Module 1
Lesson 1: Basic Parent Functions and their Transformations
There are too many functions in the world to try to identify which ones are
the most important or most impressive to talk about. After all, consider that
basic functions simply involve two variables and their relationship. If you
try, you can probably list many yourselves. All you have to do is name two
variables that depend on each other…and you’ve found probably a function.
Because there are so many, what we will do instead is to create some basic
graphs of functions we might run into (if we haven’t already).
These graphs are called parent functions. The reason is that they’re the
main picture for their class of functions. For example, x 2 is a parabola that
has a vertex at (0, 0). That’s the basic and simplest mathematical parabola.
There are an infinite number of others. Some are over to the left of the
origin, some over to the right, some above, some below, some wide, some
steep, some even upside-down. But what they all have in common is that
original x 2 picture and value in their equations.
So here are the graphs of these functions. Learn them. Someday you’ll be
quizzed and/or tested on them. Learn them. Someday you’ll need to know
them quickly. You’ll need to remember what they look like without having to
take the time to re-graph them. Learn them.
y
f(x)=x
x
-9 -8 -7 -6 -5 -4 -3 -2 -1
1. Linear 𝑓(𝑥) = 𝑥
2. Quadratic 𝑓(𝑥) = 𝑥 2
y
f(x)=x^2
x
-9 -8 -7 -6 -5 -4 -3 -2 -1
1
2
3
4
3. Cubic 𝑓(𝑥) = 𝑥 3
5
6
7
8
9
1
2
3
4
5
6
7
8
9
y
f(x)=x^3
x
-9 -8 -7 -6 -5 -4 -3 -2 -1
4. Absolute value
1
2
f(x)=abs x
x
1
4
2
3
4
5
5
6
𝑓(𝑥) = | 𝑥 |
y
-9 -8 -7 -6 -5 -4 -3 -2 -1
3
6
7
8
9
5. Square root 𝑓(𝑥) = √𝑥
7
8
9
y
f(x)=sqrt x
x
6.
Cube root f ( x )  3 x
y
f(x)=x ^(1/3)
x
7.
Exponential f ( x )  e x
y
f(x)=e^x
x
8. Logarithmic f ( x )  ln x
y
f(x)=ln (x)
x
9. Reciprocal f ( x ) 
1
x
y
f(x)=1/x
x
10. Constant f ( x )  c
y
f(x)=2
x
Transformations
Terms-
Here are a couple of terms to know:
Rigid transformations- Transformations that keep the shape of the original
graph. They include vertical, horizontal and
reflections.
Non-rigid transformations-
Transformations that distort the shape of the
original graph. These include stretching and
shrinking the graph.
Vertical Translations:
y  f ( x)  c
a translation up by c units
y  f ( x)  c
a translation down by c units
Horizontal Translations:
y  f ( x  c)
a translation to the right by c units
y  f ( x  c)
a translation to the left by c units
A way to help keep straight which is which is this mnemonic device:
LASeR
which stands for:
(Left Add, Subtract Right)
Reflections across the x-axis:
y   f ( x)
is the way to find the reflection of a function across the x-axis
Notice this means take the opposite of the entire function.
change the sign of each term:
Example
We would
f ( x )  3x 2  4 x  1
Find g(x) if g(x) is a reflection of f(x) across the x-axis:
g ( x)   f ( x)
g ( x )  (3 x 2  4 x  1)
g ( x )  3 x 2  4 x  1
Reflections across the y-axis:
y  f ( x)
is the way to find the reflection of a function across the y-axis
Stretches and Shrinks-
Vertical-
y  c f ( x)
a stretch by a factor of c if c  1
a shrink by a factor of c if c  1
Horizontal -
x
y f( )
c
a stretch by a factor of c if c  1
a shrink by a factor of c if c  1
(Notice that in either case if c > 1, we always have a stretch.)
One note on stretches and shrinks. Notice that the factor, c, is placed within
the function itself when the stretch/shrink is horizontal. The factor, c, is
multiplied out front when the transformation is a vertical stretch/shrink.
For example:
y = 2(x)3 Is a vertical stretch by a factor of 2
y = ½ ( x)3 is a vertical compression ( shrink ) by ½
Now y = ( 2x )3 is a horizontal compression ( shrink ) by ½
And y = ( ½ x )3 is a horizontal stretch by 2.