Algebra 1 Workbook — Chapter 17: Exponential Functions

SOL A.F.2 (e, f, g) · Companion to Textbook Chapter 17

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 120.

PAGE 1 — Chapter opener

Chapter 17 · Exponential Functions

Standard A.F.2 (e, f, g)

In this chapter you will:

• Work in the form $y = ab^x$ with $b$ a natural number
• Name $a$ as the $y$-intercept and $b$ as the growth factor
• State domain, range, and $y$-intercept from equation or graph
• Interpret those characteristics in a real situation with units
• Graph exponentials by table-plot-connect
• Apply transformations $f(x) + k$ and $kf(x)$ ($k$ rational)
• Evaluate $f(x)$ for an exponential and explain meaning in context
• Stay honest: recovering $x$ from $f(x)$ for exponentials needs logs and is not required

Words to know: exponential function · $y = ab^x$ · natural number · initial value · growth factor · $y$-intercept · domain · range · horizontal asymptote · table-plot-connect · $f(x) + k$ · $kf(x)$ · evaluate

Convention: $b$ is a natural number ($1, 2, 3, \ldots$). Decay with base $\tfrac12$ is out of scope.

Convention: A.F.2g asks for $f(x)$ given $x$ for exponentials. It asks for $x$ given $f(x)$ for quadratics only.

Calculator. Use one to confirm table values and graphs. Do not use it to invent logarithms this chapter does not teach.

PAGE 2 — Growth table and graph

17.1 The Form $y = ab^x$

FIGURE: fig1-exponential-growth-table-and-graph.png (full width)

Fill in the blanks.

Each step in the value column ____________ by $2$.

The shaded row is $x =$ ______, and it names the ____________ $(0,\ 1)$.

1. Shaded row: $x =$ ______ Point on the graph: ______

2. Growth factor when $x$ increases by $1$: ______

3. Anatomy preview — in $y = ab^x$, $a$ is the ____________ and $b$ is the ____________.

PAGE 3 — Anatomy of $ab^x$

Naming $a$ and $b$

FIGURE: fig9-growth-factor-versus-y-intercept.png (full width)

4. For $y = 3 \cdot 2^x$: $a =$ ______ $b =$ ______ $y$-intercept = ______

5. $y(-3)$ for $y = 2^x$: $2^{-3} =$ ______ so $y(-3) =$ ______

6. Explain. Why must $b^0 = 1$ for the $y$-intercept of $y = ab^x$ to equal $a$?

   ---

PAGE 4 — Practice · $a$, $b$, and tables

Reading $a$ and $b$

7. Identify $a$, $b$, and the $y$-intercept.

8. Next output by multiplying by the growth factor.

   a) $y = 2^x$: after $(3, 8)$ comes $(4,\underline{\hspace{1.2cm}})$

   b) $y = 3 \cdot 2^x$: after $(2, 12)$ comes $(3,\underline{\hspace{1.2cm}})$

9. Evaluate $y = 4 \cdot 2^x$.

PAGE 5 — Practice · applications and errors

Growth Stories and Non-Examples

10. Apply it. $P(t) = 6 \cdot 2^t$ cells.

    $a =$ ______ $b =$ ______ $P(0) =$ ______ $P(4) =$ ______

11. Explain. Why is $y = 2x$ not exponential?

    ---

12. Error analysis. Student says intercept of $y = 5 \cdot 3^x$ is $(0, 15)$. Error: ____________ Correct: ______

13. Fit $y = ab^x$ with $b$ natural? Name $a$, $b$ or write no.

    a) $y = 2 \cdot 5^x$ ______

    b) $y = 3 \cdot \left(\tfrac12\right)^x$ ______

    c) $y = 7 \cdot 1^x$ ______

    d) $y = x^2$ ______

14. Table for $y = 3^x$ at $x = -1, 0, 1, 2$:

15. Every step multiplies by $4$; value at $x = 0$ is $2$. Equation: $y =$ ______

16. Technology. Enter $y = 2^x$. Table at $x = -2,-1,0,1,2$: ____________ Match the figure? ______

PAGE 6 — Exit ticket 17.1

Exit Ticket · 17.1

17. $y = 8 \cdot 2^x$: $a =$ ______ $b =$ ______ intercept = ______

18. $y(3) =$ ______ $y(-2) =$ ______

19. In one sentence, what does the growth factor $b$ tell you?

20. Why is $y = 5 \cdot \left(\tfrac12\right)^x$ outside A.F.2e?

PAGE 7 — Characteristics on the graph

17.2 $y$-Intercept, Domain, and Range

FIGURE: fig2-y-intercept-domain-range-of-exponential.png (full width)

21. For $y = 3 \cdot 2^x$: intercept ______ domain ______ range ______

22. Why does the range arrow start above $y = 0$?

PAGE 8 — The horizontal asymptote

Approaching $y = 0$

FIGURE: fig3-horizontal-asymptote-y-equals-zero.png (full width)

23. Labeled points with negative $x$: ______ Pattern: ____________

24. From the equation alone, $y = 2^x$: intercept ______ domain ______ range ______

25. $y = 7 \cdot 3^x$: intercept ______ domain ______ range ______

26. Explain. Why is the domain of $y = ab^x$ all real numbers, even for negative $x$?

PAGE 9 — Practice · characteristics

Reading Characteristics

27. Intercept, domain, range.

28. Match: $y = 2^x$ or $y = 3 \cdot 2^x$.

    a) intercept $(0, 1)$, asymptote $y = 0$ → ______

    b) intercept $(0, 3)$, asymptote $y = 0$ → ______

29. Apply it. $R(t) = 2 \cdot 3^t$ people. Intercept ______ Meaning: ____________

30. Error. Student says range of $y = 2^x$ is all reals. Fix: ______

31. Error. Student writes intercept of $y = 5 \cdot 2^x$ as $5$ (not a point). Correct form: ______

32. Continue the asymptote pattern: $2^{-5} =$ ______ $2^{-6} =$ ______

PAGE 10 — Practice · more characteristics

Sketching and Technology

33. Sketch $y = 2 \cdot 3^x$: mark intercept, asymptote, growth direction.

(use space / grid)

34. Technology. Graph $y = 3 \cdot 2^x$. Confirm $y(0) = 3$ and no table value $\le 0$. Notes: ____________

35. Why can the range of $y = ab^x$ ($a > 0$, $b \ge 2$) not include negatives?

36. If context restricts $0 \le x \le 5$, what happens to domain and range vs. the equation?

Domain: ____________ Range: ____________

PAGE 11 — Exit ticket 17.2

Exit Ticket · 17.2

37. $y = 4 \cdot 3^x$: intercept ______ domain ______ range ______

38. Horizontal asymptote: ______ Why it matters for range: ____________

39. From the characteristics figure, intercept of $y = 3 \cdot 2^x$: ______

40. Why is "range: $y \ge 0$" wrong for $y = 2^x$?

PAGE 12 — Graphing from a table

17.3 Graphing and Transformations

FIGURE: fig7-graphing-from-a-table-of-powers.png (full width)

41. Copy $y = 3^x$ table values: ____________ $y$-intercept: ______

42. Which Chapter 10 fact fills $x = -2$? ____________

PAGE 13 — Vertical shift

Transformation $f(x) + k$

FIGURE: fig5-parent-and-vertical-shift.png (full width)

43. For $g(x) = 2^x + 3$: intercept ______ asymptote ______

44. Preview stretch: if outputs are halved, $h(3)$ vs $f(3)$: ____________

PAGE 14 — Vertical stretch

Transformation $kf(x)$

FIGURE: fig6-parent-and-vertical-stretch.png (full width)

44. (continued) $h(3) =$ ______ Why half of $f(3)$? ____________

45. Graph $y = 2^x$ on blank grid (a) — at least five points.

FIGURE: fig10-blank-exponential-grids.png (full width)

46. Explain. Why does $+k$ move the asymptote, while $\times k$ does not?

PAGE 15 — Practice · graphing and transforms

Tables, Shifts, and Stretches

47. Table and graph.

    a) $y = 2^x$ on grid (a)

    b) $y = 3^x$ on grid (b)

48. For $f(x) = 2^x$, write equation; give intercept and asymptote.

49. Match to $f(x)+5$ or $\tfrac12 f(x)$ ($f(x)=2^x$).

    a) Same asymptote $y=0$, intercept halved → ______

    b) Asymptote $y=5$, intercept up by $5$ → ______

50. Apply it. Display shows $S(t)+10$ where $S(t)=2^t$. Effect: ____________ New asymptote: ______

PAGE 16 — Practice · more transforms

Reflections, Errors, and Technology

51. Explain. What does $k = -1$ do to $f(x) = 2^x$? Range becomes: ______

52. Error. Student shifts $y=2^x$ up by $3$ but leaves asymptote on the $x$-axis. Fix: ____________

53. Graph $y = 2^x + 3$ on grid (c); mark intercept and asymptote.

54. Graph $y = \tfrac12 \cdot 2^x$ on grid (c); mark $(0,\tfrac12)$ and $(3,4)$.

55. Technology. Graph $y=2^x$ and $y=2^x+3$ together. Constant vertical gap? ______

56. Rewrite $4 \cdot 2^x$ as $k \cdot f(x)$ with $f(x)=2^x$: $k =$ ______

57. A.F.2f limits $k$ to ____________ values.

58. On grid (d), graph $y = 2 \cdot 3^x$ for $x = -1,0,1,2$.

PAGE 17 — Exit ticket 17.3

Exit Ticket · 17.3

59. Three graphing steps: ______ · ______ · ______

60. $f(x)=3^x$.

61. What does $k=\tfrac12$ do to every output? ____________

62. Why is plotting $x=-2$ for $y=3^x$ a Chapter 10 skill?

PAGE 18 — Bacteria context

17.4 Exponential Models in Context

FIGURE: fig4-context-bacteria-doubling.png (full width)

63. $a =$ ______ $b =$ ______ intercept = ______

64. $P(1) =$ ______ $P(4) =$ ______

65. Domain in context: ____________ Range in context: ____________

66. Growth factor $b=2$ in one sentence: ____________

67. $A(n)=4\cdot 3^n$ MB: intercept means ____________ $b=3$ means ____________

68. Explain. Why start the bacteria graph at $t=0$?

PAGE 19 — Practice · contexts

Stories with Units

69. $C(t)=3\cdot 2^t$ cells.

    a) Intercept and meaning: ____________

    b) $C(5) =$ ______

    c) Domain in context: ______

70. Apply it. $F(n)=2\cdot 3^n$ MB.

$F(0)$ means: ____________

71. Apply it. $R(t)=5\cdot 2^t$ people. After $4$ days: ______ Intercept means: ____________

72. Explain. Why is "range: all reals" wrong for the bacteria model?

73. Error. Student keeps domain "all reals" in context. Correct domain: ______

74. Fit A.F.2e natural-number base? yes/no

    a) $10\cdot 2^t$ ______ b) $10\cdot(\tfrac12)^t$ ______ c) $10\cdot 3^t$ ______ d) $10t+2$ ______

PAGE 20 — Practice · more contexts

More Models

75. One-sentence insect story for $y=7\cdot 2^x$ ($x$ days). Intercept means: ____________

76. Why is range in context $P\ge 5$, not merely $P>0$?

77. Technology. Graph $P(t)=5\cdot 2^t$; read $P(0)$ through $P(4)$: ____________

78. Triples each hour from $8$ cells: $P(t)=$ ______ ($a=$ ______ $b=$ ______)

79. $D(t)=2^t+5$: transformation ______ asymptote ______

80. Apply it. $P(t)=5\cdot 2^t$. First whole hour with $P>50$: check $t=3$: ______ $t=4$: ______ Answer: ______

PAGE 21 — Exit ticket 17.4

Exit Ticket · 17.4

81. Interpret $(0,5)$ for bacteria, with units: ____________

82. Domain in context: ____________ Range in context: ____________

83. $Q(t)=9\cdot 3^t$: $a$ means ____________ $b$ means ____________

84. Why is a half-life model with base $\tfrac12$ outside this chapter?

PAGE 22 — Evaluating on the graph

17.5 Evaluating $f(x)$

FIGURE: fig8-evaluating-f-of-x-on-an-exponential.png (full width)

85. Read $f(3)=$ ______ Confirm: $3\cdot 2^3=$ ______

86. Confirm $f(-2)=\tfrac34$: work ____________

87. $f(0)=$ ______ $f(1)=$ ______ $f(2)=$ ______

88. $g(-3)$ for $g(x)=4\cdot 2^x$: ______

89. $P(2)=20$ means (bacteria): ____________

90. Explain. Which words in A.F.2g limit "find $x$ given $f(x)$" to quadratics?

PAGE 23 — Practice · evaluate

Substitute and Explain

91. Evaluate.

92. Negatives (Chapter 10).

93. Apply it. $P(t)=5\cdot 2^t$: $P(0)=$ ______ $P(3)=$ ______ $P(5)=$ ______

    Explain $P(5)$: ____________

94. Apply it. $A(n)=4\cdot 3^n$ MB: $A(2)=$ ______ Meaning of $n=2$ and $A(2)$: ____________

PAGE 24 — Practice · graph path and limits

Graph Path, Errors, Technology

95. Describe the path from $x=3$ to $f(3)=24$ on the figure: ____________

96. Explain. Why may you recognize $3\cdot 2^x=24$ at $x=3$, but not be required to solve $3\cdot 2^x=10$ by logs?

97. Error. Student writes $f(-2)=-3\cdot 2^2=-12$ for $f(x)=3\cdot 2^x$. Two errors: ____________

98. Technology. Table $f(x)=3\cdot 2^x$ at $-2,0,3$: ____________

99. $f(x)=2^x+3$: $f(0)=$ ______ $f(3)=$ ______ Transform: ______

100. Explain. Why evaluating never undoes an exponent, but recovering $x$ would.

101. Technology: $2^{1/2}$ ≈ ______ (nearest hundredth)

102. $R(t)=2\cdot 3^t$: $R(2)=$ ______ Meaning of $t=2$ and $R(2)$: ____________

PAGE 25 — Exit ticket 17.5

Exit Ticket · 17.5

103. $f(4)$ for $f(x)=3\cdot 2^x$: ______

104. $f(-1)$ for $f(x)=3\cdot 2^x$: ______

105. Explain $P(3)=40$ (bacteria) with units: ____________

106. What A.F.2g requires for exponentials vs. what it leaves to later courses:

PAGE 26 — Review Part A

Chapter 17 Review · Characteristics (A.F.2e)

FIGURE: fig9-growth-factor-versus-y-intercept.png (half width) FIGURE: fig2-y-intercept-domain-range-of-exponential.png (half width)

107. Roles of $a$ and $b$; natural-number limit on $b$: ____________

108. $y=3\cdot 2^x$: intercept ______ domain ______ range ______

FIGURE: fig3-horizontal-asymptote-y-equals-zero.png (half)

109. Why range is $y>0$, not $y\ge 0$: ____________

FIGURE: fig4-context-bacteria-doubling.png (half)

110. Interpret intercept; domain/range in context; $P(4)=$ ______

111. Intercept, domain, range.

a) $8\cdot 2^x$ ____________

b) $5^x$ ____________

c) $6\cdot 1^x$ ____________

PAGE 27 — Review Part B

Review · Graphing and Transforms (A.F.2f)

FIGURE: fig7-graphing-from-a-table-of-powers.png (half) FIGURE: fig5-parent-and-vertical-shift.png (half)

112. $y=3^x$ table values: ____________ Three steps: ____________

113. $g(x)=2^x+3$: intercept ______ asymptote ______ Unchanged: ______

FIGURE: fig6-parent-and-vertical-stretch.png (half) FIGURE: fig10-blank-exponential-grids.png (half)

114. $h(x)=\tfrac12\cdot 2^x$: $h(3)=$ ______ intercept ______

115. $f(x)=2^x$:

116. Sketch $y=2\cdot 3^x$ for $x=-1,0,1,2$ on a blank grid.

PAGE 28 — Review Part C

Review · Evaluate $f(x)$ (A.F.2g)

FIGURE: fig8-evaluating-f-of-x-on-an-exponential.png (full width)

117. $f(3)=$ ______ Confirm $f(-2)=\tfrac34$: ____________

118. Evaluate.

a) $4\cdot 2^x$ at $0,3,-2$: ____________

b) $3^x$ at $4$ and $-1$: ____________

119. Apply it. $P(t)=5\cdot 2^t$: $P(0)=$ ______ $P(3)=$ ______

     Explain both with units: ____________

120. Explain. Why evaluate $f(x)$ for exponentials, but not recover $x$ by logs? Quote the standard's split.

Canva production notes

• Figures: fig1 through fig10 in virginia-math-sol/algebra-1/figures/. Prefer full-width for single graphs; half-width pairs on review pages as marked.
• Grids: Page 14 embeds fig10-blank-exponential-grids.png for items 45–58 graphing work; print large enough to plot.
• Item parity: Numbers 1–120 match the textbook exactly.
• Dollar signs: None required in math mode; use words for money if a future edit adds currency.
• Scope callouts: Pages 1, 6, 21, and 25 restate the natural-number base limit and the no-logs evaluation boundary — keep those callouts visible in Canva.