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=rt-16t^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*} values, to calculate corresponding \begin{align*}y\end{align*} values. In those cases, our input values are known as the domain, while the output values our known as the range. These are also called the independent variable and dependent variable, at times.
Here is a table that could be used to graph a quadratic equation.
\begin{align*}\underline{x}\end{align*}-values | \begin{align*}\underline{y}\end{align*}-values |
---|---|
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*}, 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? What happens if it does?
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 \begin{align*}2^{nd}\end{align*} degree.
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*} 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; \ 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*}y-6=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+4x-2\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*} from both sides. Because of this, 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.
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*}y-8=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=rt-16t^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*} degree in standard form-a 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*}3y-9=3x^2\end{align*}
Solution
First, we have to get the \begin{align*}y\end{align*} value alone. Let's add 9 to both sides to start.
\begin{align*}3y-9+9 &= 3x^2+9 \\ 3y &= 3x^2+9\end{align*}
Next, we need to get \begin{align*}y\end{align*} alone. We can do this by dividing both sides by 3.
\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*} 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*}
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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
Here you'll recognize a quadratic function as an equation in two variables with a specific form.