Chapter 8 — Linear Functions and $y = mx + b$

Standard: 8.PFA.3 — The student will represent and solve problems, including those in context, by using linear functions and analyzing their key characteristics (the value of the y-intercept (b) and the coordinates of the ordered pairs in graphs will be limited to integers).

By the end of this chapter you will be able to:

• Describe how adding a constant $b$ to a proportional relationship $y = mx$ translates its line vertically (8.PFA.3a)
• Name the key characteristics of a linear function — slope $m$, y-intercept $b$, and the independent and dependent variables (8.PFA.3b)
• Graph a linear function from an equation, a table, or a situation in context (8.PFA.3c)
• Create a table of values for a linear function from a graph, from $y = mx + b$, or from a context (8.PFA.3d)
• Write the equation $y = mx + b$ from a graph, a table, or a situation in context (8.PFA.3e)
• Invent a plausible real-world context for a linear function you are handed (8.PFA.3f)

Lessons: 8.1 Adding a Constant: Translating $y = mx$ · 8.2 Key Characteristics of a Linear Function · 8.3 Graphing a Linear Function from an Equation · 8.4 Tables of Values for Linear Functions · 8.5 Writing $y = mx + b$ from a Graph or a Table · 8.6 Linear Functions in Context · 8.7 Creating a Context for a Linear Function

Integer note. This standard puts a bound on the pictures: the value of the y-intercept $b$ and the coordinates of every plotted ordered pair are limited to integers. The slope $m$ may be a fraction — that is exactly what "rise over run" is for — but when you choose points to plot, choose the ones that land on grid corners. Lesson 8.4 shows you how to pick those $x$-values on purpose.

Numbering note. Item numbers run straight through the chapter, from 1 in Lesson 8.1 to 172 at the end of the review. They do not restart at each lesson.

Lesson 8.1 — Adding a Constant: Translating $y = mx$

Where this chapter starts

In Grade 7 you studied proportional relationships. Their equation is

$$y = mx,$$

and their graph is always a straight line through the origin, because substituting $x = 0$ gives $y = m(0) = 0$. The number $m$ is the slope, the constant rate of change: the amount $y$ changes for each increase of $1$ in $x$.

This chapter takes that line and adds one number to it. That single change produces the linear function

$$y = mx + b,$$

which is called slope-intercept form. Everything in Grade 8 algebra readiness runs through this equation, so the first job is to see exactly what the new number $b$ does.

Adding $b$ slides the whole line up or down

Start with $y = 2x$ and compare it to $y = 2x + 3$ at the same $x$-values.

Every entry in the third column is exactly $3$ more than the entry beside it. That is not a coincidence — it is arithmetic. Whatever $2x$ works out to, adding $3$ raises the result by $3$.

Since the $y$-coordinate of every point rises by $3$ while its $x$-coordinate does not move at all, every point slides straight up $3$ units. Sliding every point of a figure the same distance in the same direction is a translation. So:

Adding a constant $b$ to $y = mx$ translates the line vertically — up $b$ units when $b$ is positive, down $|b|$ units when $b$ is negative.

The figure shows all three lines. The blue line is $y = 2x$ translated up $3$; the red line is $y = 2x$ translated down $4$, because $y = 2x - 4$ is $y = 2x + (-4)$.

The translation does not change the slope

Look at the three lines again. They never cross. They cannot cross, because at every single $x$-value the blue line sits exactly $3$ above the black one and the red line exactly $4$ below it — the gap never shrinks.

Lines with the same slope and different y-intercepts are parallel. Translating a line vertically moves it without turning it, so the slope survives the move untouched: all three lines still rise $2$ for every run of $1$.

Here is the same idea with a fractional slope, $m = \tfrac{1}{2}$.

Every one of those lines rises $1$ for every run of $2$. Notice which points are marked: only the ones at even $x$-values, because those are where a line of slope $\tfrac{1}{2}$ passes through a grid corner. A fractional slope is allowed; a plotted point with a fractional coordinate is not.

Where the new line crosses the y-axis

Substitute $x = 0$ into $y = mx + b$:

$$y = m(0) + b = b.$$

So the line passes through $(0, b)$ no matter what $m$ is. That point is the y-intercept, and $b$ is its $y$-coordinate. In the first figure the three y-intercepts are $(0, 3)$, $(0, 0)$, and $(0, -4)$ — read them straight off the equations.

This also tells you when a linear function is not proportional. A proportional relationship must contain $(0,0)$. If $b \neq 0$, the line misses the origin, so $y = 2x + 3$ is linear but not proportional.

Worked examples

Example 1 — Naming the translation

Describe how the graph of $y = 4x + 6$ is related to the graph of $y = 4x$.

Both have slope $4$. The constant $+6$ raises every $y$-value by $6$.

Answer: It is the line $y = 4x$ translated up $6$ units.

Example 2 — A negative constant

Describe how the graph of $y = -x - 5$ is related to the graph of $y = -x$.

Rewrite the constant as an addition: $y = -x + (-5)$. Adding $-5$ lowers every $y$-value by $5$.

Answer: It is the line $y = -x$ translated down $5$ units.

Example 3 — Writing the equation of a translated line

The line $y = \tfrac{2}{3}x$ is translated down $7$ units. Write the equation of the image.

Translating down $7$ means subtracting $7$ from every $y$-value, so $b = -7$. The slope does not change.

Answer: $y = \tfrac{2}{3}x - 7$

Example 4 — How far apart are two parallel lines?

How far apart, vertically, are $y = 2x + 3$ and $y = 2x - 4$?

At any $x$, the difference of the $y$-values is $(2x + 3) - (2x - 4) = 7$.

Answer: $7$ units, at every $x$-value. The lines are parallel.

Example 5 — Deciding whether a relationship is proportional

Is $y = 5x - 1$ proportional? Justify your answer.

Substituting $x = 0$ gives $y = -1$, so the graph passes through $(0, -1)$, not the origin. A second check: at $x = 1$, $\tfrac{y}{x} = \tfrac{4}{1} = 4$, but at $x = 2$, $\tfrac{y}{x} = \tfrac{9}{2}$. The ratio is not constant.

Answer: No. It is linear but not proportional, because $b \neq 0$.

Guided practice

1. The graph of $y = 3x$ is translated up $5$ units. Describe the resulting line and write its equation.
2. Describe how the graph of $y = -x - 6$ is related to the graph of $y = -x$.
3. What proportional line was translated to produce $y = \tfrac{1}{2}x + 4$, and how far did it move?
4. Write the equation of the line produced by translating $y = 4x$ down $7$ units.
5. Does translating a line vertically change its slope? Explain in one sentence.
6. Give the y-intercept of $y = 2x - 9$ as an ordered pair.

Independent practice

7. For each equation, name the proportional parent line $y = mx$ and describe the translation. a) $y = 5x + 2$ b) $y = -2x - 8$ c) $y = \tfrac{2}{3}x + 6$ d) $y = x - 1$
8. Write the equation of the line produced by translating $y = -3x$ up $4$ units.
9. Write the equation of the line produced by translating $y = \tfrac{1}{4}x$ down $3$ units.
10. Look at the figure of $y = 2x$, $y = 2x + 3$, and $y = 2x - 4$. Which line is highest at $x = 1$? Give the ordered pair that shows it.
11. How far apart vertically are $y = 2x + 3$ and $y = 2x - 4$, and does that distance depend on $x$? Show the subtraction that settles it.
12. Copy and complete the table, then state how the two output columns are related.

13. Is $y = 2x + 3$ a proportional relationship? Give two reasons.
14. Reasoning. Explain why the graphs of $y = \tfrac{1}{2}x + 3$ and $y = \tfrac{1}{2}x - 2$ never meet, no matter how far the grid is extended.
15. Application. In Grade 7 a pool filling from empty at $2$ inches per hour was modeled by $y = 2x$. Today the pool already holds $6$ inches of water before the hose is turned on. Write the new equation and describe what happened to the graph.
16. Error analysis. A student says the graph of $y = 3x - 5$ is the graph of $y = 3x$ translated $5$ units to the left. Explain the mistake and describe the translation correctly.

Exit ticket 8.1

17. Write the equation of the line produced by translating $y = 6x$ up $2$ units.
18. Name the parent line and the translation for $y = -x - 7$.
19. Give the y-intercept of $y = \tfrac{3}{5}x + 4$ as an ordered pair.
20. Explain why a vertical translation cannot change the slope of a line.

Lesson 8.2 — Key Characteristics of a Linear Function

The four things to name

A linear function is a function whose graph is a straight line. Written in slope-intercept form,

$$y = mx + b,$$

it has four key characteristics, and this standard asks you to describe all of them by name.

The slope $m$. The coefficient of $x$. It is the rate of change: the amount $y$ changes for each increase of $1$ in $x$. As a fraction it is

$$m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x}.$$

The y-intercept $b$. The $y$-coordinate of the point where the line crosses the $y$-axis, which is $(0, b)$. Substituting $x = 0$ into the equation gives it immediately.

The independent variable. The input, $x$. You choose it, or the situation hands it to you. It goes in the left column of a table and along the horizontal axis.

The dependent variable. The output, $y$. Its value depends on the value of $x$. It goes in the right column of a table and along the vertical axis.

The sentence "$y$ is a function of $x$" is another way of saying that $x$ is independent and $y$ is dependent. In a context, ask which quantity is being decided by the other: the total cost of a taxi ride depends on how far you go, so the miles are independent and the cost is dependent, never the reverse.

For $y = 2x - 3$: $m = 2$, so the line rises $2$ for every run of $1$; $b = -3$, so it crosses the $y$-axis at $(0, -3)$; $x$ is the independent variable and $y$ the dependent variable.

Reading $m$ and $b$ off an equation, carefully

Slope-intercept form has to actually be in that form before you read the numbers off it. Two traps come up constantly.

A subtraction sign belongs to the number after it. In $y = 4x - 1$, the y-intercept is $b = -1$, not $1$.

The terms may be written out of order. In $y = 8 - 3x$ the $x$-term comes second. Rewrite it by the commutative property of addition:

$$y = 8 + (-3x) = -3x + 8,$$

so $m = -3$ and $b = 8$. A student who reads left to right and answers "$m = 8$, $b = -3$" has the two numbers exactly backwards.

An equation with no visible $b$ has $b = 0$: $y = 7x$ is $y = 7x + 0$, and it is the proportional case. An equation with no visible $x$-term has $m = 0$: $y = 3$ is $y = 0x + 3$.

What the sign of $m$ looks like

• $m > 0$: the line rises left to right. As $x$ increases, $y$ increases.
• $m < 0$: the line falls left to right. As $x$ increases, $y$ decreases.
• $m = 0$: the line is level. Every input gives the same output, $y = b$.

All three lines in the figure have $b = 2$, so all three pass through $(0, 2)$. Two linear functions with the same y-intercept always share that one point.

Steepness is about the size of $m$, not its sign. Comparing $y = 3x + 1$ with $y = \tfrac{1}{2}x + 1$: the first climbs $3$ units per step across, the second climbs half a unit, so the first is steeper. And $y = -4x + 1$ is steeper than $y = 2x + 1$, even though it falls, because $4 > 2$.

Worked examples

Example 1 — Naming $m$ and $b$

Name the slope and y-intercept of $y = -\tfrac{2}{3}x + 5$.

The coefficient of $x$ is $-\tfrac{2}{3}$; the constant term is $5$.

Answer: $m = -\tfrac{2}{3}$, $b = 5$, so the line crosses the $y$-axis at $(0, 5)$.

Example 2 — Terms out of order

Name the slope and y-intercept of $y = 12 - 5x$.

Rewrite as $y = -5x + 12$.

Answer: $m = -5$, $b = 12$

Example 3 — Interpreting $m$ and $b$ in a situation

A gym charges $30 to join plus $15 each month. The total paid after $x$ months is $y = 15x + 30$. Name the independent variable, the dependent variable, $m$, and $b$, and say what each number means.

Answer: $x$, the number of months, is independent; $y$, the total paid, is dependent. $m = 15$ dollars per month is the rate the bill grows. $b = 30$ dollars is the amount owed at $x = 0$, before any month has passed — the joining fee.

Example 4 — Comparing steepness

Which is steeper, $y = \tfrac{3}{4}x - 2$ or $y = -2x - 2$? Which rises?

Compare sizes: $2 > \tfrac{3}{4}$, so the second is steeper. Compare signs: $\tfrac{3}{4} > 0$, so the first rises and the second falls.

Answer: $y = -2x - 2$ is steeper; $y = \tfrac{3}{4}x - 2$ is the one that rises.

Example 5 — A zero slope

Describe the graph of $y = 0x - 4$ and name its key characteristics.

Every input gives $y = -4$.

Answer: A level horizontal line through $(0, -4)$; $m = 0$ and $b = -4$. The dependent variable never changes, no matter what the independent variable does.

Guided practice

21. Name $m$ and $b$ for $y = 4x - 1$.
22. Name $m$ and $b$ for $y = -\tfrac{2}{3}x + 5$.
23. The cost of a taxi ride depends on the number of miles driven. Which quantity is the independent variable?
24. Does the graph of $y = -x + 8$ rise or fall from left to right? Name the characteristic that decides it.
25. Give the y-intercept of $y = 7x$ as an ordered pair.
26. What is the slope of $y = 0x + 3$, and what does its graph look like?

Independent practice

27. Copy and complete the table.

28. Use the figure of $y = 2x - 3$. State $m$, state $b$ as an ordered pair, and list the coordinates of three other plotted points on the line.
29. State whether each line rises, falls, or is level. a) $y = \tfrac{3}{4}x - 2$ b) $y = -5x + 1$ c) $y = 0x - 4$ d) $y = x$
30. Which is steeper, $y = 3x + 1$ or $y = \tfrac{1}{2}x + 1$? Justify your answer using rise and run.
31. A phone plan costs $y = 4x + 20$ dollars, where $x$ is the number of gigabytes used. Name the independent variable, the dependent variable, $m$, and $b$, and say what $m$ and $b$ mean in the situation.
32. Reasoning. Two linear functions have the same y-intercept but different slopes. Name one thing their graphs must have in common and one thing that must differ.
33. Reasoning. Explain why a linear function has exactly one y-intercept. Use the equation $y = mx + b$ in your explanation.
34. For $y = -2x + 5$, name $m$ and $b$, and explain what the negative slope tells you about how $y$ behaves as $x$ increases.
35. Application. A lawn service charges $12 to show up plus $9 per hour of work. Write the equation for the total charge $y$ after $x$ hours, then name $m$, $b$, and both variables.
36. Error analysis. For $y = 8 - 3x$, a student writes $m = 8$ and $b = -3$. Explain the error and give the correct values.

Exit ticket 8.2

37. Name $m$ and $b$ for $y = -\tfrac{1}{3}x + 6$.
38. Name $m$ and $b$ for $y = 12 - 5x$.
39. The distance a car has traveled depends on how long it has been driving. Name the dependent variable and explain your choice.
40. Explain what the slope of a linear function tells you that the y-intercept does not.

Lesson 8.3 — Graphing a Linear Function from an Equation

Two moves, in this order

Slope-intercept form is built for graphing, because the equation hands you a point and a direction.

Step 1. Plot the y-intercept $(0, b)$. You do not have to compute anything; $b$ is sitting in the equation.

Step 2. From that point, use $m$ as $\dfrac{\text{rise}}{\text{run}}$ to step to a second point. Move right by the run, then up or down by the rise. Then draw the line through the two points.

The figure graphs $y = -\tfrac{3}{2}x + 4$. Step 1 plots $(0, 4)$. Step 2 reads the slope as $\tfrac{-3}{2}$: run $2$ to the right, rise $-3$, which is down $3$. That lands on $(2, 1)$. Stepping again lands on $(4, -2)$. Check against the equation: $-\tfrac{3}{2}(4) + 4 = -6 + 4 = -2$. ✓

Whole-number slopes hide a denominator of 1

A slope like $m = 3$ is $\tfrac{3}{1}$: right $1$, up $3$. A slope of $m = -3$ is $\tfrac{-3}{1}$: right $1$, down $3$.

This is where the most common graphing error in the chapter lives. Given $y = -3x + 2$, a student who moves right $3$ and down $1$ has used the slope $-\tfrac{1}{3}$ and drawn the wrong line. The numerator is the rise; the denominator is the run. Say it as "rise over run" every time and the two cannot swap.

Fractional slopes and integer points

When $m$ is a fraction, stepping by rise-over-run automatically lands you on grid corners, which is exactly what the standard requires. For $y = \tfrac{2}{3}x - 2$: start at $(0, -2)$, run $3$, rise $2$, land on $(3, 0)$; again gives $(6, 2)$. Every point is a lattice point.

Stepping backwards works too, and it is often what you need to fill the left half of the grid: reverse both directions. From $(0,-2)$, run $-3$ and rise $-2$ gives $(-3, -4)$.

Always plot a third point

Two points determine a line, but three points catch mistakes. Compute one extra value straight from the equation and check that it lands on the line you drew. For $y = -\tfrac{3}{2}x + 4$ at $x = -2$:

$$y = -\tfrac{3}{2}(-2) + 4 = 3 + 4 = 7,$$

so $(-2, 7)$ should be on the line — and it is.

Worked examples

Example 1 — An integer slope

Graph $y = 2x - 3$ and list three points on it.

Plot $(0, -3)$. Slope $2 = \tfrac{2}{1}$: right $1$, up $2$ to $(1, -1)$; again to $(2, 1)$.

Answer: $(0, -3)$, $(1, -1)$, $(2, 1)$

Example 2 — A negative integer slope

Graph $y = -4x + 3$ and list three points.

Plot $(0, 3)$. Slope $-4 = \tfrac{-4}{1}$: right $1$, down $4$ to $(1, -1)$; again to $(2, -5)$.

Answer: $(0, 3)$, $(1, -1)$, $(2, -5)$

Example 3 — A fractional slope

Graph $y = \tfrac{1}{2}x + 1$ and list three points with integer coordinates.

Plot $(0, 1)$. Slope $\tfrac{1}{2}$: right $2$, up $1$ to $(2, 2)$; again to $(4, 3)$.

Answer: $(0, 1)$, $(2, 2)$, $(4, 3)$

Example 4 — Stepping backwards as well

Graph $y = -\tfrac{1}{3}x + 2$, using points on both sides of the $y$-axis.

Plot $(0, 2)$. Slope $-\tfrac{1}{3}$: right $3$, down $1$ to $(3, 1)$. Reversing, left $3$ and up $1$ to $(-3, 3)$.

Answer: $(-3, 3)$, $(0, 2)$, $(3, 1)$

Example 5 — A zero slope

Graph $y = -4$.

Rewrite as $y = 0x - 4$. Plot $(0, -4)$; the rise is $0$ for every run, so the line is level.

Answer: A horizontal line through $(0, -4)$, containing $(3, -4)$ and $(-3, -4)$.

Example 6 — Graphing a context

A candle is $12$ centimeters tall and burns down $2$ centimeters per hour. Write and graph the function, then say what the point where it meets the $x$-axis means.

The height starts at $12$, so $b = 12$; it decreases $2$ per hour, so $m = -2$. The function is $y = -2x + 12$. Plot $(0, 12)$, then step right $1$ down $2$: $(1, 10)$, $(2, 8)$, and so on to $(6, 0)$.

Answer: $y = -2x + 12$. The point $(6, 0)$ means the candle burns out after $6$ hours, so only $x$-values from $0$ to $6$ make sense in the story.

Guided practice

Graph each function. Plot the y-intercept first, then use the slope, and list three points with integer coordinates.

41. $y = 2x - 3$
42. $y = -x + 4$
43. $y = \tfrac{1}{2}x + 1$
44. $y = -\tfrac{3}{2}x + 4$
45. $y = 3x$
46. $y = -4$

Independent practice

47. Graph $y = \tfrac{2}{3}x - 2$ and list three points with integer coordinates.
48. Graph $y = -2x + 5$ and list three points with integer coordinates.
49. Graph $y = \tfrac{1}{4}x + 3$, listing one point to the left of the $y$-axis and one to the right.
50. Graph $y = -\tfrac{1}{2}x - 1$, listing one point on each side of the $y$-axis.
51. Graph $y = 5x - 6$ and list three points with integer coordinates.
52. Graph $y = x + 2$ and $y = x - 3$ on the same grid. Describe how the two lines are related and how far apart they are.
53. Reasoning. Why does plotting $b$ first make graphing $y = mx + b$ faster than building a whole table of values?
54. Application. A candle is $12$ centimeters tall and burns $2$ centimeters per hour. Write the function, graph it from $x = 0$ to $x = 6$, and explain what $(6, 0)$ means.
55. Application. A taxi charges $3 when you get in plus $2 per mile. Write the function, graph it from $x = 0$ to $x = 6$, and use the graph to find the cost of a $5$-mile ride.
56. Error analysis. To graph $y = -3x + 2$, a student starts at $(0, 2)$ and moves right $3$ and down $1$. Explain the error, name the slope the student actually used, and describe the correct move.

Exit ticket 8.3

57. Graph $y = 3x - 4$ and list three points with integer coordinates.
58. Graph $y = -\tfrac{2}{3}x + 2$ and list three points with integer coordinates.
59. Graph $y = \tfrac{1}{2}x - 3$ and list three points with integer coordinates.
60. Explain how the sign of $m$ tells you which way to move after you plot the y-intercept.

Lesson 8.4 — Tables of Values for Linear Functions

Building a table from the equation

A table of values lists inputs beside their outputs. To build one from $y = mx + b$, choose $x$-values, substitute, and record. Show the substitution rather than doing it in your head — that is where sign errors get caught.

For $y = 4x - 1$:

Two features of that table are worth naming, because Lesson 8.5 will run them in reverse.

The row at $x = 0$ shows $b$. Here $y = -1$ when $x = 0$, and $b = -1$.

Equal steps in $x$ produce equal steps in $y$. Each $x$ goes up by $1$ and each $y$ goes up by $4$, which is $m$. That constant step is what makes the function linear.

Choosing $x$-values on purpose

When $m$ is a fraction, careless $x$-values produce outputs you cannot plot. For $y = \tfrac{1}{3}x + 2$, choosing $x = 1$ gives $y = 2\tfrac{1}{3}$ — a real value, but not a plottable ordered pair under this standard.

Choose $x$-values that are multiples of the denominator of $m$. For $m = \tfrac{1}{3}$, use $x = -3, 0, 3, 6$. For $m = \tfrac{3}{4}$, use $x = -4, 0, 4, 8$.

For $y = \tfrac{1}{3}x + 2$ that gives:

Always include $x = 0$ when you can. It costs nothing and it hands you $b$.

Building a table from a graph

Reading a table off a graph is just recording the lattice points the line passes through. Start at the y-intercept and follow the slope.

The line is $y = 3x - 2$. Its table, read straight off the marked points:

Check any row against the equation: $3(2) - 2 = 4$. ✓

Testing whether a table is linear

Not every table comes from a linear function. Test it: take equal steps in $x$ and see whether the steps in $y$ are also equal.

The $y$-values change by $2$, then $3$, then $4$. The rate of change is not constant, so this table is not linear and no equation $y = mx + b$ fits it.

Worked examples

Example 1 — Table from an equation

Make a table for $y = -2x + 6$ at $x = 0, 1, 2, 3, 4$.

$-2(0) + 6 = 6$; $-2(1) + 6 = 4$; $-2(2) + 6 = 2$; $-2(3) + 6 = 0$; $-2(4) + 6 = -2$.

Answer:

Each step of $1$ in $x$ drops $y$ by $2$, matching $m = -2$.

Example 2 — Choosing $x$-values for a fractional slope

Make a table of four integer ordered pairs for $y = \tfrac{2}{3}x - 1$.

The denominator of $m$ is $3$, so use multiples of $3$: $x = -3, 0, 3, 6$.

$\tfrac{2}{3}(-3) - 1 = -3$; $\tfrac{2}{3}(0) - 1 = -1$; $\tfrac{2}{3}(3) - 1 = 1$; $\tfrac{2}{3}(6) - 1 = 3$.

Answer:

Example 3 — Table from a graph

The graph of $y = -2x + 5$ passes through the lattice points shown in the figure for Lesson 8.5. Write a table for $x = 0, 1, 2, 3$.

Follow the line: from $(0,5)$, each step right $1$ drops $2$.

Answer:

Example 4 — Table from a context

A saver starts with $25 and adds $10 every week. Make a table for weeks $0$ through $4$, and find when the total reaches $85.

Starting amount $25$ is $b$; the weekly $10$ is $m$; so $y = 10x + 25$.

Continuing the pattern: $75$ at week $5$, $85$ at week $6$. Check: $10(6) + 25 = 85$. ✓

Answer: The table above; the total reaches $85 in week $6$.

Example 5 — Deciding whether a table is linear

Is the table with $x = 0, 1, 2, 3$ and $y = 2, 4, 8, 16$ linear?

Equal $x$-steps of $1$ give $y$-steps of $2$, $4$, and $8$.

Answer: No. The rate of change is not constant, so the function is not linear.

Guided practice

61. Make a table for $y = 4x - 1$ at $x = -1, 0, 1, 2, 3$.
62. Make a table for $y = -2x + 6$ at $x = 0, 1, 2, 3, 4$.
63. Make a table for $y = \tfrac{1}{2}x + 2$ at $x = -2, 0, 2, 4$.
64. Read the graph of $y = 3x - 2$ in this lesson and write its table for $x = -1, 0, 1, 2, 3$.
65. Make a table for $y = -x + 3$ at $x = -2, -1, 0, 1, 2$.
66. In a table for a linear function the $x$-values go up by $1$ each row. What must be true of the $y$-values? Name the characteristic involved.

Independent practice

67. Make a table for $y = 3x + 2$ at $x = -2, -1, 0, 1, 2$, and state the constant change in $y$.
68. Make a table for $y = -\tfrac{3}{2}x + 4$ at $x = -2, 0, 2, 4$.
69. Make a table of four integer ordered pairs for $y = \tfrac{2}{3}x - 1$. Explain how you chose the $x$-values.
70. Make a table for $y = 6$ at $x = 0, 1, 2, 3$. What is the slope, and how does the table show it?
71. Using the graph of $y = -2x + 5$ from Lesson 8.5, write a table of four ordered pairs read from the line.
72. A bike rental charges $5 plus $3 per hour, so $y = 3x + 5$. Make a table for $x = 0$ through $4$.
73. A tank holding $20$ liters drains $4$ liters per minute, so $y = -4x + 20$. Make a table for $x = 0$ through $5$, and say what the last row means.
74. Reasoning. A table has $x = 0, 1, 2, 3$ and $y = 1, 3, 6, 10$. Is the function linear? Show the test you used.
75. Application. A saver starts with $25 and adds $10 a week. Write the equation, make a table for weeks $0$ through $4$, and find the week the total reaches $85.
76. Error analysis. Making a table for $y = \tfrac{1}{3}x + 2$, a student chose $x = 1, 2, 4, 5$ and got $y$-values that were not integers. Explain why, and suggest four better $x$-values.

Exit ticket 8.4

77. Make a table for $y = 5x - 3$ at $x = -1, 0, 1, 2$.
78. Make a table for $y = -x - 4$ at $x = -2, 0, 2$.
79. Choose four $x$-values for $y = \tfrac{3}{4}x + 1$ that give integer $y$-values, and complete the table.
80. Explain how you can find $b$ directly from a table of values.

Lesson 8.5 — Writing $y = mx + b$ from a Graph or a Table

From a graph: read $b$, then count the slope

Every graph of a linear function tells you both numbers you need.

Find $b$. Look at where the line crosses the $y$-axis and read the $y$-coordinate.

Find $m$. Pick two lattice points on the line — points at grid corners, so you are counting exactly and not estimating. Count the rise from the first to the second, then the run, and write

$$m = \frac{\text{rise}}{\text{run}}.$$

Counting down or left gives a negative rise or run.

The line crosses the $y$-axis at $(0, 5)$, so $b = 5$. From the lattice point $(1, 3)$ to the lattice point $(3, -1)$ the line runs $2$ right and rises $-4$, so

$$m = \frac{-4}{2} = -2.$$

The equation is $y = -2x + 5$. Confirm with a third point: at $x = 0$, $-2(0) + 5 = 5$. ✓

Any two lattice points give the same slope. Using $(0, 5)$ and $(1, 3)$ instead: rise $-2$, run $1$, $m = \tfrac{-2}{1} = -2$. Same answer, which is exactly what "constant rate of change" means.

From a table: divide the change in $y$ by the change in $x$

Find $m$ by comparing two rows:

$$m = \frac{\text{change in } y}{\text{change in } x}.$$

Find $b$ from the row where $x = 0$, if there is one.

Here $x$ increases by $1$ and $y$ increases by $3$, so $m = \tfrac{3}{1} = 3$. The row $x = 0$ gives $b = 4$. The equation is $y = 3x + 4$. Check the last row: $3(3) + 4 = 13$. ✓

When the table has no $x = 0$ row

Find $m$ first, then walk back to $x = 0$ one step at a time, undoing $m$ at each step.

Each step of $1$ in $x$ raises $y$ by $2$, so $m = 2$. Stepping back from $(1, 1)$ means going left $1$ and therefore down $2$: at $x = 0$, $y = 1 - 2 = -1$. So $b = -1$ and the equation is $y = 2x - 1$. Check a row that was given: $2(4) - 1 = 7$. ✓

The other route is substitution. With $m = 2$ known, put the known point $(1, 1)$ into $y = 2x + b$: $1 = 2(1) + b$, so $b = -1$. Same answer, and this is the method that carries into Algebra I.

The number-one error here is reporting the two numbers in the wrong slots. From the table above, "$y = -x + 2$" or "$y = 5x + 3$"-style mix-ups come from grabbing whichever number is handy. Name each one out loud: this is the change per step, so it multiplies $x$; that is the value at $x = 0$, so it stands alone.

Worked examples

Example 1 — From a graph

A line crosses the $y$-axis at $(0, -1)$ and passes through $(2, 3)$. Write its equation.

$b = -1$. From $(0,-1)$ to $(2,3)$: rise $4$, run $2$, so $m = \tfrac{4}{2} = 2$.

Answer: $y = 2x - 1$

Example 2 — From a table with $x = 0$

Write the equation for the table with $x = 0, 1, 2, 3$ and $y = 9, 7, 5, 3$.

Each step of $1$ in $x$ lowers $y$ by $2$, so $m = -2$. At $x = 0$, $y = 9$, so $b = 9$.

Answer: $y = -2x + 9$. Check: $-2(3) + 9 = 3$. ✓

Example 3 — From a table with a fractional slope

Write the equation for the table with $x = 0, 2, 4, 6$ and $y = 5, 6, 7, 8$.

Change in $x$ is $2$, change in $y$ is $1$, so $m = \tfrac{1}{2}$. At $x = 0$, $y = 5$, so $b = 5$.

Answer: $y = \tfrac{1}{2}x + 5$. Check: $\tfrac{1}{2}(6) + 5 = 8$. ✓

Example 4 — From a table with no $x = 0$ row

Write the equation for the table with $x = 2, 4, 6, 8$ and $y = -1, -2, -3, -4$.

Change in $x$ is $2$, change in $y$ is $-1$, so $m = -\tfrac{1}{2}$. Substituting the point $(2, -1)$:

$$-1 = -\tfrac{1}{2}(2) + b = -1 + b, \qquad b = 0.$$

Answer: $y = -\tfrac{1}{2}x$. Since $b = 0$, this one is proportional. Check: $-\tfrac{1}{2}(8) = -4$. ✓

Example 5 — From a description

A line has slope $-3$ and passes through $(0, 7)$. Write its equation.

The given point is on the $y$-axis, so it is the y-intercept: $b = 7$.

Answer: $y = -3x + 7$

Example 6 — Two students, two slope triangles

One student reads a graph using $(0, 4)$ and $(2, 6)$; another uses $(2, 6)$ and $(6, 10)$. Show that they get the same equation.

First: $m = \tfrac{6 - 4}{2 - 0} = \tfrac{2}{2} = 1$. Second: $m = \tfrac{10 - 6}{6 - 2} = \tfrac{4}{4} = 1$. Both read $b = 4$ from the axis crossing.

Answer: Both get $y = x + 4$. The slope of a line does not depend on which two of its points you count between.

Guided practice

81. Write the equation of the line in the figure for this lesson.
82. Write the equation of the line $y = mx + b$ graphed in Lesson 8.2, which crosses the $y$-axis at $(0, -3)$ and passes through $(1, -1)$.
83. Write the equation for the table with $x = 0, 1, 2, 3$ and $y = 4, 7, 10, 13$.
84. Write the equation for the table with $x = 0, 1, 2, 3$ and $y = 9, 7, 5, 3$.
85. A line passes through $(0, -1)$ and $(2, 3)$. Write its equation.
86. Write the equation for the table with $x = 0, 2, 4, 6$ and $y = 5, 6, 7, 8$.