When passing the baseball field, Mr. Travis handed the students the following problem written on a piece of paper that looked like a baseball. This is what it said:

When an object is thrown into the air with a starting velocity of

feet per second, its distance in feet, above its starting point seconds after it is thrown is about . Use a table of values to show the distance of an object from its starting point that has an initial velocity of 80 feet per second. Then graph the velocity of the ball.In this concept, you will learn to evaluate and graph quadratic functions by using tables.

### Quadratic Functions

A **function** is a relation that assigns exactly one value of the domain to each value of the range. When you say **quadratic function**, you are referring to any function that can be written in the form where and are constants and . When a quadratic function is written as it is said to be written in **standard form**.

Let’s look at quadratic functions in more detail. The word **domain** refers to input values and the word **range** refers to output values. For a function, every value has only one value.

The

-values are the result of substituting values into the function. You organize the information using a table of values or a t-table. In most cases, the input values could be any numbers. However, for our convenience, you will use negative numbers, zero, and positive numbers for the -values.Let’s look at an example.

Complete a table of values for the function

.

Evaluating a quadratic function is always the same. You substitute the

-values into the equation and solve for the -values.The values of parabola.

and have an effect on the graph of quadratic equation. Now you are going to use this information when you look at a quadratic function. What you know about the values of and help you to understand the opening of a

The graph of a quadratic function will always be a parabola. A **parabola** is a kind of “U”- shape that is always symmetrical on both sides. It can go either open upwards or downwards. Also, a parabola is not linear.

The shape of the parabola can be predicted by the value of ‘\begin{align*}a<0\end{align*} then the parabola opens downward.

’. If then the parabola opens upward. IfTake a look at the image below for some examples.

Now that you understand how these graphs look and how the equation of the graph affects its appearance, it is time to make some predictions.

Let’s look at an example.

What would you predict about the graph of

?Because the ‘\begin{align*}a\end{align*}’ value is 7, it would be very narrow.

Also, because

, it would open upward.### Examples

#### Example 1

Earlier, you were given a problem about Mr. Travis and the problem.

First, think about the information given in the problem and write an equation to model it.

Next, make a table of values.

0 | 1 | 2 | 3 | 4 | 5 | |

0 | 64 | 96 | 96 | 64 | 0 |

Finally, enter these values into a graphing calculator to create the following graph.

#### Example 2

What would you predict about the graph of

?Because the ‘\begin{align*}a\end{align*}’ value is , it would be very wide.

Also, because

, it would open downward.#### Example 3

Predict the opening of

.It will open downwards because the ‘\begin{align*}a\end{align*}’ value is negative.

#### Example 4

For the quadratic function in Example 1, where will the vertex be?

The

value is 4, therefore the vertex is (0, 4).#### Example 5

Which graph will have a wider opening one with a vertex at 0 or one with a vertex at 8?

Vertex at 0 is the wider opening.

### Review

Use your tables to graph the following functions.