12.14: Recognizing Quadratic Functions
Have you ever thought about velocity? Take a look at this dilemma.
Travis found the following equation in his math book.
\begin{align*}d=rt16t^2\end{align*}
It is an equation to calculate velocity. In fact, it is a function. Being an avid baseball 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. This is Travis' dilemma.
Can you identify this function?
Pay attention to this Concept and by the end of it, you will be able to identify the function.
Guidance
Do you remember what a parabola is?
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*}
Here is a table that could be used to graph a quadratic equation.
\begin{align*}\underline{x}\end{align*} 
\begin{align*}\underline{y}\end{align*} 

input values  output values 
domain  range 
independent variable  dependent variable 
Now let's talk about functions.
A function is a relation that assigns exactly one value of the domain to each value of the range.
So, when we say quadratic function, we are referring to any function that can be written in the form
\begin{align*}y=ax^2+bx+c\end{align*}
Why can’t \begin{align*}a\end{align*}
If the \begin{align*}a\end{align*}
Let's take a look at some.
Tell if the following equations are quadratic functions. If they are, place them in standard form and identify the \begin{align*}a, b,\end{align*}
1. \begin{align*}y=x^23x+5\end{align*}
Yes, it is a quadratic function. The standard form is \begin{align*}y=x^23x+5; \ 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; \ a=7,b=4,c=0\end{align*}
3. \begin{align*}y6=x^2\end{align*}
Yes, it is a quadratic function. The standard form is \begin{align*}y=x^2+6; \ a=1,b=0,c=6\end{align*}
4. \begin{align*}3x^2+y=3x^2+4x2\end{align*}
With this function, we have to rewrite it into standard form. Standard form would have the y value on the left side of the equals and the a, b and c values on the right side. To accomplish this task, we will have to subtract \begin{align*}3x^2\end{align*}
Write down the definition and form for a quadratic function in your notebook.
Identify whether or not each function is a quadratic function.
Example A
\begin{align*}y8=x^2\end{align*}
Solution: Yes, this is a quadratic function.
Example B
\begin{align*}y+2x^2=2x^2 + 1\end{align*}
Solution: No, this is not a quadratic function.
Example C
\begin{align*}y+4=2x^2\end{align*}
Solution: Yes, this is a quadratic function.
Now let's go back to the dilemma from the beginning of the Concept.
\begin{align*}d=rt16t^2\end{align*}
This is a quadratic function because "d" 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.
Vocabulary
 Domain
 input value, independent value
 Range
 output value, dependent value
 Function
 relation that assigns one value of the domain to each value of range
 Quadratic Function

To the \begin{align*}2^{nd}\end{align*}
2nd degree in standard forma parabola is created by a quadratic function.
Guided Practice
Here is one for you to try on your own.
Is this function a quadratic function? If so, write it into standard form.
\begin{align*}3y9=3x^2\end{align*}
Solution
First, we have to get the \begin{align*}y\end{align*}
\begin{align*}3y9+9 &= 3x^2+9 \\ 3y &= 3x^2+9\end{align*}
Next, we need to get \begin{align*}y\end{align*}
\begin{align*}y=x^2+3\end{align*}
Now looking at the form of the function, you can see that this is a quadratic function.
Video Review
Practice
Directions: Tell if the following equations are quadratic functions. If they are, identify the \begin{align*}a, b,\end{align*}

\begin{align*}y = 3x^2  x + 4\end{align*}
y=3x2−x+4 
\begin{align*}y = 2x^2 + 4\end{align*}
y=2x2+4 
\begin{align*}2y = 4x^2 + 4\end{align*}
2y=4x2+4 
\begin{align*}3y = 6x^2 + 12\end{align*}
3y=6x2+12 
\begin{align*}4y = 2x^2  12\end{align*}
4y=2x2−12  \begin{align*}3y1 = 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*}y2x^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*}
domain
The domain of a function is the set of values for which the function is defined.Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .quadratic function
A quadratic function is a function that can be written in the form , where , , and are real constants and .Range
The range of a function is the set of values for which the function is defined.Image Attributions
Description
Learning Objectives
Here you'll recognize a quadratic function as an equation in two variables with a specific form.