MAT‐101 Reference Sheet: Part 3

(for Test 3, created 2020‐05‐11 by David Grimes)
(3.4) Transformations of Functions
o There are 3 types of transformation: shifts, stretch/compressions, and reflections.

A shift is caused by adding/subtracting a number, like 𝑓 𝑥
(Shifts move the graph up/down/left/right.)
𝑥
5

A stretch/compression is caused by multiplying/dividing a number, like 𝑓 𝑥
(Stretch/compressions make the graph taller/shorter/wider/thinner.)

A reflection is caused by a negative sign, like 𝑓 𝑥
(Reflections flip the graph upside‐down or sideways.)
5𝑥
𝑥
o Every transformation has 2 versions: horizontal and vertical.

A horizontal transformation changes the 𝑥 coordinates, like 𝑓 𝑥
5𝑥
(The 5 is in parentheses with the 𝑥 and it does get squared with the 𝑥.)

A vertical transformation changes the 𝑦 coordinates, like 𝑓 𝑥
5𝑥
(The 5 is not in parentheses with the 𝑥 and it doesn’t get squared with the 𝑥.)
o The coordinates of each point change with each transformation.
(Horizontal transformations “go backwards.” They do the opposite of what’s written.)

Examples of vertical shifts:
1 shifts the graph up by 1 and changes each y‐coordinate by 1
𝑓 𝑥
𝑥
𝑓 𝑥
𝑥
1 shifts the graph down by 1 and changes each y‐coordinate by 1

Examples of horizontal shifts:
𝑓 𝑥
𝑥 1
shifts the graph left by 1 and changes each x‐coordinate by 1
𝑓 𝑥
𝑥 1
shifts the graph right by 1 and changes each x‐coordinate by 1

Examples of vertical stretch/compressions:
𝑓 𝑥
5𝑥 stretches the graph taller and changes each y‐coordinate by
𝑓 𝑥

5
compresses the graph shorter and changes each y‐coordinate by
Examples of horizontal stretch/compressions:
compresses the graph thinner and changes each x‐coordinate by
𝑓 𝑥
5𝑥
𝑓 𝑥

𝑥
5
𝑥
5
stretches the graph wider and changes each x‐coordinate by
5
Examples of reflections:
𝑓 𝑥
𝑥 reflects the graph vertically (“over x‐axis”) and changes the sign of each y‐coordinate.
𝑓 𝑥
𝑥 reflects the graph horizontally (“over y‐axis”) and changes sign of each x‐coordinate.
o You must do the transformations in the correct order:

When doing multiple transformations, do the horizontal shift first and the vertical shift last.
In between, do the other transformations in any order.
Example:
Suppose the function 𝑓 𝑥
𝑥 is transformed into 𝑔 𝑥
5 2𝑥 1
6
The three horizontal transformations are
shift 1,
compression 2,
and reflection.
The three vertical transformations are
reflection,
stretch 5,
and shift 6.
The point 3,9 was on the original graph. These transformations will move it to
The x‐coordinate will change like this:
3
The y‐coordinate will change like this:
9
⎯⎯
4
⎯⎯
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
2
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
•
9 ⎯⎯
45
2, 51
2
⎯⎯
51

(3.5) The Algebra of Functions; Composite Functions
o The following shorthand is used for adding/subtracting/multiplying/dividing functions:
𝑓 𝑔 𝑥
𝑓 𝑥
𝑔 𝑥
𝑓 𝑔 𝑥
𝑓 𝑥
𝑔 𝑥
𝑓𝑔 𝑥
𝑓•𝑔 𝑥
𝑓 𝑥 •𝑔 𝑥
𝑥
o Composite functions are more complicated:
𝑓○𝑔 𝑥
𝑓 𝑔 𝑥
represents plugging the output of 𝑔 into the function 𝑓.
This diagram shows the idea:
𝑥 ⎯ 𝑔 ⎯ 𝑔 𝑥 ⎯ 𝑓 ⎯ 𝑓 𝑔 𝑥
To find the domain, first find the domains of 𝑓 and 𝑔 separately. Example: If the domain of 𝑔 is 5 and
the domain of 𝑓 is 3 then the domain of 𝑓 ○ 𝑔 is all values of 𝑥 such that 𝑥 5 and 𝑔 𝑥
3.
After solving, suppose 𝑔 𝑥
3 when 𝑥 7. Then the domain of 𝑓 ○ 𝑔 is ∞, 5 ∪ 5,7 ∪ 7, ∞

(3.6) One‐to‐One Functions and Inverse Functions
o A function is one‐to‐one if every one y‐value is paired with only one x‐value. (It also has an inverse.)
You can test whether a function is one‐to‐one by using the horizontal line test on the graph.
o The inverse of a function named 𝑓 is represented by the notation 𝑓


Inverse functions perform opposite actions.
Example: 𝑓 𝑥
𝑓 𝑥
𝑥 5 because subtracting 5 is the opposite of adding 5.
𝑥
5 is the inverse of

In the inverse function, 𝑥 and 𝑦 are switched. Example: If 𝑓 2
7 then 𝑓
If the point 2,7 is on the graph of 𝑓, then the point 7,2 is on the graph of 𝑓

The domain of 𝑓 is the range of 𝑓

To find an inverse, rewrite the function with 𝑥 and 𝑦 in opposite places; then solve for 𝑦.
Example: 𝑓 𝑥
𝑥 5 would be rewritten as 𝑦 𝑥 5
Then we would switch the 𝑥 and 𝑦 to get 𝑥 𝑦 5 and solve for 𝑦 to get 𝑦 𝑥 5
Then we would rewrite it using the notation of inverse functions: 𝑓 𝑥
𝑥 5
7
2
and the range of 𝑓 is the domain of 𝑓
(4.1‐4.2) Quadratic Functions & Their Applications
 General Form:
𝑓 𝑥
𝑎𝑥
𝑏𝑥 𝑐
𝑓 𝑥
𝑎 𝑥
ℎ
𝑘
with vertex ℎ, 𝑘

Standard Form:

If 𝑎 is positive; the parabola opens upward and has a maximum; if 𝑎 is negative, it opens downward
and has a minimum. The maximum or minimum value is always found at the vertex.
o The axis of symmetry is a vertical line that goes through the vertex, so its equation is 𝑥 ℎ
o To find the vertex from the General Form, there are 2 strategies:
 Use the fact that ℎ
Then plug that value into the function to find 𝑘.
 Complete the square to change the General Form into Standard Form (so you can see ℎ and 𝑘).

(4.3) Polynomial Functions
o Even‐degree polynomials have end behavior that is up‐up (positive leading coefficient) or down‐down
(negative); odd‐degree polynomials’ end behavior is down‐up (positive) or up‐down (negative).
o To find a zero, do the same thing you would do to find an x‐intercept: Plug in 0 for 𝑦 and solve for 𝑥.
o The multiplicity of a zero is how many times it appears on the list of zeros. Example:
For the function 𝑓 𝑥
𝑥 5 𝑥 9 the list of zeros is 5, 5, 9 so the multiplicity of 5 is 2.
For odd multiplicity, the graph crosses the x‐axis. For even multiplicity, it touches and “bounces off.”
o At a point where the multiplicity is odd, the graph will cross the x‐axis.
At a point where the multiplicity is even, the graph will touch the x‐axis (and bounce off).