Algebra 2 Unit 1 - Functions Packet - Dam

UNIT 1:
Key Features &
Transformations of

Functions

Algebra 2

Dr. Dam – [email protected]

Period:
Name:
ID:

SUCCESS CRITERIA 

 

I can…  identify the function family of a graph/equation and label all the key features. 

 

I can explain…  how transformations affect a function and its key features.  

 

I can justify…  why the different representations of a function are equivalent. 

 

I can apply…  transformations to model a scenario. 

 

 

 

MATH FUNCTION DANCE :) 

 

 
 

TABLE OF CONTENTS 

LESSON  TITLE  ​OPTIONAL  PG # 
1.1  Definition of Functions & Function Notation  4-5
1.2  Function Families & their Parent Functions  6-10
1.3  Key Features of Functions  11-14
1.4  Domain & Range  15-16
1.5  Transformations of Functions  17-19
1.6  Piecewise Functions 20-21

2 

Lesson 1.1 - Definition of Functions & Function Notation 
Function 
Function Notation 

Example 1: Let f (x) = x2 + 3 . Evaluate f (x) at x = − 1, 0, and 2 . Make a table using these values.

Example 2: Let g(x) = √ x − 1 . What values can be inputted into g(x) ? Explain your reasoning. 

You Try: Evaluate the three functions below.

f (x) = x3 at x = − 2 and 1/2 g(x) = |x| + 2 at x = − 4 and 7 h(x) = 1 at x = − 4 and 0
x

3 

Classwork 

 

1) Let f (x) = 2x − 3 ; Find f (6) .

2) Let g(x) = x2 − 4x ; Find g(− 1) .

3) Let h(x) = |3x| − 2 ; Find h(− 2) .

4) Let v(t) = 4t + 3 ; Find v(2) .

5) Let f (x) = (x − 2)2 ; Create a table of values for x = − 1, 0, 1, 2, 3 .
Then, graph the values and draw the function.

6) In the graph below, f (4) = 1. Use the graph to determine: f (2), f (8), and f (0) .

7) Using the graph above, what values of x make f (x) = 1 ? Explain your reasoning.

 

 
4 

Lesson 1.2 - Function Families & their Parent Functions 

LINEAR FAMILY Parent: f (x) = x Parent Graph 
QUADRATIC FAMILY Parent Graph 
CUBIC FAMILY x f​(x​ ​) Parent Graph 
-2
-1
0
1
2

Parent: f (x) = x2

x f(​ x​ ​)
-2
-1
0
1
2

Parent: f (x) = x3

x f(​ ​x)​
-2
-1
0
1
2

5 

SQUARE ROOT FAMILY Parent: f (x) = √x Parent Graph 
CUBE ROOT FAMILY Parent Graph 
ABSOLUTE VALUE FAMILY x f(​ x​ )​ Parent Graph 
-1
0
1
2
4

Parent: f (x) = √3 x

x f(​ ​x)​
-8
-1
0
1
8

Parent: f (x) = |x|

x f(​ ​x​)
-2
-1
0
1
2

6 

EXPONENTIAL FAMILY Parent: f (x) = 2x     Parent Graph 
RATIONALFAMILY     Parent Graph 
RATIONAL FAMILY   Parent Graph 
x f(​ ​x​)
3
2
1
0
-1
-2

Parent: f (x) = 1  
x
 

x f(​ x​ )​

10

1

1/10

0

-1/10

-1

-10

Parent: f (x) = 1  
x2
 

x f​(​x​)

10

1

1/10

0

-1/10

-1

-10

 
7 

LOGARITHMIC FAMILY Parent: f (x) = log x Parent Graph 
SINE FAMILY Parent Graph 
COSINE FAMILY x f(​ x​ ​) Parent Graph 
-2
-1 8 
0
1
2
4
10

Parent: f (x) = sin x

x f(​ ​x)​
-π
- π/2
0
π/2
π
3π/2
2π

Parent: f (x) = cos x

x f​(​x​)
-π
- π/2
0
π/2
π
3π/2
2π

Everyday Functions 

9 

Prerequisite Skill for Lesson 1.3 - Interval & Inequality Notation 

Interval  Inequality  Number Line 

10 

Lesson 1.3 - Key Features of Functions 
x-Intercept
y-Intercept
Relative Maximum 
Relative Minimum 
Increasing 
Decreasing 
End Behavior 

How do we mathematically write these key features? 

11 

Example 1: The graph depicts the function. Identify the function’s intercepts; relative minimums and
maximums; absolute minimum and maximum; and intervals of increase/decrease.

You Try: For each graph identify the function’s intercepts; relative minimums and maximums;
absolute minimum and maximum; and intervals of increase/decrease.
If it does not exist, then write “DNE.”

12 

Key Features Review 

Domain: 
Range: 
x-intercept(s):
y- intercept(s):
Extrema: 
Asymptote(s): 
Increasing Behavior: 
Decreasing Behavior: 
End Behavior: 

Domain: 
Range: 
x-intercept(s):
y- intercept(s):
Extrema: 
Asymptote(s): 
Increasing Behavior: 
Decreasing Behavior: 
End Behavior: 

Domain: 
Range: 
x-intercept(s):
y- intercept(s):
Extrema: 
Asymptote(s): 
Increasing Behavior: 
Decreasing Behavior: 
End Behavior: 

13 

Lesson 1.4 - Domain & Range 

Domain​ All possible inputs in a function (consider what can and cannot go into a function’s input). U​ se x-values​. 
Range​ All possible outputs in a function (consider what can and cannot go into a function’s output). ​Use y-values.​  

Example 1: Find the domain & range of f (x) = √x . Example 2: Find the domain & range of f (x) = x2
.

You Try: Determine the domain & range of each of the following parent functions below.

f (x) = |x| f (x) = x

Domain: Domain:

Range: Range:

f (x) = x3 f (x) = √3 x

Domain: Domain:
Range: Range:

14 

f (x) = 2x f (x) = 1
x

Domain: Domain:
Range: Range:

Example 3: The graph shows f (x) = √x . Draw the graphs for g(x) = √x + 3 and h(x) = √x + 3 .
Determine the domain & range of g(x) and h(x) .

Consider how they changed from the parent function and why they changed.

Example 4: The graph shows f (x) = x2 . Draw the graphs for g(x) = x2 − 5 and h(x) = (x − 5)2 .
Determine the domain & range of g(x) and h(x) .
Consider how they changed from the parent function and why they changed.

15 

Lesson 1.5 - Transformations of Functions 

Translation 
Dilation 
Reflection 

Translations Description Example 
Example 
f (x) + a Example 
f (x) − a
f (x − a) 16 
f (x + a)

Dilations Description
Description
k f (x)

1 f (x)
k

f (kx)

f( 1 x)
k

Reflections
− f (x)
f (− x)

*ORDER of transformations: 1) Dilations & Reflections 2) Translations

Example 1: Identify the parent function and transformations of each function. Then sketch each function.

f (x) = (x + 4)2 h(x) = = 2(x + 3)3

g(x) = − 3 |x| v(x) = 1 √− x + 4
2

Example 2: Given the graph of the parent function f (x) , describe the transformations for each function.
Then apply the transformation to the graph and label them respectively using a, b, c.

a) 2f (x − 5)

b) − f (x) − 3

c) 1 f (x + 3) − 2
2

Example 3: Write a possible function for each graph.

 

 
17 

Transformations Review 
 
 

Suppose (6, 1) is the point of the graph y = f (x) .
For each of the following, name the point on the graph, and then name the transformation.

1. y = f (3x)

2. y = f (x + 2)

3. y = f (x) + 5

4. y= 1 f (x)
2

5. y = f (− x)

6. y = f (x − 7)

7. y = 4f (x)

8. y = f( 1 x)
2

9. y = − f (x)

 
18 

Lesson 1.6 - Piecewise Functions OPTIONAL 

Piecewise Function​ A function defined by multiple functions on different intervals 

Example 1: Determine the key features. Domain:
f (x) = Range:
Intercept(s):
Extrema:
Asymptote(s):
Increasing Behavior:
Decreasing Behavior:
End Behavior:

Example 2: Let f (x) be the parent function. Graph f (− x) + 2 .

Example 3: Let g(x) be the parent function shown in the graph. List the key features of g(x) .

Then perform the transformations described by the function 1 g(x − 3) − 2 .
2

19 

Example 4: Let h(x) be the parent function shown in the graph. List the key features of h(x) .

Then perform the transformations described by the function − 2h(x) + 5 and h(2x) .
 

You Try: Let m(x) be the piecewise function shown.
a) Make a table showing at least 4 values of m(x) .

b) Identify the domain and range of m(x) .

c) Pick any transformation(s) to perform and show it in the blank graph.
Write the equation for your transformation(s).

 

 
 

 
 
 
 
 

 

 
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21