# Range and Domain of Quadratic Functions

I know two ways to solve range and domain of quadratic functions. First is the graphical method and the second one is the algebraic method. This topic will just focus on quadratic functions, which means that the opening is either up or down.

Consider the figure:

Graphically, the range of quadratic function is from the y-coordinate to positive infinity (if the graph opens upward) or negative infinity (if the graph opens downward). The domain is the set of all real numbers always since it extends from left to right.

The second way is to solve it algebraically. Given a quadratic function,

$y=ax^2+bx+c$

We solve for y in terms of x

Rearrange the term and equate all terms to 0.

$ax^2+bx+c-y=0$

Treat y as constant. By quadratic formula we can solve for x in terms of y now.

$x=\displaystyle\frac{-b\pm\sqrt{b^2-4ac}}{2a}$    where, $a=a, b=b, c=c-y$

By direct substitution,

$x=\displaystyle\frac{-b\pm\sqrt{b^2-4a(c-y )}}{2a}$

To solve for range,

$b^2-4a(c-y)\geq$ for it to become real.

Solving for y,

$y\geq\displaystyle\frac{4ac-b^2}{4a}$

What a surprise! Isn’t it the same formula to get the y-coordinate of the vertex of quadratic equation?

Worked Example 1: 2007 MMC divison finals

Find the range of $y=9-x^2$

Solution:

Algebraic Method:

Solve for x in terms of y.

$x^2=9-y$ $x=\sqrt{9-y}$ $9-y\geq0$

$y\leq 9$ or in symbols (-∞,9]

Graphical Method:

Again get the y-coordinate of the vertex using the formula

$y=\displaystyle\frac{4ac-b^2}{4a}$ but b=0.

The formula can be simplified to

$y=c$ , the next step is look at the sign of a if the graph is opening downward or upward. Since a here is negative the graph opens downward. Therefore the range is $y\leq9$ or (-∞,9]

Sample Problem 2:

Find the range of the function $f(x)=3x^2-2x+4$

Solution:

$y=\displaystyle\frac{4ac-b^2}{4a}$ $y=\displaystyle\frac{4(3)(4)-(-2)^2}{4(3)}$

$y=\displaystyle\frac{11}{3}$ and the graph is opening upward.

The range is $y\geq\displaystyle\frac{11}{3}$ or $[\displaystyle\frac{11}{3},+\infty)$

### Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.

### 5 Responses

1. Talitha says:

Nice post. I discover some thing extra challenging on diverse blogs everyday. It will generally be stimulating to read content material from other writers and practice a bit some thing from their shop. I’d prefer to utilize some with the content material on my weblog regardless of whether you do not thoughts. Natually I’ll provide you with a link in your internet blog.

2. suba me says:

lPHEOx Your style is so unique in comparison to other people I ave read stuff from. Thanks for posting when you ave got the opportunity, Guess I will just book mark this blog.

3. ShN0b0 You ave made some really good points there. I checked on the web to find out more about the issue and found most individuals will go along with your views on this web site.

4. friv says:

114542 548285I really appreciate your piece of work, Great post. 49272

5. spinner says:

846316 140242Hmm is anyone else having issues with the images on this blog loading? Im trying to figure out if its a issue on my end or if its the blog. Any responses would be greatly appreciated. 554295