Algebra 1 Workbook — Chapter 5: Linear Functions: Characteristics and Forms

SOL A.F.1 (a, b, c) · Companion to Textbook Chapter 5

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 124.

PAGE 1 — Chapter opener

Chapter 5 · Linear Functions: Characteristics and Forms

Standard A.F.1 (a, b, c)

In this chapter you will:

• State the domain and range of a linear function, from an equation or a graph
• Find the slope from a graph, from two points, from a table, and from an equation
• Identify the $y$-intercept, the $x$-intercept, and the zero
• Interpret all five characteristics in a real situation, with units
• Recognize $y = x$ as the parent function and describe transformations of it
• Write a line in slope-intercept, standard, and point-slope form, and convert between them

Words to know: linear function · slope · rate of change · rise · run · $y$-intercept · $x$-intercept · zero · domain · range · parent function · transformation · reflection · vertical shift · slope-intercept form · standard form · point-slope form

Convention: an intercept is a point, written $(0,b)$ or $(a,0)$. A zero is a number. The zero is the $x$-coordinate of the $x$-intercept.

PAGE 2 — Domain and range of a line

5.1 Domain and Range of a Linear Function

FIGURE: fig1-domain-range-of-a-line.png (full width)

Fill in the blanks.

For a linear function with a nonzero slope, the domain is ____________________ and the range is ____________________.

The arrowheads on the ends of a drawn line say the graph ____________________ past the edge of the grid.

1. $f(x) = 2x - 4$ Domain: _______________________ Range: _______________________

2. What do the two arrowheads tell you that the drawn portion does not?

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PAGE 3 — The two special lines

Horizontal and Vertical

FIGURE: fig2-horizontal-line-domain-range.png (full width)

Complete the frame. A horizontal line has slope $m =$ ______. Its range is ______________________ and it has ______ zero. (one / no)

A vertical line $x = a$ ______ a function, and its slope is ____________________. (is / is not)

3. The line $y = 3$ Domain: _______________________ Range: _______________________

4. Is $x = -2$ a function? ______ What goes wrong?

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   Why does it have no slope? _______________________________________

5. $y = -4$ Domain: _______________________ Range: _______________________

6. Is $x = 5$ a function? ______ Its slope: ____________________

PAGE 4 — When the story cuts the domain down

Domain in a Real Situation

FIGURE: fig3-context-domain-and-range.png (full width)

Three questions to ask of any context.

Can the input be ____________? Can the input be a ____________? Where does the story ____________?

5. $V(x) = -2x + 8$ Domain: _______________ Range: _______________

6. Why is the domain not all real numbers, even though the domain of the equation is?

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7. Apply it. State the domain and the range, then say what each means about the cooler, with units.

   Domain means _______________________________________

   Range means _______________________________________

PAGE 5 — Practice · domain and range

Practice · Both Sets

7. Give both sets.

8. Apply it. A candle is $12$ in. tall and burns down $1.5$ in. per hour: $H(t) = 12 - 1.5t$.

   Domain: _______________ Range: _______________

   What fixes each endpoint? _______________________________________

9. Apply it. A shuttle van charges $2.50 per passenger and seats at most six.

   Domain: _______________________ Range: _______________________

   Why is the graph dots, not a segment? _______________________________________

10. Explain. Why is the range of a linear function with a nonzero slope always all real numbers?

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11. Find the error. A student says the range of $f(x) = 2x - 4$ is "all real numbers greater than $-4$."

    What went wrong? _______________________________________________

    Correct range: _______________________

PAGE 6 — Exit ticket 5.1

Exit Ticket · Lesson 5.1

Name: ________________________ Date: ____________

13. $f(x) = -3x + 11$ Domain: _______________ Range: _______________

14. Explain. How does the domain of a line drawn with arrowheads differ from the domain of a segment with two filled-in endpoints?

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PAGE 7 — Slope and the two intercepts

5.2 Slope and the Two Intercepts

FIGURE: fig4-slope-and-intercepts.png (full width)

Complete the formulas.

$m = \dfrac{\underline{\hspace{2cm}}}{\underline{\hspace{2cm}}} = \dfrac{y_2 - y_1}{\underline{\hspace{2cm}}}$

In $y = mx + b$: the slope is ______ and the $y$-intercept is the point ____________.

A positive slope ____________ from left to right; a negative slope ____________.

17. $f(x) = \tfrac34 x - 3$ Slope: ______ $y$-intercept: ____________ $x$-intercept: ____________

18. What do "rise $3$" and "run $4$" tell you, and how do they produce the slope?

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19. The graph crosses the $y$-axis at ____________, so $b =$ ______

PAGE 8 — Practice · finding slope

Practice · Slope

20. Through $(-1, 2)$ and $(3, 10)$: $m =$ ______

21. Through $(2, 7)$ and $(6, 1)$: $m =$ ______

22. $y = -5x + 8$ Slope: ______ $y$-intercept: ____________

23. Read each equation.

24. Find each slope.

PAGE 9 — Practice · intercepts

Practice · Both Intercepts

Frame. To find the $x$-intercept, set ______ equal to $0$ and solve for ______.

25. $y = 5x - 10$ $y$-intercept: ____________ $x$-intercept: ____________

26. $y = -\tfrac14 x + 3$ $y$-intercept: ____________ $x$-intercept: ____________

27. A table lists $x$: $0, 1, 2, 3$ and $y$: $-5, -1, 3, 7$.

    Slope: ______ How you got it: _______________________________________

    $y$-intercept: ____________ How you got it: _______________________________

28. Find the error. A student reads $y = 3 - 2x$ as slope $3$ and $y$-intercept $(0,-2)$.

    What went wrong? _______________________________________________

    Correct slope: ______ Correct $y$-intercept: ____________

PAGE 10 — Slope and intercept in context

What the Two Numbers Mean

28. Apply it. A phone plan costs $C(m) = 0.05m + 20$ dollars for $m$ minutes.

    The slope $0.05$ means _______________________________________

    The $y$-intercept $(0,20)$ means _______________________________________

29. Apply it. A tank drains: $W(t) = 240 - 15t$ gallons after $t$ minutes.

    Slope means _______________________________________

    $y$-intercept means _______________________________________

    $x$-intercept: ____________ It means _______________________________________

PAGE 11 — Exit ticket 5.2

Exit Ticket · Lesson 5.2

Name: ________________________ Date: ____________

31. $y = -\tfrac12 x + 6$ Slope: ______ $y$-intercept: ____________

32. Through $(-3, 8)$ and $(5, 4)$: $m =$ ______

33. $y = 3x + 12$ $y$-intercept: ____________ $x$-intercept: ____________

34. Apply it. $T(x) = 8x + 45$ dollars for $x$ tickets in one order.

    Slope means _______________________________________

    $y$-intercept means _______________________________________

PAGE 12 — The zero

5.3 The Zero of a Linear Function

FIGURE: fig5-zero-of-a-linear-function.png (full width)

Complete the frame. Three names for one fact:

the zero of $f$ is the input with output ______ · the $x$-intercept is the point ____________ · the solution of the equation ____________________

A zero is a ____________. An $x$-intercept is a ____________. (number / point)

35. $f(x) = -2x + 6$ Zero: ______ Point that names it: ____________

36. Why is $f(3) = 0$ the same statement as "the graph passes through $(3,0)$"?

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37. $f(x) = 4x - 20$ Zero: ______

38. $g(x) = -3x + 7$ Zero: ______

39. $f(x) = \tfrac34 x - 3$ Zero: ______ How the figure shows it: _______________________

40. Does $y = 3$ have a zero? ______ Why? _______________________________________

PAGE 13 — Practice · zeros

Practice · Finding Zeros

41. Find each zero.

42. Apply it. $B(w) = 320 - 40w$ dollars after $w$ weeks. Zero: ______

    It means _______________________________________

43. Apply it. The cooler, $V(x) = -2x + 8$. Zero: ______

    It means _______________________________________

    Why does the domain of the model end at exactly that input? _______________________________

44. Find the error. A student says the zero of $f(x) = 2x - 10$ is $-10$.

    What did the student find instead? _______________________

    Correct zero: ______ Correct $x$-intercept: ____________

PAGE 14 — Exit ticket 5.3

Exit Ticket · Lesson 5.3

Name: ________________________ Date: ____________

45. $f(x) = 6x + 18$ Zero: ______

46. $f(x) = -\tfrac14 x + 5$ Zero: ______

47. Apply it. A drone descends: $D(t) = 90 - 12t$ meters after $t$ seconds.

    Zero: ______ It means _______________________________________

48. Explain. Why are the zero, the $x$-intercept, and the solution of $mx + b = 0$ the same fact?

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PAGE 15 — The parent function

5.4 The Parent Function $y = x$

FIGURE: fig6-parent-function-y-equals-x.png (full width)

Complete the sentence. In $y = x$, each output equals its own ____________.

Written as $y = mx + b$, the parent is $y = \underline{\hspace{2cm}}x + \underline{\hspace{2cm}}$.

49. Slope: ______ $y$-intercept: ____________

50. The five marked points: _______________________________________

51. The triangle shows a rise of ______ over a run of ______, so $m =$ ______

52. Domain: _______________________ Range: _______________________

53. Zero: ______ Point that carries it: ____________

54. Explain. What is a parent function, and why is $y = x$ the parent of the linear family?

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PAGE 16 — Practice and exit ticket 5.4

Practice · The Parent

55. Complete the table for $y = x$.

56. Is $(-6,-6)$ on $y = x$? ______ Is $(2,-2)$? ______ Explain each.

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57. In $y = mx + b$, the parent has $m =$ ______ and $b =$ ______

58. Apply it. A club gives one hour of credit per hour volunteered: $C(h) = h$.

    Slope means _______________________________________

    $y$-intercept means _______________________________________

Exit Ticket · Lesson 5.4

59. For $y = x$ — slope: ______ $y$-intercept: ______ zero: ______ domain: ____________ range: ____________

60. Is $(-9,-9)$ on the parent? ______ Why? _______________________________________

61. Find the error. A student says the parent of the linear family is $y = 0$.

    What is wrong? _______________________________________ Correct parent: ____________

62. Explain. What makes $y = x$ the parent, and name one transformation of it.

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PAGE 17 — Transformations of the parent

5.5 Transformations of the Parent Function

FIGURE: fig7-transformations-of-the-parent.png (full width)

Two dials. Multiplying $x$ by a number changes the ____________ and leaves the $y$-intercept at ____________.

Adding a constant changes the ____________ and leaves the slope at ______.

If $|m| > 1$ the line is ____________ than the parent; if $0 < |m| < 1$ it is ____________.

A negative slope reflects the parent across the ______-axis.

63. Left panel — equations: ____________ and ____________ Slopes: ______ and ______ Shared $y$-intercept: ____________

64. Which is steeper than the parent? ____________ Less steep? ____________ How can you tell?

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65. Middle panel — transformation: ____________________ Slopes: ______ and ______

66. Right panel — what changed: ____________________ What stayed the same: ____________________

PAGE 18 — Practice · describing transformations

Practice · Against the Parent

67. $y = 5x$ — effect on slope: ____________________ effect on $y$-intercept: ____________________

68. $y = x - 7$ — effect on slope: ____________________ effect on $y$-intercept: ____________________

69. Describe each as a transformation of $y = x$.

70. Steeper: $y = 4x$ or $y = -6x$? ____________ What did you compare? _______________________

71. Parent with slope multiplied by $-3$, then shifted down $2$: $y =$ _______________________

72. Does reflecting the parent across the $x$-axis change the $y$-intercept? ______ Why?

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PAGE 19 — Reasoning about transformations

Why the Dials Are Independent

73. The parent shifted up $6$.

    Slope: ______ $y$-intercept: ____________ Equation: ____________ Zero: ______

74. Complete both rows, then say what the table shows.

What it shows: _______________________________________________

75. Explain. Why does changing $b$ never change the slope? Say what happens to every output.

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76. Find the error. A student says $y = -4x$ is the parent shifted down $4$.

    What went wrong? _______________________________________________

    Correct description: _______________________________________

PAGE 20 — Exit ticket 5.5

Exit Ticket · Lesson 5.5

Name: ________________________ Date: ____________

77. $y = 7x$ — slope: ______ $y$-intercept: ____________ Description: _______________________

78. $y = x - 9$ — slope: ______ $y$-intercept: ____________ Description: _______________________

79. $y = -\tfrac12 x + 4$ — slope: ______ $y$-intercept: ____________ Description: _______________________

80. Explain. In $y = mx + b$, which number does a slope-changing transformation affect, and which does a vertical shift affect? What does each leave alone?

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PAGE 21 — Three forms, one line

5.6 Three Equivalent Forms

FIGURE: fig8-three-forms-one-line.png (full width)

Complete the table.

81. The three equations in the figure:

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82. Which form shows the slope and $y$-intercept at a glance? ____________________

    Slope: ______ $y$-intercept: ____________

83. Which form names a point the other two never mention? ____________________ The point: ____________

PAGE 22 — What standard form is good for

Intercepts the Cheap Way

FIGURE: fig9-standard-form-intercepts.png (full width)

Frame. In $Ax + By = C$, let $y = 0$ to find the ______-intercept and let $x = 0$ to find the ______-intercept.

84. $3x + 4y = 12$ $x$-intercept: ____________ $y$-intercept: ____________

85. Convert $3x + 4y = 12$ to slope-intercept form: _______________________

    Slope: ______ $y$-intercept: ____________

FIGURE: fig10-vertical-line-standard-form.png (full width)

86. Why can $x = 3$ not be written in slope-intercept form?

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    In standard form: _______________________

PAGE 23 — Practice · converting

Practice · Converting Between Forms

87. Convert to slope-intercept form.

88. Convert to standard form with integer coefficients.

89. Convert to slope-intercept form.

    a) $y - 5 = 3(x - 2)$: _______________________

    b) $y + 1 = -2(x - 4)$: _______________________

    c) $y - 7 = \tfrac12(x + 6)$: _______________________

PAGE 24 — Practice · reading and checking

Practice · Reading Each Form

90. From $y - 4 = -5(x + 3)$: slope ______ point ____________

91. $5x - 3y = 15$ $x$-intercept: ____________ $y$-intercept: ____________

92. $2x + 7y = 14$ $x$-intercept: ____________ $y$-intercept: ____________

93. Check the point $(5,6)$ in all three forms of the same line.

96. Find the error. A student converts $3x - y = 6$ and writes $y = 3x + 6$.

    What went wrong? _______________________________________________

    Correct form: _______________________

PAGE 25 — Forms in context

Choosing a Form

94. Apply it. Shirts at $4 and hats at $6 toward a $120 goal: $4x + 6y = 120$.

    $x$-intercept: ____________ It means _______________________________________

    $y$-intercept: ____________ It means _______________________________________

95. Apply it. A taxi charges $3 to start plus $2.50 per mile.

    Best form: ____________________ Equation: _______________________

    The $3 is the ____________________ The $2.50 is the ____________________

PAGE 26 — Exit ticket 5.6

Exit Ticket · Lesson 5.6

Name: ________________________ Date: ____________

97. $4x - y = 10$ → _______________________ Slope: ______ $y$-intercept: ____________

98. $y = -3x + 8$ → standard form: _______________________

99. $y - 6 = 4(x - 1)$ → _______________________

100. $6x + 5y = 30$ $x$-intercept: ____________ $y$-intercept: ____________

PAGE 27 — Chapter 5 review · domain, range, zeros

Chapter 5 Review

Part A · Domain, range, and zeros

101. $f(x) = -7x + 2$ Domain: ____________ Range: ____________ Zero: ______

102. $y = 9$ Domain: ____________ Range: ____________ Zero: ______

103. Is $x = -1$ a function? ______ Slope: ____________ Why?

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FIGURE: fig3-context-domain-and-range.png (half width)

104. $V(x) = -2x + 8$ Domain: ____________ Range: ____________ Zero: ______

     Each, in context: _______________________________________________

105. Apply it. An elevator descends from $45$ m at $3$ m per second: $E(t) = 45 - 3t$.

     Zero: ______ Domain: ____________ Range: ____________

106. Explain. Why does a linear function with a nonzero slope have exactly one zero? Which kind has none?

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PAGE 28 — Chapter 5 review · slope and intercepts

Chapter 5 Review (continued)

Part B · Slope and intercepts

107. $y = 2x - 6$ Slope: ______ $y$-intercept: ____________ $x$-intercept: ____________

108. Through $(-4, 9)$ and $(2, -3)$: $m =$ ______

109. $y = -3x + 15$ $y$-intercept: ____________ $x$-intercept: ____________

110. Apply it. A pool fills: $G(t) = 30t + 150$ gallons after $t$ minutes.

     Slope means _______________________________________

     $y$-intercept means _______________________________________

FIGURE: fig4-slope-and-intercepts.png (half width)

111. Read all four from the figure, and say which part of the picture gives each.

112. Find the error. A student says the $x$-intercept of $y = 4x + 8$ is $(0,8)$.

     What went wrong? _______________________ Correct intercepts: ____________ and ____________

PAGE 29 — Chapter 5 review · parent and transformations

Chapter 5 Review (continued)

Part C · The parent function and transformations

FIGURE: fig7-transformations-of-the-parent.png (full width)

113. $y = 6x - 1$ — slope effect: ____________________ $y$-intercept effect: ____________________

114. $y = -x + 5$ — slope effect: ____________________ $y$-intercept effect: ____________________

115. For each panel, name what changed and what stayed the same.

116. Two transformations with $y$-intercept $(0,0)$ but different steepness:

     _______________________ and _______________________ Steeper: ____________

117. Explain. Why is every linear function a transformation of $y = x$? Which straight line is not covered, and why?

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PAGE 30 — Chapter 5 review · three forms

Chapter 5 Review (continued)

Part D · Three equivalent forms

118. $x + 4y = 8$ → _______________________ Slope: ______ $y$-intercept: ____________

119. $y = 5x - 3$ → standard form: _______________________

120. $y + 2 = -3(x - 1)$ → _______________________

121. $4x - 5y = 20$ $x$-intercept: ____________ $y$-intercept: ____________

FIGURE: fig10-vertical-line-standard-form.png (half width)

122. Which forms can write $x = 3$? ____________________ Which can write $y = -2$? ____________________

     The difference: _______________________________________________

PAGE 31 — Chapter 5 review · mixed application

Chapter 5 Review (continued)

Part E · Application

123. Apply it. A rideshare charges $C(m) = 1.75m + 4$ dollars for $m$ miles.

     Slope: ______ $y$-intercept: ____________ Zero: ______

     Slope means _______________________________________

     $y$-intercept means _______________________________________

     Why does the zero have no meaning here? _______________________________________

124. Apply it. Tickets at $5 and programs at $8 toward $200: $5x + 8y = 200$.

     $x$-intercept: ____________ It means _______________________________________

     $y$-intercept: ____________ It means _______________________________________

     Slope-intercept form: _______________________

     What that form reveals: _______________________________________

PAGE 32 — Blank grids

Graph Paper

FIGURE: fig11-blank-grids.png (full page)

Use these grids for any sketch a teacher assigns alongside this chapter: plotting the two intercepts of a line you have just found, drawing a slope triangle to check a rate of change, or drawing the parent $y = x$ beside one of its transformations to compare steepness.

Two reminders for every sketch:

• A line with no restricted domain is drawn all the way across the grid, with an arrowhead on each end.
• A line that models a situation with a beginning and an end is drawn as a segment with a filled-in point at each endpoint, as in the cooler graph on page 4.

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 items are placed by figure rather than by number, so that every item on a page can be answered from the figure on that page. Page 3 runs 3, 4, 14, 15; page 6 runs 13 then 16; page 9 places 30 after 27; page 16 runs 55–58 then the exit ticket. The numbers still match the textbook exactly, and the answer key is in numerical order.
• Figure widths: fig1, fig3, fig4, fig5, fig6, fig8, fig9, and fig10 are full width on first appearance. fig2-horizontal-line-domain-range.png and fig7-transformations-of-the-parent.png are wide two- and three-panel images — set them full width and do not place text beside them, or the panels shrink below readable. fig11-blank-grids.png takes a full page on its own. fig3, fig4, fig7, and fig10 repeat in the review; repeat the image rather than asking students to flip back.
• The fig4 slope triangle is the load-bearing picture of Lesson 5.2 and of item 111. Do not reduce it below half a page — the dashed rise-and-run labels and the two intercept labels all have to stay legible at once.
• Blank work space: items 6, 10, 12, 16, 18, 36, 48, 54, 62, 72, 75, 80, 103, 106, 117, and 122 are written explanations — give each at least four ruled lines, not a single blank. Items 8, 9, 11, 28, 29, 34, 42, 43, 47, 58, 94, 95, 110, 123, and 124 are applications with two or more parts; leave two lines per part.
• Points versus numbers: every blank for an intercept must be wide enough for an ordered pair with a comma and parentheses; every blank for a zero or a slope is short. Setting the two blank widths differently is a deliberate cue, since confusing the point with the number is this chapter's most common error.
• Fractions: items 23b, 26, 31, 41d, 46, 69d, 79, 85, 87b, 87c, 87d, 88c, 88d, 89c, and 118 have fractional slopes. Set them as stacked fractions, not slashes, so that $-\tfrac23$ never reads as $-2/3$ ambiguously across a line break.
• Negative signs: use the typographic minus (−), matching the figures, not a hyphen.
• Currency: dollar amounts appear on pages 5, 10, 11, 13, 25, and 31. Set them as plain text ($2.50, $20, $320, $4, $6, $120, $3, $5, $8, $200), not inside a math box.