Algebra 1 Workbook — Chapter 16: Quadratic Functions

SOL A.F.2 (b, c, d, g) · Companion to Textbook Chapter 16

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 126.

PAGE 1 — Chapter opener

Chapter 16 · Quadratic Functions

Standard A.F.2 (b, c, d, g)

In this chapter you will:

• Name the vertex, axis of symmetry, and opening of a parabola, and say whether the vertex is a max or a min
• Find zeros / $x$-intercepts, the $y$-intercept, domain, and range (including context)
• Graph using transformations $f(x) + k$ and $kf(x)$ ($k$ rational) of the parent $y = x^2$
• Connect standard and factored forms to the same graph
• Evaluate $f(x)$ and recover $x$ from $f(x) = c$; explain in context

Words to know: quadratic function · parabola · parent $y = x^2$ · vertex · axis of symmetry · maximum · minimum · zero · $x$-intercept · $y$-intercept · domain · range · vertical shift · vertical stretch / shrink / reflection · standard form · factored form · evaluate

Convention: transformations are limited to $f(x) + k$ and $kf(x)$ with rational $k$. No exponential functions (Ch 17). No family comparison (Ch 18).

PAGE 2 — Vertex and axis

16.1 Vertex, Axis, and Opening

FIGURE: fig1-quadratic-vertex-and-axis.png (full width)

1. Equation: ____________ Vertex: ____________ Axis: ____________

2. Opens up or down? ____________ How do you know from the equation?

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3. Explain. Why is the axis a vertical line through the vertex?

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PAGE 3 — Max versus min · parent

Opens Up vs Opens Down · The Parent

FIGURE: fig3-opens-up-versus-opens-down.png (full width)

3. For $y = x^2 - 4$: max or min? ____________ Coordinates: ____________

4. For $y = -x^2 + 4$: max or min? ____________ Coordinates: ____________

FIGURE: fig5-parent-quadratic-y-equals-x-squared.png (half width)

5. Parent $y = x^2$: vertex ______ axis ______ domain ______ range ______

PAGE 4 — Practice · opening and symmetry

Opening, Vertex Type, and Symmetry

7. Opens up/down? Max or min?

   a) $f(x) = 2x^2 - 3x + 1$ → ____________ b) $g(x) = -x^2 + 4$ → ____________ c) $h(x) = \tfrac12 x^2 - 7$ → ____________ d) $y = -5x^2 + x$ → ____________

8. Vertex $(4, -1)$; point $(6, 3)$ on the curve. Another point: ____________

   Why? _______________________________________________

9. Sketch $y = x^2$; mark vertex and axis.

10. Reasoning. Why can a quadratic never open sideways?

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11. Error analysis. Student says $y = -x^2 + 4$ has a minimum at $(0, 4)$:

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PAGE 5 — Application and exit · 16.1

Vertex in Context · Exit 16.1

12. Mirrors across the axis for $y = x^2 - 4x - 5$: $f(0)=$ ______ and $f(4)=$ ______

    Mirror points: ____________

13. Application. $h(t) = -16t^2 + 32t + 6$: max or min? ____________ Why?

14. How are $y = x^2$ and $y = -x^2$ related? Vertex of each?

15. Technology. Confirm vertex $(2, -9)$ for $y = x^2 - 4x - 5$. Window: ____________

16. Why $a \neq 0$? _______________________________________________

Exit ticket 16.1

17. $f(x) = 4x^2 - x - 9$: up/down? ______ max/min? ______

18. Vertex $(-2, 5)$, $a < 0$: axis ____________ max or min value? ____________

19. Parent vertex ______ axis ______

20. In one sentence, what the axis of symmetry does:

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PAGE 6 — All characteristics

16.2 Intercepts, Zeros, and the Vertex Formula

FIGURE: fig2-all-key-characteristics-labeled.png (full width)

21. Zeros: ____________ $y$-int: ____________ Vertex: ____________

    Domain: ____________ Range: ____________

22. Show $h = 1$ and $k = -4$ algebraically.

    $h = -\dfrac{b}{2a} =$ ____________ $k = f(1) =$ ____________

23. $y$-intercept of $y = -3x^2 + 4x - 8$: ____________

24. Zeros of $y = x^2 - 5x + 6$: ____________ as points: ____________

25. Vertex of $y = x^2 - 6x + 5$: ____________ max or min? ____________

26. Explain. Zero vs $x$-intercept for $y = x^2 - 2x - 3$:

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PAGE 7 — Practice · finding characteristics

Finding Characteristics from the Equation

27. $y$-intercepts:

    a) $x^2 + 9x - 2$ → ____________ b) $-4x^2 + 11$ → ____________ c) $3x^2 - x$ → ____________

28. Zeros and $x$-intercepts:

    a) $x^2 - 9$ → ____________ b) $x^2 + 2x - 15$ → ____________ c) $(x - 4)(x + 1)$ → ____________

29. Axis and vertex (max/min):

    a) $x^2 - 4x + 1$ → ____________ b) $-2x^2 + 8x - 3$ → ____________ c) $x^2 + 6x + 10$ → ____________

30. Verify figure 2 by algebra: zeros ______ $y$-int ______ vertex ______

PAGE 8 — Reasoning and exit · 16.2

Characteristics · Reasoning · Exit 16.2

31. Zeros at $-2$ and $6$. Axis? ____________ Why?

32. Error analysis. For $y = x^2 - 4x - 5$, student gets $h = -2$:

    _______________________________________________ Correct $h$: ______

33. Application. $R(x) = -x^2 + 40x$: $y$-int ______ meaning ____________

    Vertex ______ meaning ____________

34. All characteristics of $y = -x^2 + 4$: ____________

35. Technology. Confirm figure 2 zeros and minimum: ____________

36. Could $y = x^2 - 4x - 5$ have vertex $(2, -9)$ and $y$-int $(0, -5)$? Check:

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Exit ticket 16.2

37. $f(x) = x^2 - x - 6$: $y$-int ______ zeros ______

38. Vertex of that $f$: ____________ max/min? ______

39. $x$-intercepts $(-3, 0)$ and $(5, 0)$: axis ____________

40. How is $h = -\dfrac{b}{2a}$ related to the axis?

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PAGE 9 — Domain and range · context

16.3 Domain and Range, Including Context

FIGURE: fig4-context-parabola-domain-range.png (full width)

41. From figure 2: domain ____________ range ____________

42. From figure 3 ($y = -x^2 + 4$): domain ____________ range ____________

43. Projectile in context: domain ____________ range ____________

44. Meaning of $(0, 0)$ ____________ $(1.5, 36)$ ____________ $(3, 0)$ ____________

45. Why equation domain ≠ context domain:

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46. Parent domain ______ range ______

PAGE 10 — Practice · domain and range

Domain and Range Practice

47. Domain and range (unrestricted):

    a) $x^2 + 6x + 5$ → ____________ b) $-2x^2 + 8x$ → ____________ c) $x^2 - 9$ → ____________ d) $-x^2 - 1$ → ____________

48. Application. Interpret domain, range, and vertex of $h(t) = -16t^2 + 48t$ (one sentence each):

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49. Application. Arch $y = -x^2 + 6x$ for $0 \le x \le 6$: vertex ______

    Domain in context ______ Range in context ______

50. Reasoning. Why is range of an upward parabola never all reals?

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51. Error analysis. Student says range of $y = x^2 - 2x - 3$ is $y \ge 0$:

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PAGE 11 — More context · exit 16.3

Context Domains · Exit 16.3

52. $R(x) = -x^2 + 40x$, $0 \le x \le 40$: vertex ______

    Domain in context ______ Range in context ______

53. $y = x^2 - 4$ unrestricted: D ______ R ______

    Restricted to $-2 \le x \le 2$: D ______ R ______

54. Technology. Confirm max near $(1.5, 36)$ for $h = -16t^2 + 48t$.

55. Sketch $y = -x^2 + 9$; shade the range.

56. Explain. How context changes domain without changing the equation:

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Exit ticket 16.3

57. $f(x) = x^2 - 8x + 12$: D ______ R ______

58. $g(x) = -x^2 + 5$: D ______ R ______

59. Projectile domain in context; meaning of each endpoint:

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60. Why unrestricted quadratics always have domain all reals:

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PAGE 12 — Vertical shifts

16.4 Transformations $f(x) + k$ and $kf(x)$

FIGURE: fig6-vertical-shift-of-parent.png (full width)

61. How $y = x^2 + 3$ comes from parent: ____________ Vertex: ______

    How $y = x^2 - 4$ comes from parent: ____________ Vertex: ______

FIGURE: fig7-vertical-stretch-of-parent.png (full width)

62. Match: $2x^2$ → ____________ $\tfrac12 x^2$ → ____________ $-x^2$ → ____________

63. Transformations for $y = -2x^2$: ____________

64. Transformations for $y = \tfrac12 x^2 - 3$: ____________

PAGE 13 — Graphing practice

Graphing by Transformations

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

65. On a blank grid: sketch $y = x^2 + 2$ and $y = -x^2 + 2$. Label vertices.

66. Explain. Why $y = (x - 3)^2$ is not an A.F.2c parent transformation here:

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67. Describe transformation; give vertex.

    a) $x^2 - 7$ → ____________ b) $4x^2$ → ____________ c) $-x^2 + 1$ → ____________ d) $\tfrac13 x^2 + 2$ → ____________

68. Graph on blank grids: $x^2 + 4$; $-3x^2$; $\tfrac12 x^2 - 2$.

PAGE 14 — Transformations practice · exit 16.4

More Transformations · Exit 16.4

69. How to tell narrower vs wider from $y = kx^2 + c$:

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70. Error analysis. Student says $x^2 - 4x$ is parent shifted left 4:

    _______________________________________________ Correct vertex: ______

71. Application. Mirror $y = \tfrac14 x^2$: wider or narrower? ______ Vertex: ______

72. Graph $y = x^2 - 4x - 5$ by characteristics (check figure 1).

73. Transformation sequence for $y = -x^2 + 4$; range: ____________

74. Technology. Compare $x^2$, $2x^2$, $\tfrac12 x^2$ in one window. Describe:

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75. Equation: parent reflected and shifted down 6: ____________

76. Sketch $y = 2x^2 - 3$; mark vertex and $y$-intercept.

Exit ticket 16.4

77. Obtain $y = x^2 - 5$ from parent: ____________

78. Obtain $y = \tfrac12 x^2$: ____________ Wider or narrower? ______

79. Vertex of $y = -x^2 + 9$: ______ max/min? ______

80. Why $f(x)+k$ and $kf(x)$ alone cannot make $y = x^2 - 6x + 5$:

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PAGE 15 — Standard and factored same graph

16.5 Standard Form, Factored Form, and the Graph

FIGURE: fig8-standard-and-factored-same-graph.png (full width)

81. Standard: ____________ Factored: ____________

    Zeros: ____________ $y$-intercept: ____________

82. Expand $(x - 1)(x - 5) =$ ____________

FIGURE: fig11-choosing-characteristics-from-forms.png (full width)

83. Standard form “free”: ____________ Factored form “free”: ____________

84. $y = (x + 2)(x - 4)$: zeros ______ axis ______ $y$-int ______ vertex ______

85. Factored form of $x^2 + 2x - 15$: ____________ Zeros: ____________

86. Explain. Expanding and factoring as reverse operations:

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PAGE 16 — Forms practice

Connecting Forms to the Graph

87. Zeros, axis, $y$-int, vertex:

    a) $(x - 2)(x - 6)$ → ____________ b) $(x + 3)(x + 1)$ → ____________ c) $-(x - 1)(x - 5)$ → ____________

88. Factored form; then sketch.

    a) $x^2 - 5x + 6$ → ____________ b) $x^2 - 9$ → ____________ c) $x^2 + x - 12$ → ____________

89. Reasoning. Zeros $-4$ and $2$; $y$-int $-8$. Factored form:

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90. Error analysis. Student writes $x^2 - 6x + 5 = (x - 6)(x + 5)$:

    Expand: ____________ Correct: ____________

PAGE 17 — Forms application · exit 16.5

Forms in Context · Exit 16.5

91. Application. $A(x) = x(10 - x)$: standard form ____________

    Vertex ______ Max area meaning ____________

92. Factored form of $x^2 - 2x - 3$: ____________ Matches figure 2? ______

93. $y = 2(x - 1)(x - 5)$: zeros vs figure 8? ______ Vertex $y$? ______

94. Technology. Confirm $x^2 - 6x + 5$ and $(x - 1)(x - 5)$ coincide.

95. Standard form: zeros $-1$, $4$; $y$-int $-4$: ____________

96. Sketch $y = (x + 2)(x - 4)$; label zeros, $y$-int, vertex.

Exit ticket 16.5

97. Factored form of $x^2 - 6x + 5$: ____________

98. From $(x - 1)(x - 5)$: zeros ______ axis ______

99. Vertex of $(x - 1)(x - 5)$: ____________

100. $y$-intercept free from ______ form; zeros free from ______ form.

PAGE 18 — Evaluating and finding $x$

16.6 Evaluating $f(x)$ and Finding $x$

FIGURE: fig9-evaluating-f-and-finding-x.png (full width)

101. $f(5)$ from graph ______ by substitution ______

102. Solve $f(x) = 3$: ____________

103. $f(2) =$ ______ $f(1) =$ ______

104. Solve $f(x) = 0$: ____________ Relation to $x$-intercepts:

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105. $h(1.5)$ for $h = -16t^2 + 48t$: ______ Meaning:

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106. Explain. Why $f(x) = c$ can give two answers:

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PAGE 19 — Evaluate / solve practice

Evaluate and Solve $f(x) = c$

107. $f(x) = x^2 - 6x + 5$

     a) $f(0)=$ ______ $f(3)=$ ______ $f(7)=$ ______ b) $f(x) = 5$: ____________ c) $f(x) = -4$: ____________

108. $g(x) = -x^2 + 4$

     a) $g(0)=$ ______ $g(1)=$ ______ b) $g(x) = 0$: ____________ c) $g(x) = 3$: ____________

109. Application. $h(t) = -16t^2 + 48t$: $h(0.5)=$ ______ meaning ____________

     $h(t) = 0$: ____________ Interpret: ____________

110. Application. $R(x) = -x^2 + 40x$: $R(10)=$ ______ meaning ____________

     $R(x) = 300$: ____________ Interpret: ____________

PAGE 20 — More evaluate · exit 16.6

Range Check · Context · Exit 16.6

111. Reasoning. Why $f(x) = x^2 - 4x + 3$ never equals $-5$:

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112. Error analysis. Student reports only $x = 0$ for $f(x) = 3$:

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113. Solve $f(x) = 8$ for $f(x) = x^2 - 4x + 3$: ____________

114. Technology. Confirm $f(5) = 8$ and inputs for $f(x) = 3$.

115. $h(t) = -16t^2 + 32t + 48$: $h(0)=$ ______ meaning ____________

     $h(t) = 48$: ____________ Reject? ____________

116. Sketch $y = x^2 - 4x + 3$ and $y = 3$; mark intersection inputs.

Exit ticket 16.6

117. $f(6)$ for $f(x) = x^2 - 4x + 3$: ____________

118. Solve $f(x) = 3$: ____________

119. Meaning of $h(1) = 32$ for $h = -16t^2 + 48t$:

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120. Why quadratic $f(x) = c$ can have two solutions, linear at most one:

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PAGE 21 — Chapter review A–B

Chapter 16 Review · Parts A–B

Part A — A.F.2b

121. For $y = x^2 - 2x - 3$: zeros ______ $y$-int ______ vertex ______

     Domain ______ Range ______

122. Projectile $h(t) = -16t^2 + 48t$: domain/range in context ______

     Vertex meaning ____________

Part B — A.F.2c

123. $y = -2x^2 + 3$ as transformations: ____________ Vertex: ______

124. Graph $y = \tfrac12 x^2 - 4$ by transformations; graph $y = x^2 - 4x - 5$ by characteristics. Strategy for each: ____________

PAGE 22 — Chapter review C–D

Chapter 16 Review · Parts C–D

Part C — A.F.2d

125. Factored form of $x^2 - 6x + 5$: ____________

     Same zeros? ______ Same $y$-int? ______

Part D — A.F.2g

126. $f(x) = x^2 - 4x + 3$: $f(5)=$ ______ $f(x) = 3$: ____________

     For $h(t) = -16t^2 + 48t$, meaning of each solution of $h(t) = 32$:

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