Math Functions

Table of Contents
1
I. Number & Quantity
II. Algebra
III. Functions
IV. Geometry
V. Statistics & Probability
VI. Integrating Essential Skills

III. Functions
2
A. Properties of Functions
B. Linear Functions
C. Quadratic Functions
D. Trigonometric Functions
E. Other Functions
F. Sequences
G. Functions Modeling

1. Functions Defined
 A Function is a relationship in
which each input is related to
exactly one output
 ACT Functions
 Linear
 Quadratic
 Trigonometric
 Logarithmic
 Exponential
 Questions often involve graphs
and sometimes integrate
multiple functions
 Example:
3
If 𝑓 𝑥 = 𝑥2
− 4𝑥 + 3, then f −3 =?

#1
4
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

#1
4
Answer: D
Calculator?
Drawing?

2. Domain and Range
Domain
 The set of all possible x values
that output real y values
Range
 The set of all possible output
values (y values)
5

2. Domain and Range
Domain
 The set of all possible x values
that output real y values
Range
 The set of all possible output
values (y values)
5
It can often be helpful to graph a function to “visualize” its domain and range!

3. Asymptotes
 Asymptotes are lines that
continually approach a curve but
never reach it
 On the ACT, we typically see
vertical and horizontal asymptotes
6

3. Asymptotes
never reach it
 To solve for the vertical
asymptotes, we set the
denominator equal to zero and
solve for x
6

3. Asymptotes
never reach it
 To solve for the vertical
asymptotes, we set the
denominator equal to zero and
solve for x
 To solve for the horizontal
asymptotes, we divide the leading
terms
 Think of this as solving for the
value of y when x=∞
6

#2
7
Calculator?
Drawing?

#2
7
Answer: C
Calculator?
Drawing?

III. Functions
8
B. Linear Functions
E. Other Functions
F. Sequences

Linear Functions
9
𝑓 𝑥 = −
1
2
𝑥 +
1
2

Linear Functions
 Linear Functions are those that
result in a straight line when
graphed
9
𝑓 𝑥 = −
1
2
𝑥 +
1
2

Linear Functions
graphed
 Slope:
9
𝑓 𝑥 = −
1
2
𝑥 +
1
2

Linear Functions
graphed
 Slope:
9
𝑓 𝑥 = −
1
2
𝑥 +
1
2
∆𝑦
∆𝑥
=
𝑦2 − 𝑦1
𝑥2 − 𝑥1

Linear Functions
graphed
 Slope:
 Perpendicular Line has a slope
with
 Opposite Sign
 Reciprocal
9
𝑓 𝑥 = −
1
2
𝑥 +
1
2
∆𝑦
∆𝑥
=
𝑦2 − 𝑦1
𝑥2 − 𝑥1

#3
10
Calculator?
Drawing?

#3
10
Answer: B
Calculator?
Drawing?

III. Functions
11
B. Linear Functions
E. Other Functions
F. Sequences

Quadratic Functions
 Quadratic Functions are
equations that form parabolas
when graphed in the coordinate
plane
12

Quadratic Functions
plane
 Often, quadratic function
questions on the ACT involve the
intersection of multiple functions
(i.e. a line with a parabola or a
circle with a parabola)
12

Quadratic Functions
plane
 Often, quadratic function
questions on the ACT involve the
intersection of multiple functions
(i.e. a line with a parabola or a
circle with a parabola)
 Hint: it can be helpful to graph
these functions by hand or on
the calculator!
12

#4
13
Calculator?
Drawing?

#4
13
Answer: A
Calculator?
Drawing?

III. Functions
14
B. Linear Functions
E. Other Functions
F. Sequences

1. Measuring Angles in Radians
15

 Radians are used occasionally on the ACT
 Radians are a second way of measuring angles
 Radians : Degrees :: Inch : Centimeter
15

 Conversion between angles in degrees and radians
 Use the proportion below.
 Fill in either the degree measure or radian measure you have
been given.
 Solve for the other using cross multiplication.
15

 Conversion between angles in degrees and radians
 Use the proportion below.
 Fill in either the degree measure or radian measure you have
been given.
 Solve for the other using cross multiplication.
15
45° in radians?

2. Trig Functions: The Basics
16
𝑦 = 𝐴𝑓 𝐵 𝑥 + 𝐶 + 𝐷

16
𝑦 = 𝐴𝑓 𝐵 𝑥 + 𝐶 + 𝐷
Trig. Function
(sin, cos, tan)

16
𝑦 = 𝐴𝑓 𝐵 𝑥 + 𝐶 + 𝐷
Trig. Function
(sin, cos, tan)
Amplitude

16
𝑦 = 𝐴𝑓 𝐵 𝑥 + 𝐶 + 𝐷
Trig. Function
(sin, cos, tan)
Amplitude
Period

16
𝑦 = 𝐴𝑓 𝐵 𝑥 + 𝐶 + 𝐷
Trig. Function
(sin, cos, tan)
Amplitude
Period
Left/Right Shift

16
𝑦 = 𝐴𝑓 𝐵 𝑥 + 𝐶 + 𝐷
Trig. Function
(sin, cos, tan)
Amplitude
Period
Left/Right Shift
Up/Down Shift

16
𝑦 = 𝐴𝑓 𝐵 𝑥 + 𝐶 + 𝐷
Trig. Function
(sin, cos, tan)
Amplitude
Period
Left/Right Shift
Up/Down Shift
𝑦 = 3 sin 2 𝑥 +
𝜋
2
+ 4

3. Features of Trigonometric Function Graphs
Amplitude Period
17

Amplitude
 Amplitude represents the
maximum distance away from
the “midline” that a graph
reaches
 Formula:
Period
17

Amplitude
reaches
 Formula:
 Larger leading coefficients 
larger amplitude
 y = 3 sin(x)  A = 3
 y = 1.1 cos(x)  A = 1.1
Period
17

Amplitude
reaches
 Formula:
larger amplitude
 y = 3 sin(x)  A = 3
 y = 1.1 cos(x)  A = 1.1
Period
17
A

Amplitude
reaches
 Formula:
larger amplitude
 y = 3 sin(x)  A = 3
 y = 1.1 cos(x)  A = 1.1
Period
 Period represents the minimum
amount of time (measured along
the x-axis) that it takes for a
graph to start repeating its
shape
 y = sin(4x) has a smaller
period than y = sin(x) because
the first equation repeats more
often
17
A

Amplitude
reaches
 Formula:
larger amplitude
 y = 3 sin(x)  A = 3
 y = 1.1 cos(x)  A = 1.1
Period
 Period represents the minimum
amount of time (measured along
the x-axis) that it takes for a
graph to start repeating its
shape
 y = sin(4x) has a smaller
period than y = sin(x) because
the first equation repeats more
often
17
A
Represents one period

#5
18
Calculator?
Drawing?

#5
18
Answer: B
Calculator?
Drawing?

III. Functions
19
B. Linear Functions
E. Other Functions
F. Sequences

1a. Logarithm Functions
 Logarithms exist as the inverse
(or opposite operation) of
exponential functions
 Equivalent Statements
20
Logarithmic Form
Exponential
Form

1a. Logarithm Functions
 Logarithms exist as the inverse
(or opposite operation) of
exponential functions
 Equivalent Statements
To solve a logarithmic equation, it
is often helpful to rewrite it in
equivalent form. See the example
below.
Example
Solve log2(8)=x.
Rewrite in equivalent exponential
form.
2x=8
2 raised to the power of 3 is 8.
Therefore, x = 3.
20
Logarithmic Form
Exponential
Form

1b. Basic Properties of Logarithms
 Basic properties to use in
simplification of logarithmic
expressions
1. log(ab) = log(a) + log(b)
2. log(a/b) = log(a) – log(b)
3. log(ax) = x  log(a)
4. loga(ax) = x
Examples of Each Property
1. log(3x) = log(3) + log(x)
2. log(x/4) = log(x) – log(4)
3. log(x3) = 3log(x)
4. log2(25) = 5
21

#6
22
Calculator?
Drawing?

#6
22
Answer: B
Calculator?
Drawing?

2. Exponential Growth/Decay
23
𝑦 𝑡 = 𝑦0 1 ± 0.01𝑝 𝑡

23
𝑦 𝑡 = 𝑦0 1 ± 0.01𝑝 𝑡
Initial Amount

23
𝑦 𝑡 = 𝑦0 1 ± 0.01𝑝 𝑡
Initial Amount Percent Change

23
𝑦 𝑡 = 𝑦0 1 ± 0.01𝑝 𝑡
Time

 Often, we are given an initial ”amount” of something
(i.e. population)
23
𝑦 𝑡 = 𝑦0 1 ± 0.01𝑝 𝑡
Time

(i.e. population)
 There could be a percent change increase or
decrease over a period of time
23
𝑦 𝑡 = 𝑦0 1 ± 0.01𝑝 𝑡
Time

(i.e. population)
 There could be a percent change increase or
decrease over a period of time
 Remember that we might be asked to select the
correct equation too!
23
𝑦 𝑡 = 𝑦0 1 ± 0.01𝑝 𝑡
Time

#7
24
Calculator?
Drawing?

#7
24
Answer: D
Calculator?
Drawing?

3. Equation for a Circle
The equation for a circle of radius “r” and center (h,k):
(x – h)2 + (y – k)2 = r2
What you should be able to do:
 From an equation…
 State the radius
 State the center
 Find y-intercepts (set x=0 and solve)
 Find x-intercepts (set y=0 and solve)
 Recognize a graph
 From a graph…
 Find an equation
25

#8
26
Calculator?
Drawing?

#8
26
Answer: A
Calculator?
Drawing?

III. Functions
27
B. Linear Functions
E. Other Functions
F. Sequences

1. Arithmetic Sequences
 An Arithmetic Sequence is one
in which the difference between
consecutive terms is constant
28

 When we are asked about an
arithmetic sequence, we should
know to add or subtract by the
same number between terms
28

 When we are asked about an
arithmetic sequence, we should
know to add or subtract by the
same number between terms
 When sequences involve a small
number of terms, we can often
write them out!
28

2. Geometric Sequences
 A Geometric Sequence is one
in which numbers in the
sequence differ by a factor
29

 When we are asked about a
geometric sequence, we should
know to multiply or divide
between terms
29

 When we are asked about a
geometric sequence, we should
know to multiply or divide
between terms
 Sometimes, we apply skills from
sequences to discover patterns
for other problem types as well
29

#9
30
Calculator?
Drawing?

#9
30
Answer: A
Calculator?
Drawing?

III. Functions
31
B. Linear Functions
E. Other Functions
F. Sequences

#10
32
Calculator?
Drawing?

#10
32
Answer: A
Calculator?
Drawing?

Math Functions

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