# Quadratic Equations

## Graphing Quadratic Equations in Two Variables


### 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.
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 .
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 .
The graph is not a line. This figure is called a parabola. Every quadratic equation has a graph that looks like this.
In  you will practice graphing a parabola by plotting a few points.
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 .
Notice that the only difference in the two equations is the negative sign before the  in the equation of the second graph in . When the  term is positive, the parabola opens upward, and when the  term is negative, the parabola opens downward.

### Find the Axis of Symmetry and Vertex of a Parabola

Look again at . 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 blue. See .
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 . 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 x-coordinate is . To find the y-coordinate of the vertex, we substitute the value of the x-coordinate into the quadratic equation.

### Find the Intercepts of a Parabola

When we graphed linear equations, we often used the x- and y-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.
Remember, at the  the value of  is zero. So, to find the y-intercept, we substitute  into the equation.
Let’s find the y-intercepts of the two parabolas shown in the figure below.
At an , the value of  is zero. To find an x-intercept, 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 x-intercepts of the two parabolas shown in .
First, we will find the  with equation .
Now, we will find the x-intercepts of the parabola with equation .
We will use the decimal approximations of the x-intercepts, so that we can locate these points on the graph.
Do these results agree with our graphs? See .
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 x-intercepts of the parabolas occur where .
For example:
The solutions of the quadratic equation are the  values of the x-intercepts.
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 x-intercepts of the graphs, the number of x-intercepts 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 x-intercepts there are on the graph.
Before you start solving the quadratic equation to find the values of the x-intercepts, you may want to evaluate the discriminant so you know how many solutions to expect.

### 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.
We were able to find the x-intercepts in the last example by factoring. We find the x-intercepts in the next example by factoring, too.
For the graph of , the vertex and the x-intercept 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 x-intercept, and it is the vertex of the parabola.
How many x-intercepts would you expect to see on the graph of ?
Finding the y-intercept by substituting  into the equation is easy, isn’t it? But we needed to use the Quadratic Formula to find the x-intercepts in . We will use the Quadratic Formula again in the next example.

### 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 y-coordinate of the vertex is the minimum y-value of a parabola that opens upward. It is the maximum y-value of a parabola that opens downward. See .
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.

### Key Concepts

1. The graph of every quadratic equation is a parabola.
2. Parabola Orientation For the quadratic equation , if


3. Axis of Symmetry and Vertex of a Parabola For a parabola with equation :


4. Find the Intercepts of a Parabola To find the intercepts of a parabola with equation :
5. To Graph a Quadratic Equation in Two Variables


6. Minimum or Maximum Values of a Quadratic Equation



### 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.
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 y-intercepts.
Graph Quadratic Equations in Two Variables
In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.
Solve Maximum and Minimum Applications
In the following exercises, find the maximum or minimum value.
In the following exercises, solve. Round answers to the nearest tenth.

### Everyday Math


### 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.
In the following exercises, solve using the Square Root Property.

### 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.

### 10.3 Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.
In the following exercises, determine the number of solutions to each quadratic equation.
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation.

### 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.

### 10.5 Graphing Quadratic Equations in Two Variables

In the following exercises, graph by plotting point.
In the following exercises, determine if the following parabolas open up or down.
In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex.
In the following exercises, find the x- and y-intercepts.
In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.

In the following exercises, find the minimum or maximum value.
In the following exercises, solve. Rounding answers to the nearest tenth.

### Practice Test

Solve the following quadratic equations. Use any method.
Use the discriminant to determine the number of solutions of each quadratic equation.
Solve by factoring, the Square Root Property, or the Quadratic Formula.
For each parabola, find ⓐ which ways it opens, ⓑ the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and ⓔ the maximum or minimum value.
Graph the following parabolas by using intercepts, the vertex, and the axis of symmetry.
Solve.