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:
R :  y  0, y
Another Example
• Find the domain and range of the following:
D : {xeR, x  5}
R : { yeR}
Real World Applications
• Imagine the height of a ball thrown off a building
is modelled by the equation
h(t )  0.5( x  2)  15
2
• Where t is time in seconds and h is height in
meters
• What would be an appropriate domain and range?
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:
D : {0  t  7.5, tR}
Real World Applications
Similarly, it wouldn’t make sense for the ball to go below
ground so the range would be as follows:
R : {0  h  15, hR}
Pop Quiz
• Find the Domain and Range of the
following relations.
Graph 1
Graph 2
Graph 3
Graph 4
Answers:
Graph 5
1. D : {xeR}
R : { yeR}
2.
D : {xeR}
R : { yeR, y  8}
3.
D : {xeR,1  x  9}
R : { yeR,1  y  9}
4.
D : { XeR, x  8}
R : { yeR}
5.
D : {6,3,1,2,3,4}
R : {5,3,1,3,6}
So in conclusion:
• Don’t just apply a blanket idea to
everything…..look at the circumstances as
well.