Appendix A — Answer Key, Chapter 17: Exponential Functions

SOL A.F.2 (e, f, g) · Covers textbook Chapter 17 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 120 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Conventions used in every answer below. An exponential is $y = ab^x$ with $b$ a natural number ($1, 2, 3, \ldots$). The number $a$ is the $y$-coordinate of the $y$-intercept $(0, a)$; the number $b$ is the growth factor. For $a > 0$ and $b \ge 2$, the unrestricted domain is all real numbers and the range is $y > 0$, with horizontal asymptote $y = 0$. Transformations are only $f(x)+k$ and $kf(x)$ with rational $k$. Evaluating $f(x)$ is in scope; recovering $x$ from $f(x)$ for an exponential is not (logarithms). Fractional bases (decay) are out of scope.

The figures used repeatedly in the chapter:

• Figure 1 is $y = 2^x$ as a table ($x = -2$ to $4$) beside its graph, with $(0, 1)$ shaded/marked
• Figure 2 is $y = 3 \cdot 2^x$ with intercept $(0, 3)$, domain all reals, range $y > 0$
• Figure 3 is $y = 2^x$ approaching the asymptote $y = 0$ through $(-1, \tfrac12)$, $(-2, \tfrac14)$, $(-3, \tfrac18)$, $(-4, \tfrac{1}{16})$
• Figure 4 is $P(t) = 5 \cdot 2^t$ cells, table $t = 0$ to $4$, intercept $(0, 5)$
• Figure 5 is $f(x) = 2^x$ and $g(x) = 2^x + 3$, asymptotes $y = 0$ and $y = 3$
• Figure 6 is $f(x) = 2^x$ and $h(x) = \tfrac12 \cdot 2^x$, with $(3, 8)$ and $(3, 4)$
• Figure 7 is $y = 3^x$ table-and-graph (values $\tfrac19$, $\tfrac13$, $1$, $3$, $9$, $27$)
• Figure 8 is $f(x) = 3 \cdot 2^x$ with $f(3) = 24$ and $f(-2) = \tfrac34$
• Figure 9 is the anatomy of $y = ab^x$ ($a$ intercept, $b$ growth factor)
• Figure 10 is four blank grids for student graphing

Lesson 17.1 — The Form $y = ab^x$

Guided practice

1. The shaded row is $x = 0$, value $1$. It names the $y$-intercept $(0, 1)$ on the graph.
2. Each value multiplies by $2$ (the growth factor $b = 2$).
3. $a$ is the $y$-intercept (initial value); $b$ is the growth factor.
4. $a = 3$, $b = 2$; $y$-intercept $(0, 3)$.
5. $2^{-3} = \tfrac18$, so $y(-3) = \tfrac18$.
6. Because $ab^0 = a \cdot 1 = a$ only when $b^0 = 1$. Chapter 10 forces $b^0 = 1$ for $b \neq 0$; without that fact the intercept would not equal the coefficient $a$.

Independent practice

7. a) $a = 5$, $b = 2$; $(0, 5)$ b) $a = 2$, $b = 3$; $(0, 2)$ c) $a = 10$, $b = 4$; $(0, 10)$ d) $a = 1$, $b = 3$; $(0, 1)$ (since $3^x = 1 \cdot 3^x$)
8. a) $16$ (because $8 \cdot 2 = 16$) b) $24$ (because $12 \cdot 2 = 24$)
9. a) $4 \cdot 2^0 = 4$ b) $4 \cdot 2^3 = 4 \cdot 8 = 32$ c) $4 \cdot 2^{-1} = 4 \cdot \tfrac12 = 2$ d) $4 \cdot 2^{-2} = 4 \cdot \tfrac14 = 1$
10. $a = 6$, $b = 2$, $P(0) = 6$ cells. $P(4) = 6 \cdot 2^4 = 6 \cdot 16 = 96$ cells.
11. In $y = 2x$ the input is multiplied by $2$ (linear). In an exponential the input is the exponent. Same digits, different structure.
12. The student multiplied $a$ by $b$ instead of using $x = 0$. Correct: $5 \cdot 3^0 = 5 \cdot 1 = 5$, so the intercept is $(0, 5)$.
13. a) Yes; $a = 2$, $b = 5$ b) No — base $\tfrac12$ is not a natural number c) Yes; $a = 7$, $b = 1$ d) No — quadratic, not of the form $ab^x$
14. $x = -1$: $\tfrac13$; $x = 0$: $1$; $x = 1$: $3$; $x = 2$: $9$.
15. $y = 2 \cdot 4^x$ ($a = 2$, $b = 4$).
16. Table: $\tfrac14$, $\tfrac12$, $1$, $2$, $4$. Yes — matches Figure 1.

Exit ticket 17.1

17. $a = 8$, $b = 2$; intercept $(0, 8)$.
18. $y(3) = 8 \cdot 2^3 = 64$; $y(-2) = 8 \cdot 2^{-2} = 8 \cdot \tfrac14 = 2$.
19. When the input increases by $1$, the output is multiplied by $b$.
20. The base $\tfrac12$ is not a natural number; A.F.2e limits $b$ to natural numbers.

Lesson 17.2 — $y$-Intercept, Domain, and Range

Guided practice

21. Intercept $(0, 3)$; domain all real numbers; range $y > 0$.
22. Because $y = 0$ is an asymptote — outputs approach $0$ but never equal $0$, so the range is strict: $y > 0$.
23. $(-1, \tfrac12)$, $(-2, \tfrac14)$, $(-3, \tfrac18)$, $(-4, \tfrac{1}{16})$. Each $y$-coordinate is half the previous (multiply by $\tfrac12$ as $x$ decreases by $1$, or equivalently approach $0$).
24. Intercept $(0, 1)$; domain all real numbers; range $y > 0$.
25. Intercept $(0, 7)$; domain all real numbers; range $y > 0$.
26. Chapter 10 defines $b^x$ for every integer $x$, and the real-number power extends to every real $x$; nothing in $ab^x$ forbids a negative input. The curve is drawn through those points without a break.

Independent practice

27. a) $(0, 6)$; all reals; $y > 0$ b) $(0, 1)$; all reals; $y > 0$ c) $(0, 10)$; all reals; $y > 0$ d) $(0, 9)$; all reals; range $\{9\}$ (constant, since $1^x = 1$)
28. a) $y = 2^x$ b) $y = 3 \cdot 2^x$
29. Intercept $(0, 2)$: at day $0$, $2$ people have heard the rumor (the starters).
30. Domain being all reals does not force the range to be. The curve stays above $y = 0$, so the range is $y > 0$.
31. A.F.2e names the $y$-intercept, which is a point. Correct: $(0, 5)$.
32. $2^{-5} = \tfrac{1}{32}$; $2^{-6} = \tfrac{1}{64}$.
33. Sketch should show intercept $(0, 2)$, asymptote $y = 0$, rising to the right (through $(1, 6)$, $(2, 18)$, \ldots).
34. Technology confirms $y(0) = 3$ and every table value positive — agrees with range $y > 0$.
35. For $a > 0$ and $b > 0$, the product $ab^x$ is always positive, so negative outputs never occur.
36. Domain becomes $0 \le x \le 5$. Range becomes the closed interval from $f(0) = a$ to $f(5) = ab^5$ (for increasing growth), not all of $y > 0$.

Exit ticket 17.2

37. Intercept $(0, 4)$; domain all reals; range $y > 0$.
38. Asymptote $y = 0$. It forces the range to exclude $0$ (and everything below).
39. $(0, 3)$.
40. Equality would require the curve to meet $y = 0$, but $y = 0$ is only an asymptote — outputs get arbitrarily close and never arrive. Write $y > 0$.

Lesson 17.3 — Graphing and Transformations

Guided practice

41. Values: $\tfrac19$, $\tfrac13$, $1$, $3$, $9$, $27$. $y$-intercept $(0, 1)$.
42. $3^{-2} = \tfrac{1}{3^2} = \tfrac19$ (negative-exponent law from Chapter 10).
43. Intercept $(0, 4)$; asymptote $y = 3$.
44. $h(3) = 4$, which is half of $f(3) = 8$, because every output is multiplied by $k = \tfrac12$.
45. Points such as $(-1, \tfrac12)$, $(0, 1)$, $(1, 2)$, $(2, 4)$, $(3, 8)$, $(4, 16)$ plotted and connected; asymptote $y = 0$.
46. Adding $k$ moves every point — including points near the old asymptote — so the line the curve approaches moves to $y = k$. Multiplying by $k$ scales heights toward (or away from) $y = 0$, so the asymptote $y = 0$ stays put.

Independent practice

47. a) Same points as item 45 on grid (a). b) Points $(-1, \tfrac13)$, $(0, 1)$, $(1, 3)$, $(2, 9)$, $(3, 27)$ on grid (b).
48. a) $2^x + 4$; intercept $(0, 5)$; asymptote $y = 4$ b) $2^x - 2$; intercept $(0, -1)$; asymptote $y = -2$ c) $3 \cdot 2^x$; intercept $(0, 3)$; asymptote $y = 0$ d) $\tfrac14 \cdot 2^x$; intercept $(0, \tfrac14)$; asymptote $y = 0$
49. a) $\tfrac12 f(x)$ b) $f(x) + 5$
50. Shifts the graph up by $10$; new asymptote $y = 10$.
51. Reflects across the $x$-axis; range becomes $y < 0$ for the parent $2^x$.
52. The asymptote must move with the shift to $y = 3$. Redraw the dashed/labeled asymptote at height $3$.
53. Curve through $(0, 4)$, $(1, 5)$, $(2, 7)$, \ldots; asymptote $y = 3$.
54. Curve through $(0, \tfrac12)$ and $(3, 4)$; asymptote $y = 0$.
55. Yes — every point of $2^x$ sits exactly $3$ units below the corresponding point of $2^x + 3$.
56. $k = 4$.
57. Rational values.
58. Points $(-1, \tfrac23)$, $(0, 2)$, $(1, 6)$, $(2, 18)$ plotted and connected.

Exit ticket 17.3

59. Build a table; plot the points; connect smoothly (approaching the asymptote).
60. $f(x)+2 = 3^x + 2$; intercept $(0, 3)$; asymptote $y = 2$. $\tfrac13 f(x) = \tfrac13 \cdot 3^x$; intercept $(0, \tfrac13)$; asymptote $y = 0$.
61. Multiplies every output by $\tfrac12$ (vertical compression toward the $x$-axis).
62. Because $3^{-2} = \tfrac19$ is the negative-exponent law — a Chapter 10 fact — not a new graphing rule.

Lesson 17.4 — Exponential Models in Context

Guided practice

63. $a = 5$, $b = 2$; intercept $(0, 5)$.
64. $P(1) = 10$ cells; $P(4) = 80$ cells.
65. Domain: $t \ge 0$ hours. Range: $P \ge 5$ cells.
66. The population doubles every hour.
67. Intercept: at generation $0$ the archive is $4$ MB. Growth factor $3$: each generation triples the size.
68. Negative hours are before the culture started; the model begins at the starting count $t = 0$.

Independent practice

69. a) $(0, 3)$: at $0$ hours there are $3$ cells. b) $C(5) = 3 \cdot 2^5 = 3 \cdot 32 = 96$ cells. c) $t \ge 0$ hours.
70. $F(0) = 2$, $F(1) = 6$, $F(2) = 18$, $F(3) = 54$ MB. $F(0)$ is the starting file size, $2$ MB.
71. $R(4) = 5 \cdot 2^4 = 80$ people. Intercept $(0, 5)$: five people know the rumor at day $0$.
72. For the unrestricted equation the range is $y > 0$; for the model with $t \ge 0$ the outputs only run from $5$ upward, so the contextual range is $P \ge 5$, still not all reals.
73. Correct domain in context: $t \ge 0$ hours.
74. a) yes b) no c) yes d) no (linear)
75. Example: An insect colony starts with $7$ insects and doubles each day. Intercept: $7$ insects at day $0$.
76. Because the story starts at $t = 0$ with $5$ cells and only grows; values between $0$ and $5$ would require negative time, which the context excludes.
77. $5$, $10$, $20$, $40$, $80$ — matches Figure 4.
78. $P(t) = 8 \cdot 3^t$; $a = 8$, $b = 3$.
79. Vertical shift $f(t) + 5$ with $f(t) = 2^t$; asymptote $y = 5$.
80. $P(3) = 40$ (not yet); $P(4) = 80$ (exceeds $50$). First whole hour: $t = 4$.

Exit ticket 17.4

81. At $0$ hours, the culture has $5$ cells.
82. Domain $t \ge 0$ hours; range $P \ge 5$ cells.
83. $a = 9$: starts with $9$ organisms. $b = 3$: triples each time unit.
84. Base $\tfrac12$ is not a natural number; A.F.2e excludes it.

Lesson 17.5 — Evaluating $f(x)$ for Exponential Functions

Guided practice

85. $f(3) = 24$; check $3 \cdot 2^3 = 3 \cdot 8 = 24$.
86. $3 \cdot 2^{-2} = 3 \cdot \tfrac14 = \tfrac34$.
87. $f(0) = 3$; $f(1) = 6$; $f(2) = 12$.
88. $g(-3) = 4 \cdot 2^{-3} = 4 \cdot \tfrac18 = \tfrac12$.
89. After $2$ hours, the culture has $20$ cells.
90. The sentence "Determine $x$ given any value $f(x)$ in the range of $f$ of a quadratic function" — the word quadratic is the limit; exponentials are named only in the evaluate-$f(x)$ sentence.

Independent practice

91. a) $2^5 = 32$ b) $5 \cdot 2^3 = 5 \cdot 8 = 40$ c) $3 \cdot 3^0 = 3$ d) $3 \cdot 3^{-1} = 3 \cdot \tfrac13 = 1$
92. a) $7 \cdot \tfrac12 = \tfrac72$ b) $7 \cdot 2^{-3} = 7 \cdot \tfrac18 = \tfrac78$ c) $2 \cdot 3^{-2} = 2 \cdot \tfrac19 = \tfrac29$
93. $P(0) = 5$; $P(3) = 40$; $P(5) = 5 \cdot 32 = 160$. After $5$ hours there are $160$ cells.
94. $A(2) = 4 \cdot 9 = 36$ MB. After $2$ generations, the archive is $36$ megabytes.
95. Start at $3$ on the $x$-axis, go up to the curve, then left/across to $24$ on the $y$-axis.
96. Recognizing a familiar power you already know ($2^3 = 8$) is evaluation in reverse by inspection. Solving $3 \cdot 2^x = 10$ requires a logarithm, which A.F.2g does not require for exponentials.
97. (1) The minus sign was applied to $3$ instead of being part of the input $x = -2$. (2) The exponent was treated as positive $2$ rather than $2^{-2} = \tfrac14$. Correct: $3 \cdot 2^{-2} = \tfrac34$.
98. $f(-2) = \tfrac34$, $f(0) = 3$, $f(3) = 24$ — matches Figure 8.
99. $f(0) = 4$; $f(3) = 8 + 3 = 11$. Transformation $f(x) + 3$ (vertical shift).
100. Evaluating substitutes a known $x$ into $ab^x$ and multiplies — Chapter 10 arithmetic. Recovering $x$ undoes the exponent, which is the definition of a logarithm.
101. About $0.71$ (since $\sqrt{2} \approx 0.707$).
102. $R(2) = 2 \cdot 9 = 18$ people. After $2$ days, $18$ people have heard the rumor.

Exit ticket 17.5

103. $f(4) = 3 \cdot 2^4 = 3 \cdot 16 = 48$.
104. $f(-1) = 3 \cdot 2^{-1} = 3 \cdot \tfrac12 = \tfrac32$.
105. After $3$ hours, the culture has $40$ cells.
106. A.F.2g requires determining $f(x)$ for an exponential (and explaining meaning in context). It requires determining $x$ from $f(x)$ only for quadratics; for exponentials that direction needs logs and is left to later courses.

Chapter 17 Review

Part A — Characteristics and context (A.F.2e)

107. $a$ is the $y$-intercept (initial value); $b$ is the growth factor; $b$ must be a natural number ($1, 2, 3, \ldots$).
108. Intercept $(0, 3)$; domain all real numbers; range $y > 0$.
109. The curve approaches $y = 0$ but never meets it; including $0$ would claim a point on the asymptote that the function never attains.
110. Intercept $(0, 5)$: $5$ cells at $0$ hours. Domain $t \ge 0$ hours; range $P \ge 5$ cells. $P(4) = 80$ cells.
111. a) $(0, 8)$; all reals; $y > 0$ b) $(0, 1)$; all reals; $y > 0$ c) $(0, 6)$; all reals; range $\{6\}$

Part B — Graphing and transformations (A.F.2f)

112. Values $\tfrac19$, $\tfrac13$, $1$, $3$, $9$, $27$. Steps: build a table; plot points; connect smoothly.
113. Intercept $(0, 4)$; asymptote $y = 3$. Growth factor $b = 2$ is unchanged.
114. $h(3) = 4$; intercept $(0, \tfrac12)$.
115. $f(x)+5 = 2^x + 5$; intercept $(0, 6)$; asymptote $y = 5$. $3f(x) = 3 \cdot 2^x$; intercept $(0, 3)$; asymptote $y = 0$.
116. Points $(-1, \tfrac23)$, $(0, 2)$, $(1, 6)$, $(2, 18)$ connected; asymptote $y = 0$.

Part C — Evaluating $f(x)$ (A.F.2g, exponential)

117. $f(3) = 24$; $3 \cdot 2^{-2} = \tfrac34$.
118. a) $f(0) = 4$; $f(3) = 32$; $f(-2) = 1$ b) $g(4) = 81$; $g(-1) = \tfrac13$
119. $P(0) = 5$ cells at $0$ hours; $P(3) = 40$ cells after $3$ hours.
120. A.F.2g says: determine $f(x)$ of a quadratic or exponential; determine $x$ given $f(x)$ of a quadratic function. The second direction names only quadratics, so exponentials stop at evaluation. Recovering $x$ for $ab^x = c$ needs logarithms, which this course does not teach.

Numeric checks used in this key (exact). $2^0=1$, $2^3=8$, $2^4=16$, $2^5=32$, $3^4=81$, $3^{-1}=\tfrac13$, $3^{-2}=\tfrac19$; $3\cdot 2^3=24$; $5\cdot 2^t$ at $t=0..4$ gives $5,10,20,40,80$; $6\cdot 2^4=96$; $8\cdot 2^3=64$; $8\cdot 2^{-2}=2$; $4\cdot 2^{-2}=1$; $P(4)=80>50$ while $P(3)=40\le 50$.