# Quadratic Equations and Functions

## Graph Quadratic Functions Using Properties


### Recognize the Graph of a Quadratic Function

Previously we very briefly looked at the function , which we called the square function. It was one of the first non-linear functions we looked at. Now we will graph functions of the form  if  We call this kind of function a quadratic function.
We graphed the quadratic function  by plotting points.
Every quadratic function has a graph that looks like this. We call this figure a parabola.
Let’s practice graphing a parabola by plotting a few points.
All graphs of quadratic functions of the form f (x) = ax2 + bx + c are parabolas that open upward or downward. See .
Notice that the only difference in the two functions is the negative sign before the quadratic term (x2 in the equation of the graph in ). When the quadratic term, is positive, the parabola opens upward, and when the quadratic 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 then 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. 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 f (x) = ax2 + bx + c is 
So to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula 
Notice that these are the equations of the dashed blue lines on the graphs.
The point on the parabola that is the lowest (parabola opens up), or the highest (parabola opens down), lies on the axis of symmetry. 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 itsx-coordinate is  To find the y-coordinate of the vertex we substitute the value of the x-coordinate into the quadratic function.

### 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 y-intercept the value of x is zero. So to find the y-intercept, we substitute x = 0 into the function.
Let’s find the y-intercepts of the two parabolas shown in .
An x-intercept results when the value of f (x) is zero. To find an x-intercept, we let f (x) = 0. In other words, we will need to solve the equation 0 = ax2 + bx + c for x.
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 we looked at. First we will find the x-intercepts of the parabola whose function is f (x) = x2 + 4x + 3.



Now we will find the x-intercepts of the parabola whose function is f (x) = −x2 + 4x + 3.



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 ax2 + bx + c = 0. We solved for x and the results were the solutions to the equation.
We are now looking at quadratic functions of the form f (x) = ax2 + bx + c. The graphs of these functions are parabolas. The x-intercepts of the parabolas occur where f (x) = 0.
For example:
The solutions of the quadratic function are the x 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 functions 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 function of the form  Now we can use the discriminant to tell us how many x-intercepts there are on the graph.
Before you 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 Functions Using Properties

Now we have all the pieces we need in order to graph a quadratic function. We just need to put them together. In the next example we will see how to do this.
We list the steps to take in order to graph a quadratic function here.
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 f (x) = −x2 + 6x − 9, 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 0 = −x2 + 6x − 9 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 f (x) = x2 + 4x + 5?
Finding the y-intercept by finding f (0) is easy, isn’t it? Sometimes we need to use the Quadratic Formula to find the x-intercepts.

### 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 function. The y-coordinate of the vertex is the minimum value of a parabola that opens upward. It is the maximum value of a parabola that opens downward. See .
We have used the formula
to calculate the height in feet, h , of an object shot upwards into the air with initial velocity, v0, after t seconds .
This formula is a quadratic function, so its graph is a parabola. By solving for the coordinates of the vertex (t, h), 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. Parabola Orientation


2. Axis of Symmetry and Vertex of a Parabola The graph of the function  is a parabola where:


3. Find the Intercepts of a Parabola


4. How to graph a quadratic function using properties.


5. Minimum or Maximum Values of a Quadratic Equation



### Practice Makes Perfect

Recognize the Graph of a Quadratic Function
In the following exercises, graph the functions by plotting points.
For each of the following exercises, determine if the parabola opens up or down.
Find the Axis of Symmetry and Vertex of a Parabola
In the following functions, find ⓐ the equation of the axis of symmetry and ⓑ the vertex of its graph.
Find the Intercepts of a Parabola
In the following exercises, find the intercepts of the parabola whose function is given.
Graph Quadratic Functions Using Properties
In the following exercises, graph the function by using its properties.
Solve Maximum and Minimum Applications
In the following exercises, find the maximum or minimum value of each function.
In the following exercises, solve. Round answers to the nearest tenth.

### Writing Exercise


### 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?