Lesson 3:
Exploring Properties of Parent Functions
Part A: Introduction: Families
Just like we are able to identify families in the animal world…
Primates
Gorillas
Monkeys
Baboons
Orangutans
different animal
but same family
Feline (Big Cats)
Lions
Tigers
Cheetahs
Leopards
…every function can be classified as a member of a family.
Linear
y=x
y = 3x
y = 2x + 1
y = ½x – 7
different function
but same family
Quadratic
y = x2
y = x2 + 3x + 2
y = 4(x – 7)2 + 2
y = (x + 5)(x – 4)
Part B: Definitions
parent function: the simplest, or base, function in a family
example: f (x )  x or g (x )  x 2
family: a collection of functions (lines or curves) sharing common
characteristics
1
ex. from f (x )  x , you can have f (x )  3x  2 or f (x )   x  4
2
2
2
from g (x )  x , you can have g (x )  x  5 or g (x )  5(x  2)2  3
asymptote: a line that the graph of a relation or function gets closer and closer to
but never meets; it can be a vertical or horizontal
Part C: Five Parent Functions
Equation Name of
of Function Function
f (x )  x
Sketch of Graph
Special Features/
Symmetry
y
x
f (x )  x 2
y
x
f (x )  x
1
parabola that opens up
vertex at the origin
y has a minimum value
y-axis is axis of
symmetry
* graph only in quadrants
1 and 2
*half parabola that opens
Right
*vertex at the origin
*x and y have minimum
values
*graph only in quadrant 1
y
x
x
f (x )  x
*
*
*
*
y
x
f (x ) 
*straight line that goes
through the origin
*slope is 1
*divides the plane exactly
in half diagonally
*graph only in quadrants
1 and 3
*hyperbola
*does not intersect x- and
y-axes
*lines y=x and y=-x are
axes of symmetry
*graph only in quadrants 1
and 3
*x and y axis are
asymptotes
y
*represents
the distance
from x to 0
x
*two straight half lines
that start at the origin
and go up
*vertex at the origin
*y has a minimum value
*y-axis is axis of symmetry
*graph only in quadrants 1
and 2
Domain
Range