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## Compare uses of different forms of quadratic equations

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Credit: Jamie McCaffrey
Source: https://www.flickr.com/photos/15609463@N03/12397735274/in/photolist-jTxCLs-5nNfia-8HAPN3-8HAQsN-8HxEp2-8HxF5r-8HAMGU-4CXYa3-4qph6C-6FY6fP-7DXAKw-8ruYr-6pUJw2-5Nnzdz-7EdPQx-5MXXyK-aGSck6-41dpym-9hDiLr-exFe7v-8HxH5P-9hDi2P-8Kq622-8Kq71K-8Kq4xv-mwSyiS-mwSmbA-8Kt8G3-8KtbKQ-8KtcKm-8Ktcis-8Kta7q-8Ktbdh-8Kt9bw-3Gocnv

Travis found the following equation in his math book.

d=rt16t2

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.

### Guidance

To graph a quadratic equation, you need input values, oftentimes x\begin{align*}x\end{align*} values, to calculate corresponding y\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 (x\begin{align*}x\end{align*}) and dependent variable (y\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 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*}. This is standard form.

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

If the ‘a\begin{align*}a\end{align*}’ value is zero, you might notice that it would make the first term ax2\begin{align*}ax^2\end{align*} disappear because anything times zero is zero. You would be left with simply y=bx+c\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 2nd 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 ‘a,b\begin{align*}a, b\end{align*} and c\begin{align*}c\end{align*}’ values.

1. y=x23x+5\begin{align*}y = x^2 - 3x + 5\end{align*}

Yes, it is a quadratic function.

The standard form is y=x23x+5\begin{align*}y = x^2 - 3x + 5\end{align*}.

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

2. y=7x2+4x\begin{align*}y =-7x^2 + 4x\end{align*}

Yes, it is a quadratic function.

The standard form is y=7x2+4x\begin{align*}y = -7x^2 + 4x\end{align*}.

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

3. y6=x2\begin{align*}y-6 = x^2\end{align*}

Yes, it is a quadratic function.

The standard form is y=x2+6\begin{align*}y = x^2+6\end{align*}.

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

4. 3x2+y=3x2+4x2\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 y\begin{align*}y\end{align*}-value on the left side of the equals and the a,b\begin{align*}a, b\end{align*} and c\begin{align*}c\end{align*} values on the right side. To accomplish this task, you will have to subtract 3x2\begin{align*}3x^2\end{align*} from both sides.

3x2 3x2+yy==3x2 3x2+4x24x2

The function is not a quadratic function because if you subtract 3x2\begin{align*}3x^2\end{align*} from both sides, your a\begin{align*}a\end{align*} value will be zero. This function is a linear function.

### Guided Practice

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

3y9=3x2

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

3y93y9 + 93y===3x23x2+ 93x2+9

Next, divide both sides by 3.

3y3y3y===3x2+93x23+93x2+3

The answer is that this is a quadratic function and the standard form is y=x2+3\begin{align*}y=x^2 +3\end{align*}.

### Examples

#### Example 1

Identify whether or not the function is a quadratic function.

y8=x2

In standard form y8=x2\begin{align*}y-8 = x^2\end{align*} becomes y=x2+8\begin{align*}y=x^2 +8\end{align*} and this is a quadratic function.

#### Example 2

Identify whether or not the function is a quadratic function.

y+2x2=2x2+1

In standard form y+2x2=2x2+1\begin{align*}y + 2x^2 = 2x^2 +1\end{align*} becomes y=1\begin{align*}y=1\end{align*} and this is not a quadratic function.

#### Example 3

Identify whether or not the function is a quadratic function.

y+4=2x2

In standard form y+4=2x2\begin{align*}y + 4 = 2x^2 \end{align*} becomes y=2x24\begin{align*}y=2x^2-4\end{align*} and this is a quadratic function.

Credit: James G
Source: https://www.flickr.com/photos/james_in_to/7953713350/in/photolist-f1rupf-e8PWYs-81gEjG-4Rpopg-d7QRSN-6Wzo2q-7UZY7E-bXyCUb-5afr3a-7PJwDb-4Gsarx-mpMdB-4UDsu7-i5kyQ-qgJdvJ-2fmdWk-6D8uqS-bGC1y-efRJwx-f4E2C-efYvzJ-wpm7q-7RcW37-bXyQ1A-7QUkpX-mcGHeb-mJ1xV-ppqrt3-LqY3-fXjcZQ-89a2Nf-brqXq2-4F1gjh-7PuGDG-mcETjr-dZdd3e-e2F7mh-c8fWiC-e2F6vG

Remember the function?

Is the function d=rt16t2\begin{align*}d=rt - 16t^2\end{align*} a quadratic function?

This is a quadratic function because “d\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.

### Explore More

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

1.  y=3x2x+4\begin{align*}y = 3x^2 - x + 4\end{align*}
2.  y=2x2+4\begin{align*}y = 2x^2 + 4\end{align*}
3.  2y=4x2+4\begin{align*}2y = 4x^2 + 4\end{align*}
4.  3y=6x2+12\begin{align*}3y = 6x^2 + 12\end{align*}
5.  4y=2x212\begin{align*}4y = 2x^2 -12\end{align*}
6.  3y1=6x2+11\begin{align*}3y -1 =6x^2 +11\end{align*}
7.  2y+2=2x2+4\begin{align*}2y +2 = 2x^2 +4\end{align*}
8.  y+2x2=2x2+4\begin{align*}y+2x^2 =2x^2 +4\end{align*}
9.  y2x2=2x2+4\begin{align*}y-2x^2 =2x^2 +4\end{align*}
10.  y=2x23x+4\begin{align*}y=2x^2 - 3x +4\end{align*}
11.  y=4x18+x2\begin{align*}y =4x - 18 + x^2\end{align*}
12.  y+x26=x2+4x5\begin{align*}y +x^2 -6 = x^2 + 4x -5\end{align*}
13.  3y+3=9x212x\begin{align*}3y +3 = 9x^2 -12x\end{align*}
14.  6y=3x5+3x4+6x+18\begin{align*}6y = 3x^5 + 3x^4 + 6x + - 18\end{align*}
15.  4y+3x=8x2+3x12\begin{align*}4y + 3x = 8x^2 + 3x - 12\end{align*}

### Vocabulary Language: English

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 $y$.

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 function is the set of $y$ values for which the function is defined.