Algebra 2 Student-Directed Review/EnrichmentAlgebra 2 Student-Directed Review/Enrichment . The following activities are all about Functions. Throughout this year, you have learned

Algebra 2 Student-Directed Review/Enrichment

The following activities are all about Functions. Throughout this year, you have learned about a variety of functions. These activities will challenge you to apply that knowledge in a combined way, drawing on concepts that you have learned about earlier this year. You may choose to work your way through all of the activities in order, or to prioritize working on activities for topics that you don’t remember as well or that you struggled with earlier in the year.

If you need extra support in any of these topics, log into Mathspace (https://bit.ly/fcpsmathspace) using your regular FCPS username and password, and navigate to the associated topic in the eBook. You will find explanations and videos there.

Contents of this Packet:

Title Focus Concepts Investigating Functions

Intercepts, Asymptotes, Discontinuity

What’s My Function

Rational Functions, Transformations

Where to Begin and End

Domain, Range, End Behavior, Continuity

Domain and Range

Domain, Range, End Behavior, Continuity

Ups and Downs

Increasing, Decreasing, Extrema

Extrema, Intervals Increasing and Decreasing Practice

Increasing, Decreasing, Extrema

Locating Extrema Given Graphs and Equations

Extrema

Exploring Inverse Functions

Inverse

Finding the Inverse of a Function

Inverse

Composition of Functions

Composition

Task -- Function of a Ride

Task – Wildfires

https://bit.ly/fcpsmathspace

Mathematics Instructional Plan – Algebra II

Virginia Department of Education ©2018 5

Investigating Functions

For each of the following functions, determine the x- and y- intercepts, discontinuous points, and vertical and horizontal asymptotes.

1. 3

2)(

x

xxf

x- and y-intercepts ____________ discontinuous points ____________

vertical and horizontal asymptotes __________________

2. 3

9)(

2

x

xxf



3. 16

1252)(

2

2

x

xxxf

x- and y-intercepts __________discontinuous points ___________


4. 6

5)(

xxf



5. 4

2)(

2

x

xxf





6. x

xf1

)(



7. xf(x) 2



8. 22)( xxf



9. 2)2()( xxf



10. )(xf log2(x)+2



11. x

xf3

)(



12. )(xf log3(x)





Extension

Using a graphing utility, graph each of the 12 functions above individually. State the domain and

range of each function, and describe the end behavior.



What’s My Function?

Write the algebraic function for each graph shown.

1. 2.

x = −4 y =

____________________________ ____________________________

3. 4.

x = 0 y = x = 1 y = 0

____________________________ ____________________________

5. 6.

x = −1 y = (Challenge)

____________________________ ____________________________



Where to Begin and End

For each function below, state the domain and range, end behavior, and continuity.

1.

2.

Domain: __________________________ Domain: __________________________

Range: __________________________ Range: __________________________

End Behavior

__________________________ End Behavior

__________________________

__________________________

__________________________

Describe the continuity of the function: Describe the continuity of the function:

3.

4.

Domain: __________________________ Domain: __________________________

Range: __________________________ Range: __________________________

End Behavior

__________________________ End Behavior

__________________________

__________________________

__________________________




5. 2

1

xf x

x

6. 53)( xxf

Domain: __________________________ Domain: __________________________

Range: __________________________ Range: __________________________

End Behavior

__________________________ End Behavior

__________________________

__________________________

__________________________


7. 2)3()( xxf 8. 22)( 2 xxf

Domain: __________________________ Domain: __________________________

Range: __________________________ Range: __________________________

End Behavior

__________________________ End Behavior

__________________________

__________________________

__________________________


9. xxxxf 96)( 23 10. 1

23

x

f x

Domain: __________________________ Domain: __________________________

Range: __________________________ Range: __________________________

End Behavior

__________________________ End Behavior

__________________________

__________________________

__________________________


Mathematics Enhanced Scope and Sequence Lesson–Algebra II


Domain and Range

For each function below, graph the function, state the domain and range, end behavior and

continuity.

Function Graph Domain and Range End Behavior Continuity

1.

3, 4

1, 4

xf x

x x

Domain:

Range:

2. 2 3y x

Domain:

Range:

3. 5

xg x

x

Domain:

Range:

4. 3 1xm x

Domain:

Range:

5. 1

2

x

y

Domain:

Range:



Ups and Downs

Use the graph of the roller coaster below to complete the following.

1. How many hills does the roller coaster have?

2. What is the highest point on the coaster? How far from the start of the coaster does the rider reach this height?

3. What is the lowest point on the coaster? How far from the start of the coaster does the rider reach this height?

4. Where is the longest drop (in linear feet) on the coaster? How long is that drop?

5. Where is the shortest drop on the coaster? How long is that drop?

6. Where is the longest hill? (The anticipation is killing you as you climb this one!)



7. How high are the cars at the start of the coaster? At the end?

Absolute Maximum: ____________________________________________________________

8. What is the absolute maximum height of the coaster? Where does it occur?

Absolute Minimum: _____________________________________________________________

9. What is the absolute minimum height of the coaster? Where does it occur?

Relative Maximum: _____________________________________________________________

10. What are the relative maximums? Where do they occur?

Relative Minimum: _____________________________________________________________

11. What are the relative minimums? Where do they occur?



Interval increasing: ____________________________________________________________

12. List the intervals where the roller coaster is increasing.

Interval decreasing: ___________________________________________________________

13. List the intervals where the roller coaster is decreasing.

Examples: Find the extrema and the interval increasing and decreasing for each function below.

1.

Absolute maximum: Absolute minimum: Relative maximum: Relative minimum: Interval increasing: Interval decreasing:

2. 22 12 16xf xx Absolute maximum: Absolute minimum: Relative maximum: Relative minimum: Interval increasing: Interval decreasing:



3. 3 29 24 22x xg xx Absolute maximum: Absolute minimum: Relative maximum: Relative minimum: Interval increasing: Interval decreasing:

4.

13

2

x

m x




5.




Extrema, Intervals Increasing and Decreasing Practice

Find the extrema, interval(s) increasing, and interval(s) decreasing for each function below.

1.

2. 2 4f x x

3.

4. 3 22 5 6x xh x x

5.

6. log 3y x

7.

8. 2 4 3p x x



Locating Extrema Given Graphs

Determine the absolute maximum or minimum of the following functions, if one exists.

1.

3.

2.

4.



5.

6.

Looking at the graphs for questions 1-6, determine the following:

7. For the functions with an absolute maximum, is the function increasing, decreasing, or constant before the maximum? What about after the maximum?

8. For the functions with an absolute minimum, is the function increasing, decreasing, or constant before the minimum? What about after the minimum?

9. If the function moves from increasing to decreasing or decreasing to increasing, then what might be the slope at the maximum or minimum?

10. Of the graphs above, which types of functions do not have a minimum or maximum?



Find the relative maximum or minimum of the following functions over the given domains.

11. Relative maximum when 𝑥: (−∞, 0)

12. Relative minimum when 𝑥 ≥ 0

13. Relative Maximum when {𝑥|0 < 𝑥 ≤ 10}

Mathematics Enhanced Scope and Sequence Lesson – Algebra II


14. Relative Minimum when {𝑥|−10 < 𝑥 ≤ 0}



Locating Extrema Given Equations

Use a graphing utility to determine the location and value of the absolute minimum or maximum, if it exists. You may round to the nearest hundredth if necessary.

1. 𝑦 = 2(𝑥 + 2)2 − 7

2. 𝑦 = −(𝑥 + 2)2 + 1

3. 𝑦 =1

3|𝑥 − 5| + 2

4. 𝑦 = −(𝑥 + 2)3 − 1

5. 𝑦 = 𝑥 − 7



Use a graphing utility to determine the location and value of the relative minimum or maximum, if it exists. You may round to the nearest hundredth if necessary.

6. 𝑦 = 2𝑥, 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑤ℎ𝑒𝑛 𝑥: (−∞, 4]

7. 𝑦 = −𝑥3 + 2𝑥2 + 5𝑥 − 6, 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑤ℎ𝑒𝑛 𝑥: [0,3]

8. 𝑦 = −𝑥3 + 2𝑥2 + 5𝑥 − 6, 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑤ℎ𝑒𝑛 𝑥: [−6,1]

9. 𝑦 = −(𝑥 + 2)3 − 1, 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑤ℎ𝑒𝑛 𝑥: [−3,2]

10. 𝑦 = 3𝑥 − 7, 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑤ℎ𝑒𝑛 𝑥: (−∞, 3.2]



Exploring Inverse Functions

1. Complete the table below showing the relationship between the length of the sides of a square and the area.

Length of Sides Area Area Length of Sides

1 1

2 4

3 9

Graph the data in the table on the same set of axis using two different colors of pen to draw each graph.

Answer the following questions:

How do you find the area of the square given the length of sides?

How do you find the length of sides given the area?

Draw a line 𝑦 = 𝑥 on the graph above. How are the graphs of the two functions related with respect to the line= 𝑥 ?

2. The table below shows the dollar-to-peso conversion.

Dollar Peso

10 500

20 1000

50 2500

100 5000

Draw a graph using the data from the table above.

Create a formula to convert dollars to pesos.

How do you convert peso back to dollar?

Using the same axis, draw a graph with pesos as the domain and dollars as the range.

Graph 𝑦 = 𝑥.



Finding the Inverse of a Function

Find the inverse of each function algebraically. Graph the function and its inverse. Determine whether the inverse is a function. Determine the domain and range of the function and its inverse.

1. 𝑓(𝑥) = 1

2𝑥 + 6

2. 𝑓(𝑥) = (𝑥 + 2)2 − 3, 𝑥 ≥ −2

3. 𝑓(𝑥) = 2𝑥3 + 3



4. 𝑓(𝑥) = √𝑥 − 3 + 1

5. 𝑓(𝑥) = √𝑥3

+ 1

Given the graphs below, graph the inverse of the function. Explain whether the inverse is a function or not. Write the domain and range of the given graph and its inverse.

y y y



Composition of Functions

Given 2

1

2

1)(and ,)(,12)( 2 xxhxxgxxf , evaluate the following:

1. f (−6) 2. 2. g (−3) 3. h(4)

4. f [g(2)] 5. )]8([gh 6. (g f )(5)

7. (g h)(7) 8. )]([ cgf 9. )]5([hf

10. )]([ rfh 11. )]]3([[ fgh 12. (f g h)(3)

Given (𝑥) = 2𝑥2 + 4 , (𝑥) = √𝑥 − 4 , and ℎ(𝑥) = 4𝑥 − 2 , evaluate the following:

13. 𝑓(𝑔(𝑥)) 14. 𝑔(𝑓(𝑥))

15. 𝑓(ℎ(𝑥)) 16. 𝑔(ℎ(𝑥))



Refer to the maps below to determine the domain and range for each of )(xfg and )(xgf .

Name the points included in each composition of functions.

17. (g f )(x) (f g)(x)

domain_____________ domain _____________

range _____________ range ______________

points _____________ points ______________

f g

18. (g f )(x) (f g)(x)

domain_____________ domain _____________

range _____________ range ______________

points _____________ points ______________

f g

19. How do you find the domain of the composite function?

0

2

3

6

−4

0

2

8

0

1

2

3

0

1

4

9

3

4

5

6

−1

−7

0

4

0

2

4

6

1

2

7

8



More Composition of Functions

Use the graph of 𝑓(𝑥) 𝑎𝑛𝑑 𝑔(𝑥) to find the composition of functions.

1. Find a. 𝑓(𝑔(𝑥)) b. 𝑔(𝑓(𝑥))

2. Evaluate a. 𝑓(𝑔(2)) b. [𝑓 𝑔](0)

c. 𝑔(𝑓(2)) d. [𝑔 𝑓](−1)

e. 𝑔(𝑔(4)) f. [𝑓 𝑓](3)

g. 𝑓 (𝑔(𝑓(1))) h. 𝑔(𝑓( 𝑔(0))



3. For what value(s) of x is 𝑓(𝑔(𝑥)) = −3?

4. For what value(s) of x is 𝑔(𝑓(𝑥)) = −3?

5. For what value(s) of x is [𝑓 𝑔](𝑥) = −4?

The table below shows values of 𝑓(𝑥) at selected values of 𝑥. The function 𝑔(𝑥) is shown in

the graph below.

Graph of g(x)

Let h be the function defined by ℎ(𝑥) = 2|𝑥 − 4|.

Find:

1. 𝑦 = ℎ(𝑓(2))

2. 𝑦 = ℎ(𝑔(3))

3. 𝑦 = 𝑔(𝑓(−2))

𝑥 𝑓(𝑥) -3 10

-2 5

-1 2

0 1

1 2

2 5

3 10



4. 𝑦 = 𝑓(𝑔(−3))

5. 𝑦 = 𝑔(𝑓(ℎ(3)))

6. Find ℎ(𝑓(𝑔(0)))

7. Let 𝑚(𝑥) be defined by 𝑚(𝑥) = ℎ(𝑓(𝑥)). What is 𝑚(−2)?

8. Let 𝑛(𝑥) be defined by 𝑛(𝑥) = ℎ (𝑔(𝑥)). What is 𝑛(3)?

9. For what value(s) of 𝑥 is 𝑔(𝑓(𝑥)) = 3?

10. For what value(s) of 𝑥 if 𝑓(𝑔(𝑥) = 2?

Virginia Department of Education © 2019 Algebra II Task

Name_____________________________________ Date_____________________

Function of a Ride

Below you will find a graph comparing the horizontal and vertical distances of a portion of the roller coaster

track, with key points labeled. Consider the point A to be the beginning of the roller coaster track. Also

consider curves that look like parabolas, are parabolas (assume the curves are smooth).

stakeholder management | Productivity Steps

https://images.app.goo.gl/3sieQyMtH3sbWoRk6

View this graph in Desmos

1. What is the domain and range of the function?

Domain Range 2. Find the intervals where the function in increasing and decreasing.

Increasing Decreasing

How did you find the intervals?

https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fimages.app.goo.gl%2f3sieQyMtH3sbWoRk6&c=E,1,cLsLksti1aCOCOVUeUPS7PcVAsrEKOxz1uBeSsLzKWatfevA2zxLCj-707cnGgZc7zD0wg8vEb5W_60k4lpJ9n2MrgB1PnwtmlncZL_VxGKdkIzltfVqhpHNEQ,,&typo=1

https://www.desmos.com/calculator/opjkgsgtma


3. At what point on the coaster would you be going the fastest? The slowest? Explain why you chose these points.

Fastest at Slowest at

4. What are the maxima and minima of the function?

Maxima Minima

5. Where would you scream? Describe your ride as you travel the roller coaster. Include in your description

your trip from point to point, whether you are moving up or down, and discuss what is happening to your

speed.

6. Write the equation of the first “scream”! Find the equation of the first hill – the complete curve up and

down again, from point A – C. Show all your work, with explanations when needed.

7. Looking at the first hill and its equation, how HIGH off the ground would you be after you have traveled 5

meters horizontally. Show your work and explain how you got the answer. How does the predicted height

compare to the actual height of the roller coaster at that point?


Name______________________________________________________Date_____________________

Wildfires

Wildfires burn millions of acres every year. Wildfires burn at a rapid speed and can consume everything in their paths. Fire trucks are used to contain wildfires such as those experienced by people living in California.

The height of a stream of water from the nozzle of a fire hose is modeled by

h(x) =−0.03𝑥2 + 𝑥 + 48

where h(x) is the height in feet, of the stream of water x feet from the fire truck.

1. What is the maximum height the water from this nozzle can reach? What is the maximum distance

from the firetruck a firefighter can stand and still reach the fire?

2. When the stream of water from the nozzle is 32 feet above ground, how much farther must the

water travel before it hits the ground?

3. If the wildfire is located 48 feet from the firetruck. Based on the original function provided, will the

firemen be able to put out the fire? Explain why or why not.

4. Based on the original function, if a wildfire is located 63 feet away from the firetruck, will the

firemen able to put out the fire? Explain why or why not.

Documents

Algebra 2 Student-Directed Review/EnrichmentAlgebra 2 Student-Directed Review/Enrichment . The following activities are all about Functions. Throughout this year, you have learned