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# Quadratic Functions and Their Graphs

## Identify the intercepts, vertex, and axis of symmetry

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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 r\begin{align*}r\end{align*} feet per second, its distance d\begin{align*}d\end{align*} in feet, above its starting point t\begin{align*}t\end{align*} seconds after it is thrown is about d=rt16t2\begin{align*}d = rt - 16t^2\end{align*}. 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.

### Guidance

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 y=ax2+bx+c\begin{align*}y=ax^2+bx+c\end{align*} where a,b,\begin{align*}a, b,\end{align*} and c\begin{align*}c\end{align*} are constants and a0\begin{align*}a \neq 0\end{align*}. When a quadratic function is written as y=ax2+bx+c\begin{align*}y=ax^2+bx+c\end{align*} 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 x\begin{align*}x\end{align*} value has only one y\begin{align*}y\end{align*} value.

The y\begin{align*}y\end{align*}-values are the result of substituting x\begin{align*}x\end{align*} 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 x\begin{align*}x\end{align*}-values.

Let’s look at an example.

Complete a table of values for the function y=x2+3x+2\begin{align*}y=x^2+3x+2\end{align*}.

Evaluating a quadratic function is always the same. You substitute the x\begin{align*}x\end{align*}-values into the equation and solve for the y\begin{align*}y\end{align*}-values.

The values of a,b,\begin{align*}a, b,\end{align*} and c\begin{align*}c\end{align*} 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 a,b,\begin{align*}a, b,\end{align*} and c\begin{align*}c\end{align*} help you to understand the opening of a parabola.

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 ‘a\begin{align*}a\end{align*}’. If a>0\begin{align*}a>0\end{align*} then the parabola opens upward. If a<0\begin{align*}a<0\end{align*} then the parabola opens downward.

Take 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 y=7x2\begin{align*}y=7x^2\end{align*}?

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

Also, because a>0\begin{align*}a > 0\end{align*}, it would open upward.

### Guided Practice

What would you predict about the graph of y=14x2\begin{align*}y=- \frac{1}{4}x^2\end{align*}?

Because the ‘a\begin{align*}a\end{align*}’ value is 14\begin{align*}- \frac{1}{4}\end{align*}, it would be very wide.

Also, because a<0\begin{align*}a < 0\end{align*}, it would open downward.

### Examples

#### Example 1

Predict the opening of y=3x2+4\begin{align*} y=-3x^2+4\end{align*}.

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

#### Example 2

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

The c\begin{align*}c\end{align*} value is 4, therefore the vertex is (0, 4).

#### Example 3

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.

Credit: Eli Christman
Source: https://www.flickr.com/photos/gammaman/8621208650/in/photolist-f1rupf-e8PWYs-5afr3a-81gEjG-7PJwDb-4Rpopg-d7QRSN-4Gsarx-mpMdB-4UDsu7-i5kyQ-qgJdvJ-2fmdWk-6D8uqS-6Wzo2q-7UZY7E-bGC1y-efRJwx-f4E2C-efYvzJ-wpm7q-7RcW37-bXyCUb-bXyQ1A-7QUkpX-brqXq2-4F1gjh-7PuGDG-mcGHeb-mJ1xV-mcETjr-dZdd3e-ppqrt3-LqY3-fXjcZQ-89a2Nf-e2F7mh-c8fWiC-e2F6vG

Remember Mr. Travis and the problem?

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

rd==8080t16t2

Next, make a table of values.

 Time t (sec)\begin{align*}\text{Time} \ t \ (sec)\end{align*} 0 1 2 3 4 5 Distance d (ft.)\begin{align*}\text{Distance} \ d \ (ft.)\end{align*} 0 64 96 96 64 0

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

### Explore More

Use your tables to graph the following functions.

1. y=x28\begin{align*}y=x^2-8\end{align*}
2. y=3x2x+4\begin{align*}y=3x^2-x+4\end{align*}
3. y=2x2+4\begin{align*}y=2x^2+4\end{align*}
4. 2y=4x2+4\begin{align*}2y=4x^2+4\end{align*}
5. 3y=6x2+12\begin{align*}3y=6x^2+12\end{align*}
6. 4y=2x212\begin{align*}4y=2x^2-12\end{align*}
7. 3y1=6x2+11\begin{align*}3y-1=6x^2+11\end{align*}
8. 2y+2=2x2+4\begin{align*}2y+2=2x^2+4\end{align*}
9. y=2x2+5x\begin{align*}y=-2x^2+5x\end{align*}
10. y=x2+3x7\begin{align*}y=-x^2+3x-7\end{align*}
11. y=23x2+2x1\begin{align*}y=\frac{2}{3}x^2+2x-1\end{align*}
12. y=x2+8\begin{align*}y=x^2+8\end{align*}
13. y=2x2+5x1\begin{align*}y=-2x^2+5x-1\end{align*}
14. y=x2+3x1\begin{align*}y=-x^2+3x-1\end{align*}
15. y=3x2+2x+1\begin{align*}y=3x^2+2x+1\end{align*}

### Vocabulary Language: English

Coefficient

Coefficient

A coefficient is the number in front of a variable.
Dilation

Dilation

To reduce or enlarge a figure according to a scale factor is a dilation.
domain

domain

The domain of a function is the set of $x$-values for which the function is defined.
Function

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
Horizontal shift

Horizontal shift

A horizontal shift is the result of adding a constant term to the function inside the parentheses. A positive term results in a shift to the left and a negative term in a shift to the right.
Parabola

Parabola

A parabola is the characteristic shape of a quadratic function graph, resembling a "U".

A quadratic function is a function that can be written in the form $f(x)=ax^2 + bx + c$, where $a$, $b$, and $c$ are real constants and $a\ne 0$.
Range

Range

The range of a data set is the difference between the smallest value and the greatest value in the data set.
standard form

standard form

The standard form of a quadratic function is $f(x)=ax^{2}+bx+c$.
Symmetry

Symmetry

A figure has symmetry if it can be transformed and still look the same.
Vertex

Vertex

The vertex of a parabola is the highest or lowest point on the graph of a parabola. The vertex is the maximum point of a parabola that opens downward and the minimum point of a parabola that opens upward.
vertical axis

vertical axis

The vertical axis is also referred to as the $y$-axis of a coordinate graph. By convention, we graph the output variable on the $y$-axis.