Appendix A — Answer Key, Chapter 18: Comparing Linear, Quadratic, and Exponential Functions

SOL A.F.2 (h) · Covers textbook Chapter 18 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 108 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Conventions used in every answer below: the three parents are $f(x) = x$, $g(x) = x^{2}$, and $h(x) = b^{x}$ with $b$ a natural number (usually $2$). An intercept is a point. First differences $\Delta y$ assume equal input steps of $1$ unless stated otherwise. Currency amounts are written in plain text.

The figures used repeatedly in the chapter, for reference:

• Figure 1 is $y = x$, $y = x^{2}$, and $y = 2^{x}$ on one window, with shared points including $(2, 4)$
• Figure 2 is the same three parents one panel at a time, with intercepts labeled
• Figure 3 is end-behavior arrows on the shared window
• Figure 4 is three tables with first-difference columns for $x$, $x^{2}$, and $2^{x}$
• Figure 5 is second differences for $x^{2}$ and common ratios for $2^{x}$
• Figure 6 is unit-step chords on the three parents
• Figure 7 is the long-run race with shared points $(2, 4)$ and $(4, 16)$
• Figure 8 is the characteristic comparison board
• Figure 9 is Contexts A (walker), B (square area), and C (doubling culture)

Lesson 18.1 — Three Families on One Window

Guided practice

1. $f(x) = x$ (straight line), $g(x) = x^{2}$ (U-shaped parabola), $h(x) = 2^{x}$ (curve above the $x$-axis that steepens to the right).
2. Shared by $x^{2}$ and $2^{x}$: $(2, 4)$ (also $(4, 16)$ off the right edge of this window). Shared by $x$ and $x^{2}$: $(0, 0)$ or $(1, 1)$.
3. $y = x$: $(0, 0)$; $y = x^{2}$: $(0, 0)$; $y = 2^{x}$: $(0, 1)$.
4. $y = 2^{x}$ (any $b^{x}$ with $b > 1$). A positive base to any real power is positive, so the output never hits $0$ and the graph never meets the $x$-axis.
5. All three outputs go to $+\infty$. In the long run the exponential grows fastest.
6. $x \to -\infty$; $x^{2} \to +\infty$; $2^{x} \to 0$ (from above).

Independent practice

7. a) straight line b) parabola (U shape) c) curve that steepens (exponential growth curve)
8. Domain of each is all real numbers. Each rule accepts every real input — there is no forbidden substitution in $x$, $x^{2}$, or $b^{x}$.
9. $x$: all real numbers; $x^{2}$: $y \ge 0$; $2^{x}$: $y > 0$.

11. It violated the fact that $b^{x}$ never equals $0$ and never crosses the $x$-axis. The left end of $2^{x}$ approaches the axis as a horizontal asymptote; it does not cross it.
12. At $x = 0$, both $0$ and $0^{2}$ equal $0$, so $(0, 0)$ lies on both graphs. But $2^{0} = 1$, so the exponential meets the $y$-axis at $(0, 1)$, not at the origin.
13. $y$-intercept $(0, 1)$; no $x$-intercept (same reason as item 4).
14. Above the others at $x = -2$: $x^{2} = 4$. Below the $x$-axis: $x = -2$ (the only parent that takes negative values).
15. Domain and range are different questions. A rule can accept every input and still miss some outputs. Correct ranges: $x$ all reals; $x^{2}$ is $y \ge 0$; $2^{x}$ is $y > 0$.
16. $3^{2} = 9$ and $2^{2} = 4$, so $3^{x}$ is larger by $5$ at $x = 2$.
17. $y = x^{2}$. It is never negative, and it equals $0$ at exactly one input ($x = 0$). The line is negative for negative inputs; the exponential never hits $0$.
18. Sketches should show a line through the origin, a parabola with vertex at the origin, and $2^{x}$ through $(0, 1)$, with the three named points marked. (Grid A on fig10 is appropriate.)

Exit ticket 18.1

19. $(0, 0)$; $(0, 0)$; $(0, 1)$.
20. $y = 2^{x}$ (more generally $y = b^{x}$).
21. Shared examples: both have domain all reals; both go to $+\infty$ as $x \to +\infty$; both are non-negative for $x \ge 0$. Separators: $y$-intercepts $(0, 0)$ vs $(0, 1)$; $x^{2}$ has an $x$-intercept and $2^{x}$ does not; as $x \to -\infty$, $x^{2} \to +\infty$ while $2^{x} \to 0$; table fingerprints differ (second differences vs common ratio).
22. Some characteristics (shape, intercepts, end behavior) are easiest on a graph; others (constant differences vs constant ratios) are easiest in a table. A.F.2h names both representations because a complete comparison uses both.

Lesson 18.2 — Growth Patterns in Tables

Guided practice

23. Linear $\Delta y$: $+1$, $+1$, $+1$, $+1$. Quadratic: $+1$, $+3$, $+5$, $+7$. Exponential: $+1$, $+2$, $+4$, $+8$. Only the linear column is constant.
24. Both quadratic and exponential tables have growing first differences. You need second differences (quadratic) or common ratios (exponential) to finish the identification.
25. $+2$.
26. $2$ (each output is twice the previous).
27. Outputs $0$, $1$, $2$, $3$, $4$; $\Delta y$ all $+1$.
28. Outputs $0$, $1$, $4$, $9$, $16$; $\Delta y$: $+1$, $+3$, $+5$, $+7$; $\Delta^{2}y$: $+2$, $+2$, $+2$.

Independent practice

29. Outputs $1$, $2$, $4$, $8$, $16$; ratios all $\times 2$.
30. a) Linear (first differences $+3$) b) Quadratic (second differences $+2$) c) Exponential (common ratio $3$)
31. $\Delta y$: $+3$, $+5$, $+7$, $+9$; $\Delta^{2}y$: $+2$, $+2$, $+2$.
32. Ratios $3$, $3$, $3$, $3$.
33. Linear. The plant grows $3$ centimeters every week — a constant additive rate.
34. Exponential; common ratio $3$.
35. Outputs $1$, $3$, $9$, $27$, $81$; each ratio equals $3$.
36. Growing first differences that themselves double are the signature of an exponential table with ratio $2$, not of a quadratic. Quadratic first differences grow by a constant second difference (the odd numbers $+1$, $+3$, $+5$, $+7$ for $x^{2}$), not by doubling. Correct family: exponential.
37. On $y = mx + b$, increasing $x$ by $1$ changes $y$ by $m(x+1)+b - (mx+b) = m$. So every first difference equals the slope.
38. Quadratic. Example: $0$, $2$, $8$, $18$, $32$ (first differences $+2$, $+6$, $+10$, $+14$; second differences all $+4$), or any vertical stretch of $x^{2}$ with second difference $2a$.
39. Yes — exponential with common ratio $\tfrac12$ (decay).
40. Linear. The first differences are already constant at $+12$ dollars per hour, which is exactly the linear fingerprint; second differences would all be $0$ and add no new information.

Exit ticket 18.2

41. Linear: constant first differences. Quadratic: constant second differences. Exponential: constant common ratio.
42. Linear (first differences $+3$).
43. $\Delta y$: $+1$, $+3$, $+5$; $\Delta^{2}y$: $+2$, $+2$. Quadratic.
44. Exponential; common ratio $2$.

Lesson 18.3 — Rates of Change Compared

Guided practice

45. $+1$ each time.
46. $+1$, $+3$, $+5$, $+7$.
47. $\times 2$.
48. $(2, 4)$ and $(4, 16)$.
49. $6$, $36$, and $64$. Largest: $2^{x} = 64$.
50. "Exponential grows fastest" is a statement about $x \to +\infty$, not about every finite input. Between $x = 2$ and $x = 4$ the parabola is temporarily higher; after $x = 4$ the exponential overtakes and stays ahead. A short window can miss that overtaking.

Independent practice

51. a) additive b) changing c) multiplicative
52. Rises: $+1$, $+3$, $+5$, $+7$, $+9$. Second differences: $+2$, $+2$, $+2$, $+2$.
53. Ratios all $3$: $3/1$, $9/3$, $27/9$, $81/27$.
54. $x = 1$: $2^{x}$ larger ($2 > 1$). $x = 2$: equal ($4 = 4$). $x = 3$: $x^{2}$ larger ($9 > 8$). $x = 4$: equal ($16 = 16$). $x = 5$: $2^{x}$ larger ($32 > 25$).
55. Additive; linear.
56. Multiplicative; exponential.
57. The first differences of a quadratic themselves change by a constant (the second difference). That constant change-of-the-change is the "rate of change of the rate of change."
58. Accept any clear three-row display, for example:
    • $y = x$: rises $+1$, $+1$, $+1$
    • $y = x^{2}$: rises $+1$, $+3$, $+5$
    • $y = 2^{x}$: ratios $\times 2$, $\times 2$, $\times 2$
59. Quadratic steps get bigger by adding a constant to the previous rise (additive acceleration). Exponential steps get bigger by multiplying the previous output by a constant (multiplicative growth). "Bigger steps" alone does not name the mechanism.
60. $2^{x} > 50$ once $x$ is past $5$ (since $2^{5} = 32$ and $2^{6} = 64$), so somewhere between $5$ and $6$, about $x \approx 5.7$. For $x^{2} = 50$, $x = \sqrt{50} \approx 7.1$. The exponential crosses $50$ at a smaller input.
61. As $x \to +\infty$: exponential grows fastest, then quadratic, then linear — all go to $+\infty$.
62. $x \to -\infty$; $x^{2} \to +\infty$; $2^{x} \to 0$.

Exit ticket 18.3

63. $y = x$ has a constant additive rate of $+1$ per unit input. $y = 2^{x}$ has a constant multiplicative rate of $\times 2$ per unit input.
64. $4^{2} = 16$ and $2^{4} = 16$, so they agree at $(4, 16)$. After $x = 4$, $2^{x}$ is larger and pulls away.
65. It says the additive rate itself changes by the same amount at every step — the rises form an arithmetic sequence.
66. Tables make the numeric pattern (constant add, changing add, multiply) obvious; graphs make the geometric consequence (line, U, steepening curve) obvious. A.F.2h wants the comparison in both languages.

Lesson 18.4 — Choosing the Model That Fits

Guided practice

67. Linear: $\Delta y$ constant. Quadratic: $\Delta^{2}y$ constant. Exponential: common ratio.
68. Linear; constant first difference $+3$ (adds $3$ miles each hour).
69. Quadratic; outputs match $n^{2}$, with constant second difference $+2$.
70. Exponential; common ratio $2$ (doubles each hour).
71. End behavior as $x \to -\infty$ (line falls; parabola rises).
72. $y$-intercept row: $(0, 0)$ versus $(0, 1)$.

Independent practice

73. a) Linear b) Exponential c) Quadratic
74. $y = x$. Confirming rows: shape (straight line) and $y$-intercept / $x$-intercept both $(0, 0)$; also constant additive rate / $\Delta y$ constant.
75. $y = x^{2}$.
76. $y = b^{x}$ (for example $y = 2^{x}$).
77. Exponential. Common ratio $2$; a matching rule is $y = 100 \cdot 2^{x}$ if $x$ counts steps from the $100 balance, or more simply the growth pattern is $y = 2^{x}$ scaled by $100$.
78. Quadratic. The $1$, $4$, $9$, $16$ pattern is $n^{2}$ (area grows with the square of a length).
79. Linear (a horizontal line, slope $0$). The constant first difference is $0$ — still constant, so still linear.
80. "Gets bigger faster and faster" describes both quadratic and exponential growth. Context C multiplies by $2$ each hour, which is the exponential fingerprint (common ratio), not constant second differences.
81. Both have growing first differences, so both accelerate. Second differences separate them: constant second differences mean quadratic; a constant common ratio means exponential.
82. Accept any clear sketches, for example: linear — walking at steady speed; quadratic — area of a square garden as side length grows; exponential — a rumor or bacteria culture that doubles on a fixed schedule.
83. Shared as $x \to +\infty$: all three go to $+\infty$; all eventually increase for large positive $x$. Disagreements as $x \to -\infty$: line $\to -\infty$, parabola $\to +\infty$, exponential $\to 0$.
84. Linear; first difference $-4$ GB per week.

Exit ticket 18.4

85. Quadratic.
86. Exponential.
87. Linear.
88. Linear: constant $\Delta y$. Quadratic: constant $\Delta^{2}y$. Exponential: constant common ratio.

Chapter 18 Review

Part A — Graphs and characteristics

89. $(0, 0)$; $(0, 0)$; $(0, 1)$.
90. $y = 2^{x}$ (any $b^{x}$ with $b > 1$). Positive outputs only, so the curve never meets the $x$-axis.
91. $x \to -\infty$; $x^{2} \to +\infty$; $2^{x} \to 0$.
92. $(2, 4)$ (or $(4, 16)$). After $x = 4$, $2^{x}$ pulls away above $x^{2}$.
93. Domain of each: all real numbers. Ranges: all reals; $y \ge 0$; $y > 0$.

Part B — Tables and fingerprints

94. Linear: constant first differences. Quadratic: constant second differences. Exponential: constant common ratio.
95. Linear (first differences $+3$).
96. Quadratic. $\Delta y$: $+3$, $+5$, $+7$, $+9$; $\Delta^{2}y$: $+2$, $+2$, $+2$.
97. Exponential; common ratio $3$.
98. $\Delta y$: $+1$, $+2$, $+3$ (not constant). $\Delta^{2}y$: $+1$, $+1$ (constant) — so the table is quadratic, not exponential. Ratios $2$, $2$, $1.75$ are not constant. The early doubling of first differences was a coincidence, not a common-ratio test on the outputs.

Part C — Rates and long-run growth

99. $y = x$: constant additive rate $+1$ per unit $x$. $y = 2^{x}$: constant multiplicative rate $\times 2$ per unit $x$.
100. At $x = 3$: $x^{2} = 9 > 8 = 2^{x}$. At $x = 5$: $2^{x} = 32 > 25 = x^{2}$.
101. They are the inputs where the quadratic and exponential agree. Between them the parabola is ahead; after the second shared point the exponential stays ahead. The points mark the transition in the long-run race.
102. Exponential fastest, then quadratic, then linear.

Part D — Choosing a model

103. a) Linear — constant first differences / constant additive rate b) Quadratic — $n^{2}$ pattern / constant second differences c) Exponential — common ratio $2$ / doubles each hour
104. Exponential ($y = b^{x}$).
105. Linear. First difference $-6$: the warehouse loses $6$ crates each day.
106. Quadratic. "Gets bigger faster" is also true of exponential tables; the constant second difference (or the $n^{2}$ match) is what completes the justification.
107. The parents are the simplest member of each family, so a comparison of characteristics is not tangled with slopes, vertical stretches, or shifts. Once you can contrast $x$, $x^{2}$, and $b^{x}$, the same contrasts apply to other members of each family. A.F.2h names those three on purpose.
108. (i) Linear — constant additive charge per mile (plus a starting value). (ii) Quadratic — height vs time under gravity is a parabolic model. (iii) Exponential — constant multiplicative growth (about $\times 4$ each hour).

Workbook-only items

Page 2, fill in the blanks. The three parents are $f(x) = x$, $g(x) = x^{2}$, and $h(x) = 2^{x}$ (in the figure). At $x = 2$ they share $(2, 4)$.

Page 3, intercept frame. $y = x$: $(0,0)$ and $(0,0)$. $y = x^{2}$: $(0,0)$ and $(0,0)$. $y = 2^{x}$: $(0,1)$ and none.

Page 4, end-behavior frame. As $x \to +\infty$, all three $\to +\infty$; fastest long-run: exponential. As $x \to -\infty$: $x \to -\infty$, $x^{2} \to +\infty$, $2^{x} \to 0$.

Page 8, first-difference definition. $\Delta y$ is the change in output when the input increases by one equal step. Constant first differences name a linear table.

Page 9, fingerprint table. Linear: $\Delta y$ constant. Quadratic: $\Delta^{2}y$ constant. Exponential: common ratio constant.

Page 14, rate blanks. Linear: constant additive rate — each step adds the same amount. Quadratic: changing additive rate — steps grow (by a constant second difference). Exponential: constant multiplicative rate — each step multiplies by $b$.

Page 28, blank grids and tables. Any assigned comparison sketch or table. Expected conventions: mark intercepts as points; compute $\Delta y$ in the third column; use second differences or ratios only after first differences fail to stay constant; on a shared window, read both ends of the $x$-axis.