Travis found the following equation in his math book.

\begin{align*}d = rt - 16 t^2\end{align*}

It is an equation to calculate velocity. In fact, it is a function. Being an avid sports player, Travis was very interested in figuring out how to use the equation, but he isn’t even sure what kind of a function it is. Can you identify this function?

In this concept, you will learn to recognize a quadratic function as an equation in two variables with a specific form.

### Quadratic Functions

A **parabola** is a U shaped figure whose equation is a quadratic equation. Let’s start with quadratic equations and standard form.

To graph a quadratic equation, you need input values, oftentimes \begin{align*}x\end{align*} values, to calculate corresponding \begin{align*}y\end{align*} values. Your input values are known as the **domain**, while the output values are known as the **range**. These are also called the **independent variable** (\begin{align*}x\end{align*}) and **dependent variable** (\begin{align*}y\end{align*}).

A **function** is a relation that assigns exactly one value of the domain to each value of the range.

So, when you say **quadratic function**, you are referring to any function that can be written in the form \begin{align*}y=ax^2 + bx +c\end{align*}, where \begin{align*}a, b\end{align*}, and \begin{align*}c\end{align*} are constants and \begin{align*}a \neq 0\end{align*}. This is **standard form**.

Why can’t ‘\begin{align*}a\end{align*}’ equal zero?

If the ‘\begin{align*}a\end{align*}’ value is zero, you might notice that it would make the first term \begin{align*}ax^2\end{align*} disappear because anything times zero is zero. You would be left with simply \begin{align*}y = bx + c\end{align*}. Although this is still a function, it is no longer quadratic. This is a linear function. All quadratic functions are to the 2^{nd} degree.

Let’s take a look at some examples.

Identify if the following equations are quadratic functions. If they are, place them in standard form and identify ‘\begin{align*}a, b\end{align*} and \begin{align*}c\end{align*}’ values.

1. \begin{align*}y = x^2 - 3x + 5\end{align*}

Yes, it is a quadratic function.

The standard form is \begin{align*}y = x^2 - 3x + 5\end{align*}.

\begin{align*}a = 1, b = -3, c = 5\end{align*}.

2. \begin{align*}y =-7x^2 + 4x\end{align*}

Yes, it is a quadratic function.

The standard form is \begin{align*}y = -7x^2 + 4x\end{align*}.

\begin{align*}a =-7, b = 4, c = 0\end{align*}.

3. \begin{align*}y-6 = x^2\end{align*}

Yes, it is a quadratic function.

The standard form is \begin{align*}y = x^2+6\end{align*}.

\begin{align*}a = 1, b = 0, c = 6\end{align*}.

4. \begin{align*}3x^2 + y = 3x^2 + 4x -2\end{align*}

With this function, you have to rewrite it into standard form. Standard form would have the \begin{align*}y\end{align*}-value on the left side of the equals and the \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} values on the right side. To accomplish this task, you will have to subtract \begin{align*}3x^2\end{align*} from both sides.

\begin{align*}\begin{array}{rcl} 3x^2 {\color{red}- \ 3x^2} + y &=& 3x^2 {\color{red}- \ 3x^2} + 4x -2 \\ y &=& 4x-2 \end{array}\end{align*}

The function is not a quadratic function because if you subtract \begin{align*}3x^2\end{align*} from both sides, your \begin{align*}a\end{align*} value will be zero. This function is a linear function.

### Examples

#### Example 1

Earlier, you were given a problem about the function.

Is the function \begin{align*}d=rt - 16t^2\end{align*} a quadratic function?

This is a quadratic function because “\begin{align*}d\end{align*}” is dependent on the right side of the function. One value will impact the others. The quadratic equation will have one value in the range for each value in the domain. This will make it a quadratic function.

#### Example 2

Is this function a quadratic function? If so, write it in standard form.

\begin{align*}3y - 9 = 3x^2\end{align*}

First, you have to get the \begin{align*}y\end{align*} value alone. Let’s add 9 to both sides to start.

\begin{align*}\begin{array}{rcl} 3y - 9 &=& 3x^2 \\ 3y - 9 \ {\color{red}+ \ 9} &=& 3x^2 {\color{red} + \ 9} \\ 3y &=& 3x^2 + 9 \end{array}\end{align*}

Next, divide both sides by 3.

\begin{align*}\begin{array}{rcl} 3y &=& 3x^2 + 9 \\ \frac{3y}{3} &=& \frac{3x^2}{3} + \frac{9}{3} \\ y &=& x^2 + 3 \end{array}\end{align*}

The answer is that this is a quadratic function and the standard form is \begin{align*}y=x^2 +3\end{align*}.

#### Example 3

Identify whether or not the function is a quadratic function.

\begin{align*}y-8 = x^2\end{align*}

In standard form \begin{align*}y-8 = x^2\end{align*} becomes \begin{align*}y=x^2 +8\end{align*} and this is a quadratic function.

#### Example 4

Identify whether or not the function is a quadratic function.

\begin{align*}y + 2x^2 = 2x^2 +1\end{align*}

In standard form \begin{align*}y + 2x^2 = 2x^2 +1\end{align*} becomes \begin{align*}y=1\end{align*} and this is not a quadratic function.

#### Example 5

Identify whether or not the function is a quadratic function.

\begin{align*}y + 4 = 2x^2 \end{align*}

In standard form \begin{align*}y + 4 = 2x^2 \end{align*} becomes \begin{align*}y=2x^2-4\end{align*} and this is a quadratic function.

### Review

Identify whether the following equations are quadratic functions. If they are, identify the ‘\begin{align*}a, b\end{align*} and \begin{align*}c\end{align*}’ values.

- \begin{align*}y = 3x^2 - x + 4\end{align*}
- \begin{align*}y = 2x^2 + 4\end{align*}
- \begin{align*}2y = 4x^2 + 4\end{align*}
- \begin{align*}3y = 6x^2 + 12\end{align*}
- \begin{align*}4y = 2x^2 -12\end{align*}
- \begin{align*}3y -1 =6x^2 +11\end{align*}
- \begin{align*}2y +2 = 2x^2 +4\end{align*}
- \begin{align*}y+2x^2 =2x^2 +4\end{align*}
- \begin{align*}y-2x^2 =2x^2 +4\end{align*}
- \begin{align*}y=2x^2 - 3x +4\end{align*}
- \begin{align*}y =4x - 18 + x^2\end{align*}
- \begin{align*}y +x^2 -6 = x^2 + 4x -5\end{align*}
- \begin{align*}3y +3 = 9x^2 -12x\end{align*}
- \begin{align*}6y = 3x^5 + 3x^4 + 6x + - 18\end{align*}
- \begin{align*}4y + 3x = 8x^2 + 3x - 12\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.14.