A Lesson on Domain and Range

Blog 3

Domain and Range of a quadratic function

Hi I’m here to teach about domain and range— specifically of the quadratic function!

Let’s first start with the definition of “domain” and “range”

Domain is the “input” of a function, or the possible x-values

Range is the “output” of a function, or the possible y-values

For example: (3,2) (7,5) (6,0)

Domain (x): 3,7,6

Range (y): 2,5,0

Another example that is a bit trickier is:

F (x) = 5/4-x

We know that: the numerator cannot equal 0, so that limits the domains to be any number BUT 0

To mathematically solve it, it would look like:

4-x=0

-x =-4

x =4

Since 4 is the number that makes the numerator of the equation F (x) = 5/4-x not possible, 4 is the limiting domain.

So, if you were to be asked for the domain and range of this equation it would look like:

Domain: (x ≠ 4) – All numbers except for 4

Range: (-∞<y< ∞)

We can look at another equation that uses a quadratic function to find the domain and range.

Ex. F (x)= x^2+4x+4

Just by graphing the function you can tell that the range (y) is limited to only positive numbers.
So when asked the range and domain you can graph the quadratic function and write:

Domain: (-∞<x<∞)

Range: (0<y<∞)

A method to find the Domain and Range of a quadratic function is finding the min. and max. by finding the vertex

The min. is the lowest point of the graph, and the max is the highest point of the curve, these points are called the vertex of a graph

Ex. F(x)= -3x^2+6x-2

Recall that to find the vertex in standard form it would be –b/2a

Vertex would be -6/-2(3) = 1

Then you would plug the vertex point (1) as the x-value

3(1)^2+6(1)-2=5

So the points would be (1,5)

To figure out if it is the min or max you check the sign in front of “a” in the standard form

In this case it is (-) negative so the vertex is a max point

Now, you might be a little confused and think that since its negative it should be the min point…

Here is a little trick so you wont be confused: Think of two faces, one happy and the other sad. Now, imagine that the mouth of these two faces represent the parabola. If you take a look at the happy person the bottom of the smile is on the bottom, while the sad person’s frown is going from the top to the bottom. SO, when you forget how to figure out if it’s the min/ max think just smile or frown!!!  J L

So now you know that the vertex (1,5) is the max point of the parabola you can solve for the range! Recall that the range is the y- value (5).

So the range is anything below 5!

You would write:

Domain: All real numbers

Range: (-∞<y<5)

You can even double-check your work by graphing the equation!

Now you all know about domain and range of a quadratic function !!