Domain and Range of Quadratic Functions

What patterns do we see?

• When we are trying to figure out the domain of any function the question we should ask ourselves is:

• What possible values could this function take on for x?

• We can ask the same question for range.• What possible values could this function

take on for y?

What patterns do we see?

• Sometimes people get confused and state domain and range in terms of what a function cannot be. THIS IS WRONG!!

• Always state the domain and range in terms of what can be!!

What patterns do we see?

• Unless a parabola has dots at its end or we are specifically told that it does not continue to extend indefinitely, we can make the presumtion that it will always have the same domain: x

Why is this?

• Think about it: As we move up from the vertex of (0,0) we notice that the parabola continues to get wider and wider. This pattern will continue forever. So there will be no restriction on what x can possibly be.

What about the range?

• The range will change from graph to graph• We can see from the previous graph that it

will never go below the y value of “0”. • Therefore, y can only be greater than 0. • It will still belong to real numbers because

there is an unbroken line connecting all the points. In short:

yyR ,0:

Another Example

• Find the domain and range of the following:

}{:

}5,{:

yeRR

xxeRD

Real World Applications

• Imagine the height of a ball thrown off a building is modelled by the equation

• Where t is time in seconds and h is height in meters

• What would be an appropriate domain and range?

15)2(5.0)( 2 xth

Real World Applications

We can see that the x intercepts are approximately –3.5 and +7.5. However, it is not realistic to have negative time therefore we would modify the domain to:

},5.70{: RttD

Real World Applications

},150{: RhhR

Similarly, it wouldn’t make sense for the ball to go below ground so the range would be as follows:

Pop Quiz

• Find the Domain and Range of the following relations.

Graph 1 Graph 2

Graph 3 Graph 4

Graph 5

Answers:

1.

2.

3.

4.

5.

}{:

}{:

yeRR

xeRD

}8,{:

}{:

yyeRR

xeRD

}91,{:

}91,{:

yyeRR

xxeRD

}{:

}8,{:

yeRR

xXeRD

}6,3,1,3,5{:

}4,3,2,1,3,6{:

R

D

So in conclusion:• Don’t just apply a blanket idea to

everything…..look at the circumstances as well.