Algebra 1 Workbook — Chapter 7: Graphing and Evaluating Linear Functions

SOL A.F.1 (f, g, h) · Companion to Textbook Chapter 7

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 7 · Graphing and Evaluating Linear Functions

Standard A.F.1 (f, g, h)

In this chapter you will:

• Graph a line by hand three ways: slope stepping, both intercepts, and a table you choose
• Graph $y = c$ and $x = a$, and graph a context on axes with two different scales
• Graph with technology, choosing the viewing window from what you already know
• Given $x$, determine $f(x)$ — from the equation and from the graph
• Given $f(x)$, recover $x$ — from the equation and from the graph
• Compare linear functions written algebraically, graphically, in tables, and in contexts

Words to know: slope stepping · rise · run · intercept method · table of values · scale · viewing window · standard window · table feature · determine $f(x)$ · recover $x$ · representation · rate of change · initial value

Convention: slope is always a fraction before you step. An integer slope $m$ is $\tfrac{m}{1}$, so the run is $1$.

Convention: the number inside $f(\ \ )$ is an input. In "$f(x) = 10$" the $10$ is outside the parentheses, so it is an output — set the rule equal to it and solve.

Convention: the calculator confirms, it does not decide. If your algebra and your graph disagree, one is wrong and you are not finished.

PAGE 2 — Slope stepping

7.1 Graphing from Slope-Intercept Form

FIGURE: fig1-graphing-from-slope-intercept.png (full width)

Fill in the procedure.

1. Plot the $y$-intercept, the point ( ______ , ______ ).

2. Write the slope as a fraction. The denominator is the ____________ and the numerator is the ____________.

3. Step, and plot. Step ____________ and plot again.

4. Draw through all three points, with ____________ on both ends.

5. $f(x) = \tfrac23 x - 4$ First point: ____________ Slope: ______ Other two points: ____________ and ____________

6. What do "up 2" and "right 3" refer to, and which is the denominator?

   ---

7. Verify $f(6) = 0$: _______________________ Which intercept is that point? ____________

PAGE 3 — Stepping both ways

Negative Slopes Go Two Directions

FIGURE: fig2-two-directions-from-the-intercept.png (full width)

Complete the frame. For $h(x) = -2x + 3$, the slope written as a fraction is $\dfrac{\underline{\hspace{2cm}}}{\underline{\hspace{2cm}}}$, so the step is right ______ and down ______.

Reversing both signs gives left ______ and up ______, and lands on the ____________ line.

4. Write the slope of $h$ as a fraction and state the step: _______________________

5. What point does "left 1, up 2" produce? ____________ Verify: _______________________

6. Graph $y = 3x - 2$. First point: ____________ Two more: ____________ and ____________

PAGE 4 — Practice · first point and step

Practice · Plot, Then Step

7. Complete the table.

11. Four points on $y = -\tfrac34 x + 2$, using inputs on both sides of the $y$-axis:

    ---

PAGE 5 — Graph them

Graph on the Grids

FIGURE: fig12-blank-graphing-grids.png (full width)

8. Graph $y = \tfrac12 x - 3$ on Grid A. Three integer points: ____________ ____________ ____________

9. Graph $y = -3x + 6$ on Grid B. Three integer points: ____________ ____________ ____________

10. Graph $y = 2x$ and $y = 2x + 3$ on Grid C.

    First point of each: ____________ and ____________

    Same: _______________________ Different: _______________________

11. Graph $y = -2x + 1$ on Grid D. $y$-intercept: ____________ Two more: ____________ ____________

PAGE 6 — Practice · reasoning and errors

Practice · Think It Through

10. Find the error. A student graphs $y = \tfrac25 x + 1$ by going up $5$ and right $2$.

    What went wrong? _______________________________________________

    What did the student actually graph? ____________ Correct step: ____________

11. Explain. Why does stepping left and up by the reversed slope land on the same line? Use $h(x) = -2x + 3$.

    ---

12. Find the error. A student says the first point of $y = 3x - 4$ is $(0,3)$.

    What went wrong? _______________________ Correct first point: ____________ Step: ____________

PAGE 7 — Slope stepping in context

Graphing a Real Model

13. Apply it. A printer starts with $500$ sheets and uses $25$ per job: $P(j) = 500 - 25j$.

    First point: ____________ Step: ____________

    Why is one sheet per cell a bad grid? _______________________________________

14. Apply it. A battery falls from full: $B(t) = 100 - 8t$ percent after $t$ hours.

    First point: ____________ Step: ____________

    How far right until the graph reaches $0$? ____________

PAGE 8 — Exit ticket 7.1

Exit Ticket · Lesson 7.1

Name: ________________________ Date: ____________

17. $y = -\tfrac43 x + 5$ First point: ____________ Step: ____________

18. Three integer points on $y = \tfrac13 x - 1$: ____________ ____________ ____________

19. Why are two points enough, and why plot a third anyway?

    ---

PAGE 9 — The intercept method

7.2 Graphing from Both Intercepts

FIGURE: fig3-graphing-with-intercepts.png (full width)

Complete the frame. To find the $y$-intercept, let $x =$ ______. To find the $x$-intercept, let $y =$ ______.

21. $3x - 4y = 12$ The two substitutions: _______________________ The two points: ____________ and ____________

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

    Does the slope agree with the two plotted points? ______ How? ____________________

PAGE 10 — Choosing table inputs

Tables You Choose on Purpose

FIGURE: fig4-table-of-values-graph.png (full width)

The rule. Choose inputs that are multiples of the ____________________ of the slope.

23. Which five inputs were chosen, and why those?

    ---

24. What would go wrong with the inputs $1$, $3$, and $5$?

    ---

25. Four whole-number-output rows for $f(x) = \tfrac23 x + 2$:

How did you choose? _______________________________________

29. Four whole-number-output rows for $f(x) = -\tfrac14 x + 5$:

PAGE 11 — The two special lines

No Stepping Required

FIGURE: fig5-horizontal-and-vertical-as-functions.png (full width)

25. How do you graph $y = -3$? _______________________________________

    How do you graph $x = 4$? _______________________________________

    Which is a function? ____________

26. Graph $y = -1$ and $x = 3$ on one grid.

39. $y = 6$ and $x = -5$: which is a function? ____________ Slope of each: ______ and ______

PAGE 12 — Practice · intercept method

Practice · Two Points, One Line

27. Give both intercepts.

33. Decide. For $4x - 5y = 20$, which is faster — intercepts or converting first?

    Intercepts: ____________ and ____________ Faster method: ____________ Why? ____________________

34. Find the error. A student graphs $2x + 3y = 12$ from "slope $2$, intercept $3$."

    What went wrong? _______________________ A correct graph, as two points: ____________ and ____________

35. $5x + 2y = 20$ Both points: ____________ and ____________

36. Four inputs for $f(x) = \tfrac34 x - 1$ with whole outputs: ____________ Outputs: ____________

PAGE 13 — Choosing a scale

Graphing a Context

FIGURE: fig6-context-graph-with-a-scale.png (full width)

The routine. (1) Find the inputs the story ____________. (2) Find the largest and smallest ____________. (3) Pick a ____________ that fits and keeps points on corners. (4) Label both axes with the quantity and its ____________. (5) Draw a ____________, not a line.

26. One cell across = ____________ One cell up = ____________ Cost of a $4$-hour rental: ____________

27. Apply it. $C(3) =$ ____________ $C(6) =$ ____________

    Why is $5$ dollars per cell better than $4$? _______________________________________

PAGE 14 — Practice · scale in context

Practice · Pick the Scale

32. Apply it. Car rental: $D(m) = 0.30m + 45$ for $0 \le m \le 200$.

    $x$-axis: ______ to ______, one cell = ____________

    $y$-axis: ______ to ______, one cell = ____________

    Endpoints: ____________ and ____________

33. Apply it. Wristbands at $8 and shirts at $12 toward $240: $8x + 12y = 240$.

    $x$-intercept: ____________ It means _______________________________________

    $y$-intercept: ____________ It means _______________________________________

34. Apply it. $W(t) = 400 - 25t$ gallons for $0 \le t \le 16$.

    $x$-axis one cell = ____________ $y$-axis one cell = ____________

PAGE 15 — Exit ticket 7.2

Exit Ticket · Lesson 7.2

Name: ________________________ Date: ____________

40. Apply it. A pool fills: $G(t) = 12t + 30$ gallons, $0 \le t \le 20$.

    $x$-axis: one cell = ____________ $y$-axis: one cell = ____________

    Endpoints: ____________ and ____________

PAGE 16 — The viewing window

7.3 Graphing with Technology

FIGURE: fig7-choosing-a-calculator-window.png (full width)

Set the window from the algebra, in this order.

1. Find the ____________-intercept. 2. Find the ____________. 3. Set the window generously around ____________, and choose a ____________ for each axis.

2. Why is the standard window useless for $f(x) = 25x - 140$?

   ---

3. $y$-intercept: ____________ Zero: ____________ Which window number did each fix? ____________________

4. One cell across = ____________ One cell up = ____________

5. A window showing both intercepts of $f(x) = 12x + 96$:

   $x$: ______ to ______, scale ______ $y$: ______ to ______, scale ______

   Intercepts: ____________ and ____________

PAGE 17 — The table feature

Bracketing a Zero

45. Table for $f(x) = 25x - 140$, stepping by $1$:

What does the change tell you? _______________________________________

46. What can a graph verify, and what can it not establish exactly?

    ---

47. Table for $f(x) = -6x + 15$:

The zero is between ______ and ______.  Exactly, the zero is ____________.

PAGE 18 — Practice · choosing windows

Practice · Set the Window

47. Complete the table. State the intercepts you used.

51. Find the error. A student graphs $f(x) = 0.1x - 4$ in the standard window and writes "no $x$-intercept."

    What went wrong? _______________________ The zero: ____________

52. A calculator reports the zero of $f(x) = 3x - 10$ as $3.3333333$.

    Exact zero: ____________ What is the difference? _______________________________________

PAGE 19 — Practice · algebra first, screen second

Practice · Verify, Don't Guess

49. The screen shows $y = 2x + 5$ falling from left to right. Most likely cause?

    _______________________ How would you check? _______________________

50. Solve $4x - 7 = 13$: ____________ What point should be on the graph? ____________

51. Explain. Two students graph the same function and get different-looking pictures. Why?

    _______________________ What should they compare first? ____________________

52. Explain. Why must you know something about the function before choosing a window? Name the two numbers to compute first.

    ---

PAGE 20 — Technology in context

Apply It with Technology

48. Apply it. Tuition savings: $T(w) = 35w + 400$ for $0 \le w \le 52$.

    Window: $x$ ______ to ______, scale ______ $y$ ______ to ______, scale ______

    $T(52) =$ ____________

49. Apply it. Moving service: $C(m) = 1.2m + 30$ dollars for $m$ miles.

    Mileage costing $90: ____________ The graphical check: _______________________________________

PAGE 21 — Exit ticket 7.3

Exit Ticket · Lesson 7.3

Name: ________________________ Date: ____________

57. Window for $f(x) = -15x + 120$: $x$ ______ to ______ $y$ ______ to ______ Intercepts used: ____________

58. $f(7) =$ ____________ The graphical check: _______________________________________

59. The rule when algebra and a graph disagree: _______________________________________

    Two things to check first: ____________________ and ____________________

60. Find the error. A calculator says the zero of $f(x) = 5x - 15$ is $2.9999999$, and the student writes that.

    The problem: _______________________ Exact zero: ______

PAGE 22 — Given $x$, find $f(x)$

7.4 Determining $f(x)$

FIGURE: fig8-reading-f-of-x-from-a-graph.png (full width)

The graphical path. From the ____________-axis, travel ____________ to the line, then ____________ to the ____________-axis.

61. $f(4) =$ ______ Describe the two travels: _______________________________________

62. Confirm algebraically: $f(4) =$ _______________________

63. $f(-1) =$ _______________________

64. $f(x) = -3x + 8$

68. $f(x) = \tfrac12 x - 6$

PAGE 23 — Given $f(x)$, recover $x$

Recovering the Input

FIGURE: fig9-recovering-x-from-f-of-x.png (full width)

The graphical path. From the ____________-axis, travel ____________ to the line, then ____________ to the ____________-axis.

64. Which input produced $f(x) = -7$? ______ Describe the path: _______________________

65. Confirm by solving: _______________________________________

66. How do items 61 and 64 differ, and what do the answers share?

    ---

67. $f(x) = -3x + 8$. Solve for $x$.

70. $g(x) = 4x + 1$. Solve for $x$.

PAGE 24 — Practice · both directions off a graph

Practice · Read It Both Ways

71. From the graph of $f(x) = 2x - 3$: $f(0) =$ ______ $f(1) =$ ______ $f(2) =$ ______

    Confirm all three algebraically: _______________________________________

72. From that same graph: $f(x) = 1$ at $x =$ ______ $f(x) = -3$ at $x =$ ______

    Which axis did you start from? ____________________

73. Find the error. Asked for $x$ when $f(x) = 10$ for $f(x) = 2x + 4$, a student answers $24$.

    What went wrong? _______________________ Which question was answered? ____________________ Correct $x$: ______

74. Explain. Why is recovering $x$ always solving an equation, while finding $f(x)$ never is?

    ---

PAGE 25 — Both directions in context

Apply It Both Ways

73. Apply it. Rideshare: $C(m) = 2.5m + 3.5$ dollars for $m$ miles.

    $C(12) =$ ____________ Sentence: _______________________________________

    A $28.50 fare was ______ miles. Sentence: _______________________________________

74. Apply it. A room cools: $T(h) = 68 - 4h$ degrees after $h$ hours.

    $T(6) =$ ____________ (units: ____________) Reaches $20$ degrees at $h =$ ______ (units: ____________)

75. Technology. $f(x) = 0.75x - 2.5$. $f(8) =$ ____________

    How the table feature confirms it: _______________________________________

76. Apply it. From the bike-rental graph, $C(h) = 5h + 20$.

    $C(5) =$ ____________ $55 buys ______ hours. Confirm each algebraically: ____________________

PAGE 26 — Exit ticket 7.4

Exit Ticket · Lesson 7.4

Name: ________________________ Date: ____________

79. $f(x) = -2x + 9$. $f(-3) =$ ______ $f(x) = 1$ at $x =$ ______

80. From the figure: $f(4) =$ ______ $f(x) = 5$ at $x =$ ______

    Why does one marked point answer both? _______________________________________

81. Apply it. $B(w) = 250 - 18w$ dollars after $w$ weeks. $B(9) =$ ____________ $34 at week ______

82. Describe the two dashed paths: which axis each starts from and ends at.

    ---

PAGE 27 — One function, four ways

7.5 Comparing the Four Representations

FIGURE: fig11-four-representations.png (full width)

83. Algebraic rule: _______________________

84. What does the table make visible that the equation does not?

    ---

85. What does the graph show faster than the other three?

    ---

86. $W(10) =$ ____________ Representation used: ____________ Why not the others? ____________________

PAGE 28 — Comparing two models

Two Plans on One Grid

FIGURE: fig10-comparing-two-plans.png (full width)

87. At $2$ hours: Plan A costs ____________, Plan B costs ____________. Cheaper: ______

    At $6$ hours: Plan A costs ____________, Plan B costs ____________. Cheaper: ______

88. Confirm the crossing with one equation in one variable:

    _______________________________________________ Crossing: ____________

89. Larger $y$-intercept: ____________ It means _______________________________________

    Why does that plan still win in the long run? _______________________________________

90. One sentence a customer could act on, with units:

    ---

PAGE 29 — Practice · build all four

Practice · All Four Forms

89. A lawn service charges a $15 trip fee plus $6 per hour.

    Equation: _______________________

Graph: _______________________________________

Contextual sentence naming the slope with units: _______________________________________

96. Apply it. A scooter costs $4 to unlock plus $0.25 per minute.

    Equation: _______________________

Graph, with a scale for each axis: _______________________________________

The slope means _______________________________________

PAGE 30 — Practice · compare across forms

Practice · Compare and Contrast

90. $f(x) = 4x + 20$ against the table $(0,30)$, $(2,38)$, $(4,46)$, $(6,54)$.

Do the graphs meet? ______  Why? _______________________________________

91. $f(x) = -5x + 40$ against a line falling from $(0,25)$ to $(5,0)$.

    Slopes: ______ and ______ Initial values: ______ and ______ What follows? ____________________

92. Rank by rate of change: the equation $y = 3x - 1$; a table rising $4$ per $1$; a seller earning $2.50 per item.

    Ranking: _______________________ What made each rate visible? ____________________

93. Find the error. From the table $(1,7)$, $(2,10)$, $(3,13)$, a student reports the $y$-intercept $(0,7)$.

    What went wrong? _______________________ Slope: ______ Intercept: ____________ Equation: ____________

94. A table says $f(3) = 11$; the equation $f(x) = 3x + 1$ says $10$. Conclude what, and do what?

    ---

PAGE 31 — Practice · choosing a representation

Which Form Would You Reach For?

93. Apply it. Gym A: $25 a month plus $5 per class. Gym B: the table $(0,10)$, $(2,30)$, $(4,50)$, $(6,70)$.

    Gym B's equation: _______________________

    $3$ classes: A costs ____________, B costs ____________

    They agree at ______ classes, at a cost of ____________

    Better for $6$ classes a month: ____________ Why? ____________________

94. Decide. Zero, exactly: ____________________ Trend, instantly: ____________________

    Justify each: _______________________________________

PAGE 32 — Exit ticket 7.5

Exit Ticket · Lesson 7.5

Name: ________________________ Date: ____________

99. $f(x) = -4x + 18$

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

100. $y = 7x + 2$ against the table $(0,9)$, $(1,14)$, $(2,19)$.

     Greater rate: ______ Greater start: ______ They agree at $x =$ ______

101. Apply it. Which representation for each person, and why?

     Exact cost of $17$ hours: ____________________

     When two plans cross: ____________________

102. One thing each of the four shows better than the other three:

     Algebraic: ____________________ Graphical: ____________________

     Tabular: ____________________ Contextual: ____________________

PAGE 33 — Chapter 7 review · graphing without technology

Chapter 7 Review

Part A · Without technology

103. $y = \tfrac23 x - 4$ First point: ____________ Step: ____________ Two more: ____________ ____________

104. $4x - 3y = 12$ Both intercepts: ____________ and ____________

105. Four whole-output rows for $f(x) = -\tfrac25 x + 3$:

 How chosen? _______________________

106. $y = -4$ and $x = 1$

107. Apply it. $S(d) = 60d + 120$ miles over a $10$-day route.

     $x$-axis one cell = ____________ $y$-axis one cell = ____________ Endpoints: ____________ and ____________

PAGE 34 — Chapter 7 review · with technology

Chapter 7 Review (continued)

Part B · With technology

108. Window for $f(x) = 18x - 144$: $x$ ______ to ______ $y$ ______ to ______

     Intercepts used: ____________ and ____________

109. Solve $-3x + 11 = 2$: ______ What to look for on the graph: _______________________________________

110. Find the error. Algebra gives $x = 7$; the graph looks like $x = 3$; the student writes "about $5$."

     What is wrong? _______________________________________

     What should the student do? _______________________________________

PAGE 35 — Chapter 7 review · both directions

Chapter 7 Review (continued)

Part C · Determining $f(x)$ and recovering $x$

111. $f(x) = 5x - 12$. $f(4) =$ ______ $f(-2) =$ ______

112. Same function. $f(x) = 13$ at $x =$ ______ $f(x) = -12$ at $x =$ ______

FIGURE: fig9-recovering-x-from-f-of-x.png (half width)

113. $f(-2) =$ ______ $f(x) = -7$ at $x =$ ______ Why one point answers both: ____________________

114. Apply it. Profit $R(x) = 9.5x - 40$ dollars on $x$ items.

     $R(20) =$ ____________ (units: ____________) A $245 profit takes ______ items.

115. Explain. Why are the two directions the same picture travelled in opposite directions?

     ---

PAGE 36 — Chapter 7 review · comparing, and mixed application

Chapter 7 Review (continued)

Part D · Comparing representations

116. One characteristic each representation displays fastest:

117. $f(x) = 2x + 30$ against the table $(0,45)$, $(3,51)$, $(6,57)$.

     Rates: ______ and ______ Initial values: ______ and ______ Do they meet? ______

Part E · Mixed application

119. Apply it. Courier: $D(x) = 2.25x + 6$ dollars for $x$ pounds, $0 \le x \le 20$.

     Scale and by-hand graph: _______________________________________

     $D(12) =$ ____________ A $42 package weighs ______ pounds.

     Slope means ____________________ $y$-intercept means ____________________

120. Apply it. A candle is $15$ in. tall and burns $0.75$ in. per hour: $H(t) = 15 - 0.75t$.

 Zero: ______  It means _______________________________________

 Scale for a by-hand graph: ____________________

 How a calculator confirms the zero: _______________________________________

 Best representation for the height at exactly $7$ hours: ____________  Why? ____________________