Algebra II Items to Support Formative Assessment
Unit 2: Exponential and Logarithmic Functions
Analyze functions using different representations.
F.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple
cases and using technology for more complicated cases.* (Cross-cutting)
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric
functions, showing period, midline, and amplitude.
F.IF.C.7e Task
Part A: For each of the following functions, identify the y-intercept, asymptote, and end behavior using
a graphing calculator. How do those features relate to the equations?
a. y = 3x
b. y = 2(3)x
c. y = 2(3)x + 5
d. y = -2(3)x + 5
Part B: Use what you learned from part 1 to identify the y-intercept, asymptote, and end behavior
without the use of a graphing calculator. Sketch a graph using this information. Once you have
completed all the graphs, check your work using the graphing calculator.
a. y= 4x
c. y = 3(4)x -7
b. y = 3(4)x
d. y = -3(4)x -7
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Part C: Summarize how you find the y-intercept, asymptote and end behavior of an exponential function
from the equation, and use them to sketch a graph.
Part D: For each of the following functions, identify the x-intercept, asymptote, and end behavior using
a graphing calculator. How do those features relate to the equations?
a. y = log2x
b. y = log2(x - 5)
c. y = log2(x - 5) + 2
d. y = - log2(x - 5) + 2
Part E: Use what you learned from part 4 to identify the y-intercept, asymptote, and end behavior without
the use of a graphing calculator. Sketch a graph using this information. Once you have completed all the
graphs, check your work using the graphing calculator.
a. y = log3x
b. y = log3(x - 2)
c. y = log3(x - 2) -3
d. y = - log3(x - 2) -3
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Part F: Summarize how you find the y-intercept, asymptote and end behavior of an exponential function
from the equation, and use them to sketch a graph.
Answers:
Part A:
a. y-intercept y = 1, Asymptote y = 0
EB: as x→∞, g(x) →∞ as x→ - ∞, g(x) → 0
c. y-intercept y = 7, Asymptote y = 5
EB: as x→∞, g(x) →∞ as x→ - ∞, g(x) → 5
b. y-intercept y = 2, Asymptote y = 0
EB: as x→∞, g(x) →∞ as x→ - ∞, g(x) → 0
d. y-intercept y = 3, Asymptote y = 5
EB: as x→∞, g(x) →-∞ as x→ - ∞, g(x) → 5
Part B:
Answers may vary. Be sure that students make corrections to any graphs that do not match prior to
moving on to the summary.
a. y-intercept y = 1, Asymptote y = 0
b. y-intercept y = 3, Asymptote y = 0
EB: as x→∞, g(x) →∞ as x→ - ∞, g(x) → 0
EB: as x→∞, g(x) →∞ as x→ - ∞, g(x) → 0
c. y-intercept y = -4, Asymptote y = -7
d. y-intercept y = -10, Asymptote y = -7
EB: as x→∞, g(x) →∞ as x→ - ∞, g(x) → -7
EB: as x→∞, g(x) →-∞ as x→ - ∞, g(x) → -7
Part C:
Summary: To find the y - intercept students can substitute x for 0. They can also add the coefficient of
the exponential and the constant. The asymptote and end behavior are determined by the sign of the
coefficient of the exponential to determine if the function goes to infinity to the right or to the left. The
asymptote is the constant, and it is the end behavior for the side that levels off.
Part D:
a. x-intercept x = 1, Asymptote x = 0
EB: as x→∞, g(x) →∞ as x→ 0, g(x) → -∞
c. x-intercept x = 5.25, Asymptote x = 5
EB: as x→∞, g(x) →∞ as x→ 5, g(x) → -∞
b. x-intercept x = 6, Asymptote x = 5
EB: as x→∞, g(x) →∞ as x→ 5, g(x) → -∞
d. x-intercept x = 9, Asymptote x = 5
EB: as x→∞, g(x) →-∞ as x→5, g(x) → ∞
Part E:
Answers may vary. Students should discover that the x-intercept cannot be determined easily once the
function has been shifted up or down.
a. x-intercept x = 1, Asymptote x = 0
b. x-intercept x = 3, Asymptote x = 2
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EB: as x→∞, g(x) →∞ as x→ 0, g(x) → -∞
c. x-intercept x = 29, Asymptote x = 2
EB: as x→∞, g(x) →∞ as x→ 2, g(x) →-∞
EB: as x→∞, g(x) →∞ as x→ 2, g(x) →-∞
d. x-intercept x =2.037 , Asymptote x = 2
EB: as x→∞, g(x) →-∞ as x→ 2, g(x)→ ∞
Part F:
Summary: To find the x-intercept: if the graph does not have a vertical shift (k = 0), then the x- intercept
is the value of x that makes the argument of the logarithm equal to 1. (x-h=1). You may want to discuss
that the logn(1)=0. The vertical asymptote is the value of h from log (x-h). The sign of the coefficient of
the logarithm determines if the graph approaches positive infinity or negative infinity as x approaches
positive infinity.
F.IF.C.7e Item 1
Given the function g(x) = 2(3)x -10 , identify any asymptotes, x- and y-intercepts, and identify the
end behavior. Use the information to graph the function.
Answer:
x-intercept x = 1.465, y-intercept y = -8, asymptote y = -10
End behavior: as x→∞, g(x) →∞
as x→ - ∞, g(x) → -10
F.IF.C.7e Item 2
Given the function h(x) = log4(2x), identify any asymptotes, x- and y-intercepts, and identify the
end behavior. Use this information to graph the function.
Answer:
x - intercept x = .5 , y-intercept: none , asymptote x = 0
End behavior: as x→∞, h(x) →∞
as x→ 0, h(x) → -∞
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F.IF.C.7e Item 3
Given the function g(x) = 8(2)-x +20 , identify any asymptotes, x-intercept and y-intercept, and
identify the end behavior. Use the information to graph the function.
Answer:
x-intercept- None, , y-intercept y = 28, asymptote y = 20
End behavior: as x→∞, g(x) →20
as x→ - ∞, g(x) → ∞
F.IF.C.7e Item 4
Given the function h(x) = ln(10 - x), identify any asymptotes, x- and y-intercepts, and identify the
end behavior. Use this information to graph the function.
Answer:
x - intercept x = 9, y-intercept y = 2.303 , asymptote x = 10
End behavior: as x→10, h(x) →-∞
as x→ - ∞, h(x) → ∞
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F.IF.B.6 Task
In groups of three, write an exponential function that has at least two transformations from the parent
function f (x) = d x . Select a value for d.
a. Find the average rate of change on a closed interval [a,b].
b. Graph the function and be sure to include the interval [a,b]. Plot the points (a, f(a)) and (b, f(b)), and
connect the line segment between these two points. Find the slope of this line. What do you notice about
the slope of this line and the average rate of change?
c. Choose c between a and b and find the average rate of change on [a, c] and [b, c]. How do these two
average rates of change compare?
Answer:
a. Answers will vary. Look for a slope formula
f (b) - f (a)
.
b- a
b. Answers will vary. Look for correct plotting of points. The slope of the line should be the same as the
average rate of change found in part a.
c. Depending on the value of d and if there is a reflection over either or both axes, students should
correctly state whether the average rates of change decrease or increase as the function decreases or
increases.
F.IF.B.6 Item 1, Part c uses A.REI.D.11
Mandy is baking pies for the annual neighborhood pie competition. She records the temperature of the
pie (in minutes) because the pies taste best with vanilla ice cream when served at a temperature of 100͒ F.
Time (min)
0
3
6
8
11
16
17
24
Temperature
(F)
350͒ F
303͒ F
268͒ F
224͒ F
182͒ F
139͒ F
135͒ F
98͒ F
a. Find the average rate of change between 3 minutes and 11 minutes.
b. During which time interval, [0,3] or [6,17], is the average temperature change the greatest?
c. Find an exponential function that best models the temperature of the pie. Define variables for time and
temperature, and use this function to determine when Mandy should eat a piece of pie with vanilla ice
cream.
Answers:
a. (182-303)/(11-3) = -15.125
b. -15.67, -12.09, so the average temperature change is greatest during [0,3]. Note: although -15.67<12.09, the slope is steeper during [0,3].
c. y = 352.047(0.946)x The intersection of y = 352.047(0.946)x and y = 100 is x = 22.672 minutes
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F.IF.B.6 Item 2
x
a. Based on the graph shown above, estimate the average rate of change during the interval [1, 3.25].
b. Is the estimated rate of change greater during [0, 2] or [1.5, 2.5]. Show the work that leads to your
answer.
Answers:
a. (80-34)/(3.25-1) = 20.444
b. (49-25)/(2) = 12; (59-40)/(2.5-1.5) = 19. The average rate of change is greater during [1.5, 2.5].
F.IF.B.6 Item 3
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a. Based on the graph shown above, estimate the average rate of change during the interval [3, 5].
b. Is the estimated rate of change greater during [4, 6] or [3, 10]. Show the work that leads to your
answer.
Answer: Note: y-values are approximations only.
a. (5-30)/(5-3)= -12.5
b. (3-11)/(6-4)= -4; (2-30)/(10-3) = -4. The average rate of change is equal. Note: Teaching opportunity
regarding the fact that although the average rate of changes are equal, if the exact y values are found by a
grapher, the average rates of change are not actually equal.
F.IF.B.6 Item 4
x
æ1 æ
Find the average rate of change of f (x) = 18 æ æ - 11 on [1.6, 8.9].
æ3 æ
Answer: -0.425
Analyze functions using different representations.
F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). (Cross-cutting)
F.IF.C.9 Task (with connections to F.IF.B.5)
Part A: Group activity
Group 1:
Graph the functions f (x) = log2(x) and g (x) = 2x on the same axis.
a. State the domain and range of each.
b. What similarities and differences do you notice between the graphs?
c. How do these functions compare to functions you have seen before?
d. What can you conclude about these two functions?
Group 2:
Complete the table of values for each function.
f (x) = log2(x)
x
y
1
2
4
9
16
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g (x) = 2x
x
y
0
1
2
3
4
a. State the domain and range of each.
b. What similarities and differences do you notice between the tables?
c. How do these functions compare to functions you have seen before?
d. What can you conclude about these two functions?
Group 3: (no graphing)
f (x) = log2(x) and g (x) = 2x
a. State the domain and range of each.
b. What similarities and differences do you notice between the functions?
c. How do these functions compare to other functions you have seen before?
d. What can you conclude about these two functions?
Part B: Whole Class
Have an “expert” from each group rotate to a new group with their materials. Have the expert explain
their findings and discuss the extension questions. Compare the expert’s answers to the group’s answers.
Then, have the experts rotate one more time and repeat the discussion.
Allow the experts to return to their own group and start a class discussion about the relationship between
the logarithmic function and the exponential function.
a. Will this relationship always hold true? Why or why not? (Hint: What will happen if the bases are not
the same?)
b. Have the class make a general statement about the relationship between logarithmic functions and
exponential functions.
Answers:
Group 1
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a. f(x): Domain: (0,∞) Range: (-∞,∞)
g(x): Domain: (-∞,∞) Range: (0,∞)
b. Similar: Both have asymptotes, both are functions; Different: f(x) has a vertical asymptote at x = 0
where g(x) has a horizontal asymptote at y = 0, f(x) has an infinite range where g(x) has an infinite
domain.
b. Answers will vary
c. Opposites, inverses, flipped
Group 2
f(x)
x
y
1
0
2
1
4
2
9
3
16
4
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g(x)
x
y
0
1
1
2
2
4
3
9
4
16
(Answers may vary)
a. f(x): Domian: (0,∞) Range: (-∞,∞)
g(x): Domain: (-∞,∞) Ragne: (0,∞)
b. Similar: Both have a 2 and an x, Different: f(x) has log, g(x) has an x in the exponent
c. Answers will vary
Group 3:
a. f(x): Domain: (0,∞) Range: (-∞,∞)
g(x): Domain: (-∞,∞) Range: (0,∞)
b. Similar: same values; different: the x-values and y-values are switched
c. Answers will vary
d. Opposites, inverses, flipped
Part B:
a. Not always. It will only hold true if the bases are the same.
b. If the bases are the same, a logarithmic function is the inverse of an exponential function.
F.IF.C.9 Item 1 (with connections to standard F.IF.B.5)
Select all that are true about the functions f(x) = log3(x) and g(x) = 3x.
A. f(x) and g(x) are inverses.
B. f(x) and g(x) have the same asymptote.
C. f(x) passes through the point (1,0) and g(x) passes through the point (0,1).
D. f(x) and g(x) have the same domain and range.
E. The domain of f(x) is the range of g(x).
Answer: A, C, E
F.IF.C.9 Item 2
If f (x) = x2 + 8x - 10 and g(x) = 12(4 x ) - 2 , select all statement(s) that are true from the following list:
A. One function has a horizontal asymptote.
B. Both functions have horizontal asymptotes.
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C.
D.
E.
F.
G.
H.
One function has a vertical asymptote.
Both functions have vertical asymptotes.
The functions have the same y-intercept.
The functions have the same x-intercept.
The functions have the same end behavior as x → ∞.
The functions have the same end behavior as x → −∞.
Answer: A, E, G
F.IF.C.9 Item 3
Which of the following functions are even functions? Select all that apply.
A. f (x) = 3x2 - 9
B. f (x) = 4x2 + 6x - 10
C. f (x) = 6 x - 2
D. f (x) = 8 + 4x3
E.
F.
Answer: A, F
F.IF.C.9 Item 4
Which of the following functions are odd functions? Select all that apply.
A. f (x) = log 8 (x + 2)
B. f (x) = 3x2 + 5x - 7
C. f (x) = 7 x - 11
D. f (x) = 4x3
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E.
F.
G.
Answer: D, E, G
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