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1 Name: Date: Period: Algebra 2 Trimester 1 Review Learning Goals Page 1.1 Evaluate and interpret functions from an equation, table, mapping, or graph. 2 1.2 Perform operations with functions and interpret them in context. 3 1.3 Perform compositions of functions 4 1.4 Find inverses of functions both graphically and algebraically 5 1.5 Determine if a relation is a function, find the domain, and determine if it is 1-1 6 1.6 Describe and interpret key features of a graph in context. 7 1.7 Calculate and interpret the average rate of change of a function. 8 2.1 Describe growth as exponential, linear or quadratic. 9 2.2 Write functions using a recursive formula. 10 2.3 Multiply polynomials and simplify fully. 11 2.4 Factor polynomials using various factoring techniques 12 2.5 Solve polynomial equations by factoring and find roots on a graph. 13 2.6 Solve a system of equations using graphs, tables, and algebraic techniques 14 2.7 Set up a systems of equations from a word problem. 15 3.1 Perform operations with rational expressions and determine when they are undefined 16 3.2 Solve rational equations and identity extraneous solutions 17 3.3 Solve absolute value equations and inequalities. 18 Resources: Mesamath.weebly.com Go to Algebra 1 Trimester Review to find: Class notes Videos Answer Keys

Name: Date: Period: Algebra 2 Trimester 1 Review · Algebra 2 Trimester 1 Review ... 2.2 Write functions using a recursive formula. 10 ... 2.7 Set up a systems of equations from a

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1

Name: Date: Period: Algebra 2 Trimester 1 Review

Learning Goals Page

1.1 Evaluate and interpret functions from an equation, table, mapping, or graph. 2

1.2 Perform operations with functions and interpret them in context. 3

1.3 Perform compositions of functions 4

1.4 Find inverses of functions both graphically and algebraically 5

1.5 Determine if a relation is a function, find the domain, and determine if it is 1-1 6

1.6 Describe and interpret key features of a graph in context. 7

1.7 Calculate and interpret the average rate of change of a function. 8

2.1 Describe growth as exponential, linear or quadratic. 9

2.2 Write functions using a recursive formula. 10

2.3 Multiply polynomials and simplify fully. 11

2.4 Factor polynomials using various factoring techniques 12

2.5 Solve polynomial equations by factoring and find roots on a graph. 13

2.6 Solve a system of equations using graphs, tables, and algebraic techniques 14

2.7 Set up a systems of equations from a word problem. 15

3.1 Perform operations with rational expressions and determine when they are undefined 16

3.2 Solve rational equations and identity extraneous solutions 17

3.3 Solve absolute value equations and inequalities. 18

Resources: Mesamath.weebly.com Go to Algebra 1 Trimester Review to find:

• Class notes • Videos • Answer Keys

2

Set 1.2: 1) The graph of a function g is given:

Find the following:

a) 𝑔 −4 = b) 𝑔 2 = c) 𝑔 4 = d) Find the value of x when 𝑔 𝑥 = −2

2) The graphs of functions f and g are shown below:

a) Which is larger, 𝑓(0) or 𝑔(0)? b) Which is larger, 𝑓(−3) or 𝑔(−3)? c) For which values of x is 𝑓(𝑥) = 𝑔(𝑥)?

Set 1.3: Use the function to evaluate the indicated expressions and full simplify: 1) For the function 𝑓 𝑥 = 6𝑥 − 18, find:

a) 𝑓 𝑥! =

b) 𝑓 !!

=

c) 𝑓 𝑥 + 2 =

d) 𝑓 3𝑏

2) For the function 𝑝 𝑥 = 𝑥!

a) 𝑝 3

b) 𝑝 4𝑥

c) 𝑝 𝑥 + 3

d) 𝑝 5𝑥!

Learning Goal 1.1

3

Learning Goal 1.2

4

1) ℎ 𝑥 = 𝑥! and 𝑝 𝑥 = 2𝑥 − 3 a) (ℎ ∘ 𝑝)(2) b) (𝑝 ∘ ℎ)(2) c) (𝑝 ∘ ℎ)(𝑥) d) (ℎ ∘ 𝑝)(𝑥)

3) a)

b)

c) 4) 𝑓 𝑥 = 𝑥 + 5 and 𝑔 𝑥 = 4𝑥 a) 𝑓 (𝑔 2 ) b) (𝑔 ∘ 𝑓)(0) c) (𝑓 ∘ 𝑔)(0) d) (𝑔 ∘ 𝑓)(𝑥) 5) 𝑓 𝑥 = 𝑥 + 2 and 𝑔 𝑥 = 𝑥! + 4 a) 𝑓 (𝑔 3 ) b) 𝑔 (𝑓 3 ) c) 𝑓 (𝑓 5 ) d) 𝑔(𝑓 𝑥 )

6) a)

b)

c)

( f ! g)(4)

(g ! f )(1)

f (g(−1))

(g ! f )(−3)

( f ! g)(4)

f (g(0))

6

4

2

-2

- 4

- 6

-10 - 5 5 10

f(x)

6

4

2

-2

- 4

- 6

-10 - 5 5 10

g(x)

6

4

2

-2

- 4

- 6

-10 - 5 5 10

g(x)

6

4

2

-2

- 4

- 6

-10 - 5 5 10

f(x)

Learning Goal 1.3

5

Example 1: The accompanying graph shows the relationship between the cooling time of magma and the size of the crystals produced after a volcanic eruption. On the same graph, sketch the inverse of this function.

Example 2: The function, f, is drawn on the set of axes. On the same set of axes, sketch the graph of 𝑓!!, the inverse of f.

Set 1.4: Find the inverse of each of the following functions:

a) 𝑓 𝑥 = 𝑥 − 3 b) 𝑝 𝑥 = !!𝑥 + 5

c) 𝐴 𝑥 = 3 − !!!

d) 𝐺 𝑥 = 𝑥! + 1

e) 𝑞 𝑎 = 2 𝑎 − 5 ! f) 𝑚 𝑟 = 3 𝑟 − 2 + 5(𝑟 + 3)

Learning Goal 1.4

6

1)

2)

3)

4)

5)

6)

a) Which ones of the functions above are one-to-one? Write the numbers below: b) Will function number 4) have an inverse that is a function? Why or why not? c) Wlil function number 3) have an inverse that is a funtion? Why or why not? (Graph to check!) d) State the domain and range of funciton 5) e) State the domain and range of function 6) f) State the domain and range of function 1) g) For function number 4) Find the the values where the function is Increasing: Decreasing: h) Find the local minimum and local maximum for function 4): 12. The graph shows the average baseball salary of a player as a function of time

Learning Goal 1.5

7

a) Determine the time interval(s) on which the function is increasing:

b) Determine the time interval(s) on which the function is decreasing in interval notation:

c) In what year were average players’ salaries the highest from 1989 to 1995?

d) From 1989 to 1998, what was the highest average baseball salary?

e) Why do you think the graph looks like this? Explain one possible conclusion you could

draw:

13. The graph below shows a distance versus time graph of a person running.

a) Is the person running faster after 16 seconds or 9.5 seconds? Explain: c) Write a brief story to describe this person’s run:

Learning Goal 1.6

8

Practice Set: 1) Find the average rate of change on the indicated time intervals a) Interval [1, 5]

b) Interval [−1, 5]

2) The graph shows the depth of water W in a reservoir over a one-year period as a function of the number days x since the beginning of the year. What was the average rate of change of W between x=100 and x=200?

3) Find the average rate of change from the tables:

4) The table gives the population in a small coastal community for the period 1997-2006. Figures shown are for January 1 in each year. a) What was the average rate of change of population between 1998 and 2001? b) What was the average rate of change of population between 2001 and 2002? c) For what period of time was the population increasing? d) For what period of time was the population decreasing?

x y -10 20 -5 35 0 47 5 90

10 120 15 90 20 80 25 105

a) Find the average rate of change on the interval −10 ≤ 𝑥 ≤ 15 b) Find the average rate of change on the interval [15, 25] b) Find an interval where the average rate of change is 0!

Learning Goal 1.7

9

5) Find the average rate of change of the function 𝑔 𝑥 = 5𝑥 − 3 on the interval [2, 4]

6) Find the average rate of change of the function 𝑞 𝑥 = 𝑥! + 3𝑥 on the interval −1 ≤ 𝑥 ≤ 3

1. Identify the following functions as exponential, quadratic, or linear. Explain your choice briefly: a)

x y 0 2 1 4 2 8 3 16 4 32 5 64

Type of function:__________________ Why?:

b) x y 0 10 1 9 2 6 3 1 4 -6 5 -15

Type of function: _____________ Why?:

Learning Goal 2.1

10

c) 𝑦 = 𝑥! − 9𝑥 + 12 Type of function:__________________ Why?:

d)

Type of function:__________________ Why?:

e) x y 0 25 1 30 2 35 3 40 4 45 5 50

f) x y 0 1000 1 100 2 10 3 1 4 .1 5 .01

3. Write recursive definitions for the functions below: a)

x y

0 125

1 25 2 5

3 1

4 1/5 5 1/25

Recursive Definition:

b) x y

0 1

1 4

2 7

3 10

4 13

Recursive Definition:

c) x y

0 20

1 18

2 16

3 14

4 12

Recursive Definition:

Learning Goal 2.2

11

Learning Goal 2.3

12

Remember your steps! Make sure you are factoring completely!!

1)

2)

3)

4)

5)

6) 20𝑥!𝑝 − 125𝑦!𝑝

Learning Goal 2.4

13

a)

b)

c)

d)

e)

Learning Goal 2.5

14

Learning Goal 2.6

15

Learning Goal 2.7

16

1. 4𝑥2𝑥𝑦

−56𝑥

2. 𝑥 + 2

𝑥! − 𝑥 − 2∙𝑥! + 2𝑥 + 1

𝑥 + 1

3. 6

𝑦 − 5−

𝑦 + 5𝑦! − 25

4. 𝑥 + 10

𝑥! + 16𝑥 + 60÷

𝑥 − 82𝑥 + 12

5. 3

𝑥 − 1−2𝑥

Learning Goal 3.1

17

6. 9𝑥! − 3𝑥 − 2

3𝑥 + 1÷3𝑥 − 2𝑥 + 5

7. Solve for all values of x 1𝑥!−6𝑥 + 6𝑥!

=1𝑥

8. Solve for all values of x 23𝑥

+ 5 =4𝑥

9. Solve for all values of x 𝑥

𝑥 + 5+

9𝑥 − 5

=50

𝑥! − 25

Learning Goal 3.2

18

Learning Goal 3.3