Algebra / Geometry II: Unit 6-Analyze Functions

SUCCESS CRITERIA:

  1. Be able toidentify x & y-intercepts and average rate of change using graphs, tables, & equations.
  1. Be able to identify and describe key features of graphs, tables and equations.
  1. Be able to analyze the transformations of functions given graphs or equations.

INSTRUCTOR: Craig ShermanHidden Lake High School

Westminster Public Schools

EMPOWER Recorded TARGETSCALE THEME

MA.11.EE.07.04Graph Systems of Different Types of Equations

MA.11.F.01.04Graph Different Types of Equations

MA.11.F.02.04Interpret Key Features of Graphs

PROFICIENCY SCALE:

SCOREREQUIREMENTS

4.0 In addition to exhibiting Score 3.0 performance, in-depth inferences and applications that go BEYOND what was taught in class.

Score 4.0 does not equate to more work but rather a higher level of performance.

3.5 In addition to Score 3.0 performance, in-depth inferences and applications with partial success.

3.0The learner exhibits no major errors or omissions regarding any of the information and processes (simple or complex) that were explicitly taught.

oBe able to identify x & y-intercepts and average rate of change using graphs, tables, & equations, AND

oBe able to identify and describe key features of graphs, tables and equations, AND

oBe able to analyze the transformations of functions given graphs or equations.

2.0Can do one or more of the following skills / concepts:

There are no major errors or omissions regarding the simpler details and processes as the learner…

oIdentify x & y-intercepts on graphs,OR

o Identify x & y-intercepts from tables, OR

oIdentify x & y-intercepts given an equation, OR

oIdentify average rate of change on graphs,OR

o Identify average rate of change from tables, OR

oIdentify average rate of change given an equation, OR

o Identify and describe key features of graphs, OR

oIdentify and describe key features from tables, OR

oIdentify and describe key features given an equation, OR

oIdentify transformationsgiven a graph, OR

oIdentify transformationsgiven an equation.

1.0 Know and use the vocabulary

  • Identify the Basic Elements
  • With help, a partial understanding of some of the simpler details and process

KEY FEATURES: Intercepts

INSTRUCTION 1 :KHAN ACADEMYINSTRUCTION 2 SOPHIA

Class Work

Find the x and y intercepts for the following graphs. Write your answers as coordinates.

  1. 2.

3.4.

Find the x and y intercepts algebraically. Write your answers as coordinates.

  1. y = 2x3 – 5x2 – 3x 8. f(x) = 3x2 – 13x + 4
  1. g(x)= -4x + 39. y = -2x2 +3x - 1
  1. .y = x3 + 2x2 + 4x10. h(x) =

Home Work

Find the x and y intercepts for the following graphs. Write your answers as coordinates.

11 12.

  1. 14.

Find the x and y intercepts algebraically. Write your answers as coordinates.

1518. y = 4x3 – 4x2 + x

  1. y = x2 + x – 2019. g(x) = 2x2 + 3x – 1
  1. h(x) = 8x3 + 6x2 – 9x20. m(x) = -3x + 5

KEY FEATURES: Increasing and Decreasing Functions

INSTRUCTION 1 :KHAN ACADEMYINSTRUCTION 2 SOPHIA

Class Work

Use the graph of f(x) to answer the following questions.

  1. Interval(s) on which is increasing
/
  1. Interval(s) on which is decreasing.
/
  1. -value of any local maxima

  1. -value of any local minima
/
  1. -value of any absolute maximum
/
  1. -value of any absolute minimum

Use the table to answer the following questions. The table represents the scores one student received on practice math exams leading up to the SAT’s.

Week / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9
Score / 510 / 520 / 550 / 560 / 530 / 550 / 560 / 580 / 590
  1. During what interval(s) were the scores increasing?
/
  1. State any relative minimum scores.

  1. What was the greatest rate of change and when did it occur?

Home Work

Use the graph of f(x) to answer the following questions.

  1. Interval(s) on which is increasing
/
  1. Interval(s) on which is decreasing.
/
  1. -value of any local maxima

  1. -value of any local minima
/
  1. -value of any absolute maximum
/
  1. -value of any absolute minimum

Use the table to answer the following questions. The table represents the number of assignments one student received in math class for a marking period

Week / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9
Assignments / 80 / 120 / 130 / 140 / 145 / 135 / 120 / 130 / 135
  1. During what interval(s) was the number of assignments increasing?
/
  1. State any relative minimum assignment weeks.

  1. What was the greatest rate of change and when did it occur?

END BEHAVIOR: Odd and Even Functions

INSTRUCTION 1 :KHAN ACADEMYINSTRUCTION 2 SOPHIA

Class Work

Is the equation given an odd function, an even function, or neither? Show work.

Is the graphed function odd, even, or neither? Explain why.

Home Work

Is the equation given an odd function, an even function, or neither? Show work.

Is the graphed function odd even or neither? Explain why.

END BEHAVIOR: Positive or Negative

INSTRUCTION 1 :KHAN ACADEMYINSTRUCTION 2 SOPHIA

Class Work

Find the end behavior of the following functions. Use appropriate notation.

  1. 58..
  1. 59.
  1. 60.

Home Work

Find the end behavior of the following functions. Use appropriate notation.

  1. 64.
  1. 65.
  1. 66.

Analyzing Functions: Key Features

Class Work

Analyze the following functions. Use appropriate notation.

67..
Domain:
Range:
Minimum (Min):
Maximum (Max):
x-intercepts:
y-intercepts:
Increasing:
Decreasing:
Odd or Even Function:
End Behavior: / f(x) = -x4 – 2x3
68.
Domain:
Range:
Minimum (Min):
Maximum (Max):
x-intercepts:
y-intercepts:
Increasing:
Decreasing:
Odd or Even Function:
End Behavior: /

Home Work

Analyze the following functions. Use appropriate notation.

69.
Domain:
Range:
Minimum (Min):
Maximum (Max):
x-intercepts:
y-intercepts:
Increasing:
Decreasing:
Odd or Even Function:
End Behavior: / f(x) = -x3 – 1

70.

Domain:
Range:
Minimum (Min):
Maximum (Max):
x-intercepts:
y-intercepts:
Increasing:
Decreasing:
Odd or Even Function:
End Behavior: /
Function Transformations
±a function ( ± b X ± c ) ± d
ACTIONS / DIRECTION
Reflection ( - ) / Vertical (outside)
Stretch/Shrink (ab) / Horizontal (inside)
Phase Shift ( ± cd)

PARENT FUNCTIONS:

LINEARQUADRATIC ABSOLUTE VALUE

SQUARE ROOT

TRANSFORMATIONS:Vertical Shifts

INSTRUCTION 1 :KHAN ACADEMYINSTRUCTION 2 SOPHIA

Class Work

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

71.. a) move up 3 72. a) move down 3

b) move down 2 b) move up 1

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

Homework

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

78. a) move up 3 79. a) move down 5

b) move down 1 b) move up 2

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

TRANSFORMATIONS:Horizontal Shifts

Class Work

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

85. a) move right 2 86. a) move right 2

b) move left 5 b) move left 4

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

Homework

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

92. a) move right 3 93. a) move left 2

b) move left 2 b) move right 4

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

TRANSFORMATIONS:Reflections

INSTRUCTION 1 :SOPHIA

Class Work

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

99. a) reflect over x-axis 100. a) reflect over x-axis

b) reflect over y-axis b) reflect over y-axis

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

Homework

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

106. a) reflect over x-axis 107. a) reflect over x-axis

b) reflect over y-axis b) reflect over y-axis

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

TRANSFORMATIONS:Vertical Stretches and Shrinks

INSTRUCTION 1 :SOPHIA

Class Work

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

112. a) Vertical Stretch of 3 112. a) Vertical Shrink of

b) Vertical Shrink of b) Vertical Stretch of 2

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

Homework

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

118. a) Vertical Shrink of 119. a) Vertical Stretch of 4

b) Vertical Stretch of 3 b) Vertical Shrink of

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

3

TRANSFORMATIONS:Horizontal Stretches and Shrinks

Class Work

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

125.. a) Horizontal Shrink of 3 126. a) Horizontal Stretch of

b) Horizontal Stretch of b) Horizontal Shrink of 2

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

Homework

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

132. a) Horizontal Shrink of 3 133. a) Horizontal Stretch of

b) Horizontal Stretch of b) Horizontal Shrink of 2

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

TRANSFORMATIONS:Combining Transformations

Class Work

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

139. a) f(x) = -2g(x) + 2 140. a) f(x) = -g(x – 3) +1

b) f(x) = b) f(x) = 3g(x + 2) – 2

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

Homework

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

146. a) 147. A) f(x) = -2g(x) – 2

b) f(x) = g(x + 3) + 2 b) f(x) = g(-2x)

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

PMI-NJ Center for Teaching & Learning ~1~ NJCTL.org

ANALYZE FUNCTIONS UNIT Review

Multiple Choice – Determine the best answer for each question.

  1. What is the domain of the graph to the right?
  1. or
  2. or or

In Questions 2 – 4, refer to the graph on the right:

  1. There is a local minimum at:

/
  1. There is no local minimum

  1. The rate of change from to is the same as the rate of change from

  1. to .
/
  1. to

  1. to
/
  1. to

  1. There is an absolute maximum at:

/
  1. There is no absolute maximum

  1. In the table to the right, the rate of change between and is

  1. 2
/
  1. 1
/
  1. 0.5
/
  1. -1

In Questions 7 – 9, consider the following graph to the right:

  1. The rate of change from to is

  1. 3
/
  1. 0.75

  1. 0
/
  1. -3

  1. The greatest rate of change is between

  1. and
/
  1. and

  1. and
/
  1. and

  1. The rate of change from to is the same as the rate of change from

  1. to
/
  1. to

  1. to
/
  1. to

In Questions 10 – 13, refer to the graph below:

  1. There is a local max at

/ / /
  1. 1

  1. In terms of concavity, the point at is

  1. concave up.
/
  1. concave down.
/
  1. a local minimum.
/
  1. none of the above.

  1. The rate of change is positive on the interval
  1. Given , the function is

  1. an odd function
/
  1. an even function

  1. neither an odd or even function
/
  1. both an odd and even function

  1. Identify the transformations on the function .

  1. shifted up 15; shifted left 3
/
  1. shifted up 3; shifted right 15

  1. vertical stretch; shifted up 15
/
  1. vertical shrink; shifted up 15

14.Describe the transformation of the parent function f(x) = x2 to g(x) = x2 – 1

  1. shift left 1
  2. shift right 1
  3. shift down 1
  4. shift up 1

15. Describe the transformation of the parent function f(x) = |x| to g(x) = | x+1|

  1. shift left 1
  2. shift right 1
  3. shift down 1
  4. shift up 1

16. Describe the transformation of the parent function f(x)= [x] to g(x)= [2x]

  1. horizontal stretch of scale factor 2
  2. horizontal stretch of scale factor 1/2
  3. vertical stretch of scale factor 2
  4. vertical stretch of scale factor 1/2

17. Describe the transformation of the parent function f(x)= to g(x)=

  1. horizontal stretch of scale factor 2
  2. horizontal stretch of scale factor 1/2
  3. vertical stretch of scale factor 2
  4. vertical stretch of scale factor ½

18. Describe the transformation of the parent function f(x)= log(x) to g(x)= log(-x)

  1. horizontal reflection
  2. vertical reflection
  3. does not affect f(x) since it is symmetrical
  4. not possible because log(x) is undefined for negatives

, find such that b(x) is continuous

  1. -8
  2. -4
  3. 4
  4. 8

Extended Response – Completely answer each question showing all work.

20.The number of people entering the exciting new amusement park, “Math World – HD in 3D” is given by the following equation, where is the amount of time, in hours, after the park opens.

  1. If the park opened at 10 am, how many people entered at 1 pm?
  1. During what hour did the most number of people enter the park? How many people entered during that hour?
  1. What is the rate of change in people entering from 12 pm to 2pm?

21.Let

  1. Describe the end behaviors of .
  1. Describe the intervals of increase and decrease of .

22. People enter a park at a rate of where t is the number of hours after opening.

People leave the park at a rate of . The park is open 12 hours a day.

  1. Write an, P(t) equation for the rate of change in the number of people in the park in terms of E(t) and L(t).
  2. Create the piecewise function for P(t).
  3. Find c so that there is no one in the park at closing.
  4. Does the answer in part c make sense? Explain.

PMI-NJ Center for Teaching & Learning ~1~ NJCTL.org