the parabola that is on the axis of symmetry is called the vertex of the parabola; it is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards.
 xintercepts of a parabola
 The xintercepts are the points on the parabola where
 yintercept of a parabola
 The yintercept is the point on the parabola where