Algebra 1 Workbook — Chapter 18: Comparing Linear, Quadratic, and Exponential Functions

SOL A.F.2 (h) · Companion to Textbook Chapter 18

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 108.

PAGE 1 — Chapter opener

Chapter 18 · Comparing Linear, Quadratic, and Exponential Functions

Standard A.F.2 (h)

In this chapter you will:

• Compare $f(x) = x$, $g(x) = x^{2}$, and $h(x) = b^{x}$ on the same window
• Read intercepts, domain, range, and end behavior across the three families
• Identify each family from a table fingerprint: constant $\Delta y$, constant $\Delta^{2}y$, or constant common ratio
• Contrast constant additive, changing additive, and constant multiplicative rates of change
• Choose which model fits a table, graph, or context, and justify the choice

Words to know: linear parent · quadratic parent · exponential parent · first difference · second difference · common ratio · rate of change · end behavior · $y$-intercept · $x$-intercept

Convention: compare the parents named in A.F.2h unless an item names another member of the family. The exponential base $b$ is a natural number; this chapter mostly uses $b = 2$.

PAGE 2 — Three families on one window

18.1 Three Families on One Window

FIGURE: fig1-three-families-same-window.png (full width)

Fill in the blanks.

The three parents are $f(x) =$ ______ , $g(x) =$ ______ , and $h(x) =$ ______ .

At $x = 2$, the quadratic and the exponential share the point ____________ .

1. Name the three parent equations shown, and say which curve is which by shape.

   ---

2. Name one point on both $x^{2}$ and $2^{x}$: ____________ Name one point on both $x$ and $x^{2}$: ____________

PAGE 3 — Intercepts of the three parents

Intercepts, Family by Family

FIGURE: fig2-parent-trio-intercepts.png (full width)

Complete the frame.

3. Give the $y$-intercept of each parent.

   $y = x$: ____________ $y = x^{2}$: ____________ $y = 2^{x}$: ____________

4. Which parent has no $x$-intercept? ______ Why can $b^{x}$ never equal $0$? _______________________________________

5. $y$-intercepts: $x$ ______ · $x^{2}$ ______ · $2^{x}$ ______

PAGE 4 — End behavior

End Behavior on the Shared Window

FIGURE: fig3-end-behavior-three-families.png (full width)

As $x \to +\infty$: all three go to ____________ . Fastest long-run growth: ____________

As $x \to -\infty$:

• $x$ → ____________
• $x^{2}$ → ____________
• $2^{x}$ → ____________

5. As $x \to +\infty$, what happens to all three outputs? Which grows fastest long-run?

   ---

6. As $x \to -\infty$, describe each parent.

   $x$: ____________ $x^{2}$: ____________ $2^{x}$: ____________

7. As $x \to -\infty$, which parent approaches $0$? ____________

PAGE 5 — Practice · graphs and characteristics

Practice · Shared Window

7. Shape of $x$: ____________ of $x^{2}$: ____________ of $2^{x}$: ____________

8. Domain of each parent: ____________ Why do the three domains agree? _______________________________________

9. Range of $x$: ____________ Range of $x^{2}$: ____________ Range of $2^{x}$: ____________

10. Evaluate at $x = -2$, $0$, $3$:

12. Why is $(0,0)$ on $y = x$ and $y = x^{2}$ but not on $y = 2^{x}$?

    ---

PAGE 6 — Practice · more graph comparison

Practice · Intercepts and Ranges

11. Apply it. A sketch of $2^{x}$ crosses the $x$-axis left of the origin. What did it violate?

    ---

12. For $y = 3^{x}$: $y$-intercept ____________ Has an $x$-intercept? ______

13. At $x = -2$, which parent is above the others? ____________ Which is below the $x$-axis? ____________

14. Error analysis. "All three have range all real numbers because the domain is." Error:

    ---

    Correct ranges: _______________________________________

15. At $x = 2$, compare $2^{x}$ and $3^{x}$: larger is ______ by ______

16. Apply it. Never negative; equals $0$ at exactly one input. Which parent?

    ____________ Justify: _______________________________________

17. Sketch the three parents; mark $(0,0)$, $(0,1)$, and $(2,4)$.

FIGURE: fig10-blank-grids-and-tables.png (use Grid A)

21. One shared characteristic of $x^{2}$ and $2^{x}$: ____________ One that separates them: ____________

22. Why does A.F.2h want both tables and graphs?

    ---

PAGE 7 — Exit ticket 18.1

Exit Ticket 18.1

19. (also on page 3)

20. (also on page 4)

21. (also on page 6)

22. (also on page 6)

PAGE 8 — First differences

18.2 Growth Patterns in Tables

FIGURE: fig4-first-differences-three-tables.png (full width)

Fill in the blanks.

A first difference $\Delta y$ is _______________________________________ .

Constant first differences name a ____________ table.

23. Copy the three $\Delta y$ columns. Which is constant? ____________

24. Why is "first differences grow" not enough to separate quadratic from exponential?

    ---

PAGE 9 — Second differences and ratios

Second Differences and Common Ratios

FIGURE: fig5-second-differences-and-ratios.png (full width)

Complete the fingerprint table.

25. Constant second difference for $y = x^{2}$: ______

26. Common ratio for $y = 2^{x}$: ______

27. Fingerprint of each family (one phrase each):

    Linear: ____________ Quadratic: ____________ Exponential: ____________

PAGE 10 — Build the tables

Build Your Own Tables

FIGURE: fig10-blank-grids-and-tables.png (use the three blank tables)

27. Table for $y = x$ at $x = 0$ to $4$, with $\Delta y$.

28. Table for $y = x^{2}$ at those inputs, with $\Delta y$ and $\Delta^{2}y$.

29. Table for $y = 2^{x}$ at those inputs, with common ratios.

PAGE 11 — Practice · identify the family

Practice · Fingerprints

30. Identify the family. a) $2$, $5$, $8$, $11$, $14$ → ____________ b) $1$, $4$, $9$, $16$, $25$ → ____________ c) $1$, $3$, $9$, $27$, $81$ → ____________

31. For 30b, show $\Delta y$ and $\Delta^{2}y$: _______________________________________

32. For 30c, show common ratios: _______________________________________

33. Outputs $0$, $3$, $6$, $9$. Family: ____________

34. Outputs $0$, $1$, $4$, $9$. Show $\Delta^{2}y$ constant. Family: ____________

35. Outputs $5$, $10$, $20$, $40$. Family: ____________ Ratio: ______

PAGE 12 — Practice · tables in context

Practice · Tables and Stories

33. Apply it. Heights $4$, $7$, $10$, $13$ cm after $0$–$3$ weeks. Family: ____________ What does the constant first difference mean? _______________________________________

34. Apply it. Rumor reaches $2$, $6$, $18$, $54$ people. Family: ____________ Ratio: ______

35. Five-row table for $y = 3^{x}$ starting at $x = 0$. Confirm ratio $3$.

36. Error analysis. Outputs $1$, $2$, $4$, $8$: student says quadratic because differences double. Error:

    ---

    Correct family: ____________

37. Reasoning. Why do first differences equal the slope $m$ when $x$ increases by $1$?

    ---

38. Constant $\Delta^{2}y = +4$, non-constant $\Delta y$. Family: ____________ Example outputs: _______________________________________

39. Outputs $16$, $8$, $4$, $2$, $1$. Exponential? ______ Ratio: ______

40. Apply it. Ticket sales rise by exactly $12 each hour. Family: ____________ Why skip second differences? _______________________________________

PAGE 13 — Exit ticket 18.2

Exit Ticket 18.2

41. (also on page 9)

42–44. (also on page 11)

PAGE 14 — Rates of change side by side

18.3 Rates of Change Compared

FIGURE: fig6-rate-of-change-three-families.png (full width)

Fill in the blanks.

Linear: constant ____________ rate — each step ____________

Quadratic: ____________ additive rate — steps ____________

Exponential: constant ____________ rate — each step ____________

45. For $y = x$, each highlighted chord adds: ______

46. Four step sizes for $y = x^{2}$: ______ , ______ , ______ , ______

47. Multiplicative factor for every chord of $y = 2^{x}$: ______

PAGE 15 — The long-run race

Long-Run Growth

FIGURE: fig7-long-run-growth-race.png (full width)

48. Shared points marked: ____________ and ____________

49. At $x = 6$, outputs: $x =$ ______ · $x^{2} =$ ______ · $2^{x} =$ ______ Largest: ____________

50. Why can a short window show $x^{2}$ above $2^{x}$ without contradicting long-run exponential growth?

    ---

51. At $x = 4$, show $x^{2} = 2^{x}$. What happens after? _______________________________________

PAGE 16 — Practice · rates

Practice · Additive vs Multiplicative

51. a) Linear: constant ____________ rate b) Quadratic: ____________ additive rate c) Exponential: constant ____________ rate

52. Unit-step rises of $y = x^{2}$ from $x = 0$ to $5$: _______________________________________ Second differences: _______________________________________

53. Successive ratios of $y = 3^{x}$ for $x = 0$ to $4$: _______________________________________

54. At $x = 1,2,3,4,5$: which is larger, $x^{2}$ or $2^{x}$? (or equal)

    $1$: ______ $2$: ______ $3$: ______ $4$: ______ $5$: ______

55. Contrast rate of change of $y = x$ vs $y = 2^{x}$:

    ---

56. A constant second difference says what about a quadratic's additive rate?

    ---

57. Why both tables and graphs for rates? _______________________________________

PAGE 17 — Practice · rates in context

Practice · Rate Stories

55. Apply it. Road crew paints $40$ m every hour. Additive or multiplicative? ______ Family: ____________

56. Apply it. Bacteria count triples every hour. Additive or multiplicative? ______ Family: ____________

57. Reasoning. Why can we say a quadratic has a "rate of change of its rate of change" that is constant?

    ---

58. Three small tables of rises/ratios from $x = 0$ to $3$ for $x$, $x^{2}$, $2^{x}$.

59. Error analysis. "Exponential is quadratic with bigger steps." Separate the ideas:

    ---

60. Roughly where does $2^{x}$ exceed $50$? ______ Where would $x^{2}$ reach $50$? ______

61. Rank as $x \to +\infty$: _______________________________________

62. As $x \to -\infty$: $x$ ______ · $x^{2}$ ______ · $2^{x}$ ______

PAGE 18 — Exit ticket 18.3

Exit Ticket 18.3

63–66. (see pages 15–16)

PAGE 19 — Comparison board

18.4 Choosing the Model That Fits

FIGURE: fig8-characteristic-comparison-board.png (full width)

67. Table-pattern row for all three families:

    Linear: ____________ Quadratic: ____________ Exponential: ____________

68. Which board row separates $y = x$ from $y = x^{2}$ as $x \to -\infty$? ____________

69. Which row separates $y = x^{2}$ from $y = 2^{x}$ at the $y$-intercept? ____________

70. Best table test for each family:

    Linear: ____________ Quadratic: ____________ Exponential: ____________

PAGE 20 — Which model fits?

Three Contexts

FIGURE: fig9-which-model-fits-contexts.png (full width)

68. Context A family: ____________ Fingerprint: ____________

69. Context B family: ____________ Fingerprint: ____________

70. Context C family: ____________ Fingerprint: ____________

71. Side length $s$, area $s^{2}$. Family: ____________

72. Population multiplies by $3$ each year. Family: ____________

73. Car at constant speed. Family: ____________

PAGE 21 — Practice · matching

Practice · Match the Family

73. a) Constant additive rate → ____________ b) Constant multiplicative rate → ____________ c) Constant second differences → ____________

74. Straight line through the origin. Parent: ____________ Two board rows that confirm: ____________ · ____________

75. U with vertex at the origin. Parent: ____________

76. $y$-intercept $(0,1)$, no $x$-intercept. Parent: ____________

77. Outputs $7$, $7$, $7$, $7$. Family: ____________ What is special about the rate? _______________________________________

PAGE 22 — Practice · choose in context

Practice · Context Choice

77. Apply it. Balances $100, $200, $400, $800. Family: ____________ Rule matching the growth: ____________

78. Apply it. Pizza areas grow like $1$, $4$, $9$, $16$ times a constant. Family: ____________ Why? _______________________________________

79. Error analysis. Context C called quadratic "because numbers get bigger faster." Error:

    ---

80. Reasoning. Why is "gets bigger faster" true of both quadratic and exponential, and what separates them?

    ---

81. One real-world sketch for each family:

    Linear: _______________________________________ Quadratic: _______________________________________ Exponential: _______________________________________

82. Two shared characteristics as $x \to +\infty$: _______________________________________ Two disagreements as $x \to -\infty$: _______________________________________

83. Apply it. Storage $64$, $60$, $56$, $52$ GB weekly. Family: ____________ Constant first difference: ______

PAGE 23 — Exit ticket 18.4

Exit Ticket 18.4

85–88. (see pages 19–20)

PAGE 24 — Chapter review · graphs

Chapter 18 Review · Graphs

FIGURE: fig1-three-families-same-window.png (half width)

89. $y$-intercepts: $x$ ______ · $x^{2}$ ______ · $2^{x}$ ______

90. No $x$-intercept: ____________ Why? _______________________________________

91. End behavior as $x \to -\infty$:

    $x$: ____________ $x^{2}$: ____________ $2^{x}$: ____________

92. Point shared by $x^{2}$ and $2^{x}$: ____________ After $x = 4$: _______________________________________

93. Domain of each: ____________ Ranges: $x$ ______ · $x^{2}$ ______ · $2^{x}$ ______

PAGE 25 — Chapter review · tables

Chapter 18 Review · Tables

94. Fingerprints: Linear ____________ · Quadratic ____________ · Exponential ____________

95. Outputs $4$, $7$, $10$, $13$, $16$. Family: ____________

96. Outputs $1$, $4$, $9$, $16$, $25$. Family: ____________ Second differences: _______________________________________

97. Outputs $2$, $6$, $18$, $54$. Family: ____________ Ratio: ______

98. Error analysis. Outputs $1$, $2$, $4$, $7$. Student says exponential. Respond:

    ---

PAGE 26 — Chapter review · rates and models

Chapter 18 Review · Rates and Models

FIGURE: fig7-long-run-growth-race.png (half width)

99. Rate of $y = x$: _______________________________________ Rate of $y = 2^{x}$: _______________________________________

100. At $x = 3$, larger is ______ . At $x = 5$, larger is ______ .

101. Why do shared points $(2,4)$ and $(4,16)$ matter?

     ---

102. Rank as $x \to +\infty$: _______________________________________

103. Match and cite a fingerprint. a) Constant speed distance → ____________ · ____________ b) Area of square side $n$ → ____________ · ____________ c) Culture doubles hourly → ____________ · ____________

104. Approaches $x$-axis on the left, $(0,1)$, climbs on the right. Family: ____________

PAGE 27 — Chapter review · applications

Chapter 18 Review · Applications

105. Apply it. Inventory $80$, $74$, $68$, $62$. Family: ____________ First difference in context: _______________________________________

106. Apply it. Areas $1$, $4$, $9$, $16$. Family: ____________ Why is "gets bigger faster" incomplete? _______________________________________

107. Reasoning. Why does A.F.2h name the specific parents $x$, $x^{2}$, and $b^{x}$?

     ---

108. Apply it. Choose a family for each; justify in one phrase. (i) Flat fee plus constant charge per mile: ____________ · ____________ (ii) Ball height vs time (quadratic in time): ____________ · ____________ (iii) Views multiply by about $4$ each hour: ____________ · ____________

PAGE 28 — Blank grids and tables

Graph and Table Paper

FIGURE: fig10-blank-grids-and-tables.png (full page)

Use Grid A or B to sketch the three parents on one window, or to compare two families after a table test. Use the blank tables to compute $\Delta y$ (and $\Delta^{2}y$ or ratios) for any equally spaced inputs a teacher assigns.

Reminders:

• Constant $\Delta y$ → linear
• Constant $\Delta^{2}y$ (with changing $\Delta y$) → quadratic
• Constant common ratio → exponential
• On a shared window, check both ends: as $x \to +\infty$ and as $x \to -\infty$

Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Item order: a few exit-ticket and review items appear beside the figure that supports them (pages 3, 4, 6, 9, 15, 19, 20) rather than strictly by number. Numbers still match the textbook; the answer key is in numerical order.
• Figure widths: fig1, fig3, fig4, fig5, fig7, fig8, and fig9 are full width on first appearance. fig2-parent-trio-intercepts.png and fig6-rate-of-change-three-families.png are three-panel images — set full width with no side text. fig10-blank-grids-and-tables.png takes a full page on page 28 and is reused at half width when a single grid or the blank tables are enough.
• Blank work space: items 11, 12, 15, 22, 24, 36, 37, 50, 57, 59, 66, 80, 81, 98, 101, 107 are written explanations — give each at least four ruled lines. Application items 17, 33, 34, 40, 55, 56, 77, 78, 84, 105, 106, 108 need two lines per part.
• Tables: leave room beside items 27–29, 31, 32, 35, 52, 53, and 58 for student-built tables; the blank tables on fig10 can be printed as a facing page if needed.
• Currency: dollar amounts on pages 12 and 22. Set them as plain text ($12, $100, $200, $400, $800), not inside a math box.
• Negative signs: use the typographic minus (−), matching the figures, not a hyphen.
• Review repeats: fig1 and fig7 repeat in the review; repeat the image rather than asking students to flip back.