Module 5 Study Guide

5.1 — Recognizing Linear Functions

Essential Question: What is a linear function?

Definition

A linear function is a function whose graph is a non-vertical straight line. A constant change in x always gives a constant change in y.

Race Car Example: y = 210x

A race car goes 210 mph. After x hours it has gone y = 210x miles.

x (hours)	y (miles)	Change in y
0	0	—
1	210	+210
2	420	+210
3	630	+210
4	840	+210
5	1050	+210

x changes by 1 every time, and y changes by 210 every time. That makes it linear.

How to Test a Table

1Check that the x-values change by the same amount each step.

2Subtract consecutive y-values.

3All the y-differences are the same → linear. If they change → not linear.

Non-Linear Examples

x	y = x²	Change
1	1	—
2	4	+3
3	9	+5
4	16	+7
5	25	+9

x	y = x³	Change
1	1	—
2	8	+7
3	27	+19
4	64	+37
5	125	+61

x changes by 1 each time, but the y changes keep growing. Not linear.

Common Mistake: Check x First!

x: −1, 0, 2, 3, 4 | y: −5, 0, 5, 10, 15
y goes up by 5 each time, so it looks linear. But x jumps by 2 between 0 and 2. A linear function would change by 10 there, not 5. Not linear.

5.1 — Why Linear Functions Grow by Equal Differences

Linear functions change by equal amounts over equal intervals.

Slope-Intercept Form

f(x) = mx + b m = slope (rate of change), b = y-intercept
Every linear function can be written this way.

The Proof

Given: x₂ − x₁ = x₄ − x₃ (two equal-size intervals), and f(x) = mx + b.
Prove: f(x₂) − f(x₁) = f(x₄) − f(x₃)

Statement	Reason
x₂ − x₁ = x₄ − x₃	Given
m(x₂ − x₁) = m(x₄ − x₃)	Multiplication Property of Equality
mx₂ − mx₁ = mx₄ − mx₃	Distributive Property
mx₂ + b − mx₁ − b = mx₄ + b − mx₃ − b	Add and subtract b (adds 0)
mx₂ + b − (mx₁ + b) = mx₄ + b − (mx₃ + b)	Distributive Property
f(x₂) − f(x₁) = f(x₄) − f(x₃)	Definition of f(x)

What It Means

Equal steps in x → equal steps in y. The step in y is always m × (step in x).

5.1 — Standard Form & Graphing

Any linear function can be written as a linear equation.

Standard Form of a Linear Equation

Ax + By = C A, B, C are real numbers; A and B are not both 0.

A solution is any ordered pair (x, y) that makes the equation true. The graph shows all the solutions.

Linear ✓

5x + y = 10
3x + 4y = 8
x/2 + y/4 = 3/2 (multiply by 4 → 2x + y = 6)
y = 3x + 2 (linear, not standard form)

Not Linear ✗

2x² + y = 6 — x squared
x + y² = 1 — y squared
3xy + y = 4 — x times y
y = 5/x — x in denominator

Example: Graph 5x + y = 10

1It is in standard form: A = 5, B = 1, C = 10 → linear.

2Solve for y: y = 10 − 5x

3Make a table, plot the points, and connect them.

x	−1	0	1	2	3
y	15	10	5	0	−5

Example: −4x + y = 11

A = −4, B = 1, C = 11. Solve for y: y = 11 + 4x

x	−4	−2	0	2	4
y	−5	3	11	19	27

Special Cases

A = 0: By = C → y = C/B → horizontal line (example: 2y = 8 → y = 4)
B = 0: Ax = C → x = C/A → vertical line: linear equation, but not a function

5.1 — Discrete vs. Continuous Models

Does the in-between make sense? That decides whether the dots connect.

Discrete

Graph is unconnected points. Inputs are whole counts: DVDs, tickets, rooms, basketballs.

Continuous

Graph is an unbroken line. Inputs can be fractions: time, distance, pounds, gallons.

Sal buys DVDs at $2 each: f(x) = 2x
Domain {0, 1, 2, 3, …} · Range {0, 2, 4, 6, …}

Elsa pays $1 per hour: f(x) = x
Domain x ≥ 0 · Range y ≥ 0

Worked Examples

Situation	Function	Type	Domain
Hans buys video games, $5 each	f(x) = 5x	Discrete	{0, 1, 2, …}
Kristoff pays $6 per umbrella sold	f(x) = 6x	Discrete	{0, 1, 2, …}
Hot air balloon at 85 mph	y = 85x	Continuous	x ≥ 0
Gallons of water over time	y = 4x	Continuous	x ≥ 0

Solving with a Model

Particle velocity: −4x + 3y = 15. When does it reach y = 30 km/hr?

−4x + 3(30) = 15 → −4x + 90 = 15 → −4x = −75 → x = 18.75

Jordan’s profits: $150, $300, $450, $600 in weeks 1–4. Each week adds $150, so f(x) = 150x and f(9) = $1350.

Careful Predicting Far Ahead

A linear model assumes the pattern never changes. In real life, customers, seasons, and prices change, so predictions far into the future are less reliable.

5.2 — Finding Intercepts

Essential Question: How can you identify and use intercepts in linear relationships?

x-intercept

Where the graph crosses the x-axis.
The point is (a, 0) — y is always 0.
To find it: set y = 0, solve for x.

y-intercept

Where the graph crosses the y-axis.
The point is (0, b) — x is always 0.
To find it: set x = 0, solve for y.

Example: 3x − 2y = 6

x-intercept (y = 0)

3x − 2(0) = 6
3x = 6
x = 2

y-intercept (x = 0)

3(0) − 2y = 6
−2y = 6
y = −3

Practice Answers

Equation	x-intercept	y-intercept
−5x + 6y = 60	−12	10
8x + 7y = 28	3.5	4
−6x − 8y = 24	−4	−3
2x − 3y = −6	−3	2
−4x − 5y = 40	−10	−8
³⁄₅x + ¹⁄₂y = 30	50	60

Shortcut for Ax + By = C

x-intercept = C/A y-intercept = C/B

Common Mistakes

Mixing up which variable to zero. Arlo found the “x-intercept” of 9x + 12y = 144 by setting x = 0 and got 12. That is the y-intercept. Correct x-intercept: 9x = 144 → 16.
(5, 0) is NOT the y-intercept. y = 0, so it’s on the x-axis: x-intercept = 5.
(0, 0) is the only point that is both intercepts.

5.2 — Interpreting Intercepts

In a real-world problem, each intercept answers a question.

y-intercept = Starting Value

The value when x (usually time) = 0: starting height, starting balance, starting temperature.

x-intercept = When It Hits 0

When the amount runs out or reaches zero: lands, arrives, paid off, sold out.

Miners Example

Miners start 90 ft underground and ride an elevator up for 3 minutes.

y-intercept −90: start 90 ft below the surface
x-intercept 3: reach the surface after 3 min

A negative y-intercept means the starting value is below zero (underground, below sea level, below 0°, owing money).

Worked Examples

Situation	Function	x-intercept	y-intercept
Sandia Peak tram	4500 − 300x	15 min to reach the peak	starts 4500 m from the peak
Hot air balloon	750 − 25x	lands after 30 min	starts 750 m up
Experiment temperature	⁵⁄₂x − 70	reaches 0°C at 28 hr	starts at −70°C
Trout in lake	350 − 14x	gone after 25 yr	350 trout at start
Air temp at sunrise	3x − 6	0°C after 2 hr	−6°C at sunrise
Brake pads	480 − 4x	enough for 120 cars	480 pads at start
Connor’s 10 km race	10 − ¹⁄₄x	finishes at 40 min	10 km to go at start
Kim owes $245, pays $35/wk	245 − 35x	paid off in 7 wk	owes $245 at start
Bank account, $8 fee/mo	360 − 8x	$0 after 45 mo	$360 at start

How to Solve (Tram)

1x-intercept: set f(x) = 0 → 0 = 4500 − 300x → 300x = 4500 → x = 15

2y-intercept: find f(0) = 4500 − 300(0) = 4500

A Graph Is Not a Picture of the Path

The tram graph shows distance from the peak vs. time. It does not show the shape of the mountain or the cable.

5.2 — Graphing with Intercepts

Two points make a line. The intercepts are the easiest two to find.

1Rewrite in standard form Ax + By = C.

2Find the x-intercept (y = 0) and the y-intercept (x = 0).

3Plot (x-int, 0) and (0, y-int). Draw a line through them.

Example: ¹⁄₂y = 3 − ³⁄₄x

Standard form: ³⁄₄x + ¹⁄₂y = 3

x-int: ³⁄₄x = 3 → x = 4

y-int: ¹⁄₂y = 3 → y = 6

Plot (4, 0) and (0, 6). (Same line as 3x + 2y = 12.)

More Examples

Equation	Standard form	x-int	y-int
18y = 12x + 108	−12x + 18y = 108	−9	6
3y = −5x − 30	5x + 3y = −30	−6	−10
−6y = −4x + 24	4x − 6y = 24	6	−4
9y = 3x + 18	−3x + 9y = 18	−6	2
−3y = 7x − 21	7x + 3y = 21	3	7
³⁄₂x = −4y − 12	³⁄₂x + 4y = −12	−8	−3

Lesson Task: Area of a Sail

The sail’s slanted edge is 10x + 4y = 240. The sail is the triangle under the line.

x-int: 240/10 = 24 ft (base)
y-int: 240/4 = 60 ft (height)

Area = ¹⁄₂ × 24 × 60 = 720 ft²

For any Ax + By = C (all positive): Area = ¹⁄₂ · ^C⁄_A · ^C⁄_B = C² / (2AB)

Choosing a Graph Scale

For 3x + 40y = 1200: x-int = 400, y-int = 30. Use an x-axis up to 400 (steps of 50) and a y-axis up to 30 (steps of 5) so both intercepts fit.

5.3 — Rate of Change

Essential Question: How can you relate rate of change and slope in linear relationships?

Rate of Change

rate of change = change in y / change in x
Compares how much y changes to how much x changes over part of the domain. Units are “y-units per x-unit”.

Postage Example

Period	Change in cost / change in years	Rate
2003–2004	(37 − 37) / (4 − 3)	0 ¢/yr
2004–2006	(39 − 37) / (6 − 4)	1 ¢/yr
2006–2008	(42 − 39) / (8 − 6)	1.5 ¢/yr
2008–2013	(46 − 42) / (13 − 8)	0.8 ¢/yr

Steeper = Greater Rate of Change

2006–2008 is the steepest segment and has the greatest rate (1.5). A flat segment (2003–2004) has rate 0. This graph is not one straight line because the rates are not constant. A linear function has the same rate of change everywhere.

Rates That Change: Hikers

Segment	Calculation	Slope (km/hr)
0.5 → 1 hr	(5 − 3) / (1 − 0.5)	4
1 → 2 hr	(7 − 5) / (2 − 1)	2
2 → 3 hr	(13 − 7) / (3 − 2)	6 (steepest)

5.3 — Slope & the Slope Formula

For a line, the rate of change is the same between any two points. That constant rate is the slope.

Slope

slope = rise / run
rise = difference in y-values
run = difference in x-values

Slope Formula

For points (x₁, y₁) and (x₂, y₂):
m = (y₂ − y₁) / (x₂ − x₁)

Example: Points (2, 1) and (3, 4)

1Rise: 4 − 1 = 3

2Run: 3 − 2 = 1

3Slope = 3/1 = 3

The dashed lines show the run (across) and the rise (up).

Four Kinds of Slope

Positive
rises left → right

Negative
falls left → right

Zero
horizontal

Undefined
vertical

Type	Graph	Why	Example
Positive	Rises left to right	rise and run have the same sign	(5, 3), (10, 8): 5/5 = 1
Negative	Falls left to right	rise and run have opposite signs	(−5, 14), (−1, 2): −12/4 = −3
Zero	Horizontal	rise = 0 → 0/run = 0	(−5, 6), (8, 6): 0/13 = 0
Undefined	Vertical	run = 0 → can’t divide by 0	(−4, −17), (−4, −3): 14/0

Practice Answers

Points	Work	Slope
(−2, 3), (1, 0)	(0 − 3) / (1 − (−2)) = −3/3	−1
(0, −3), (5, 3)	6/5	1.2
(−3, 2), (−2, −3)	−5/1	−5
(−2, −1), (2, 3)	4/4	1
(−1, 9), (2, −5)	−14/3	≈ −4.67
(2, −2), (5, 7)	9/3	3
(−6, 5), (−4, −3)	−8/2	−4
(−4, 7), (2, −8)	−15/6	−2.5
(−7, −9), (−1, 5)	14/6	7/3 ≈ 2.33
(12, −7), (2, −2)	5/(−10)	−0.5
(−3, −10), (−1, −1)	9/2	4.5

Common Mistakes

Upside down: y goes on top. For (4, 3) and (−2, 15), one student wrote (4 − (−2)) / (3 − 15) = −½. Correct: (15 − 3) / (−2 − 4) = 12/−6 = −2.
Mixed order: if you start with point 2 on top, start with point 2 on the bottom too.
Double negatives: 3 − (−1) = 3 + 1 = 4.
Same slope ≠ same line. Parallel lines have the same slope but different y-intercepts.

5.3 — Interpreting Slope

In a real-world problem, slope is a rate: how much y changes for each 1 unit of x.

Heart rate: (20, 180) and (50, 150)

slope = (150 − 180) / (50 − 20) = −30/30 = −1

For every year older, maximum heart rate goes down 1 beat per minute.

How to Say It

“For every 1 [x-unit], [y-thing] goes up/down by |slope| [y-units].”

Situation	Points	Slope	Meaning
Plant height	(30, 15), (40, 25)	1	grows 1 cm per day
°F to °C	(50, 10), (77, 25)	5/9	+1 °F = +⁵⁄₉ °C
Reservoir	(40, 3000), (60, 4000)	50	fills 50 ft³ per hour
Money earned	(4, 110), (8, 170)	15	earns $15 per hour
Peanut butter	(960, 2), (3360, 7)	1/480	1 jar per 480 peanuts
Printing	(1000, 310), (3500, 460)	0.06	6¢ per page
Bank balance	(3, 860), (8, 585)	−55	loses $55 per month
Crickets	(40, 0), (50, 40)	4	4 more chirps/min per °F

Applications

Stairs: 7.75-in riser, 10-in tread → slope = 7.75/10 = 0.775
Cog railway: slope 0.37 → rise 1 needs run = 1/0.37 ≈ 2.70
Ladder: 19 ft up, 11 ft out → slope = 19/11 ≈ 1.73. Closer to the wall → bigger slope → steeper, harder to climb.

Electric Bill: Rates That Change

kWh	0	200	400	600	1000	2000
Cost ($)	8	8	34	60	112	157

Interval (kWh)	0–200	200–400	400–600	600–1000	1000–2000
Rate ($/kWh)	0/200 = 0	26/200 = 0.13	26/200 = 0.13	52/400 = 0.13	45/1000 = 0.045

Plan: $8 flat for the first 200 kWh, then 13¢ per kWh up to 1000 kWh, then 4.5¢ per kWh after that.

Lesson Task: Internet Providers

ISP	100 → 200 MB	200 → 400 MB	Constant fee?
A ($54, $74, $94)	20/100 = $0.20/MB	20/200 = $0.10/MB	No
B ($42, $57, $87)	15/100 = $0.15/MB	30/200 = $0.15/MB	Yes
C ($60, $72, $96)	12/100 = $0.12/MB	24/200 = $0.12/MB	Yes

For heavy data use, the lowest current rate wins over time. A ($0.10/MB) ends up cheapest and B ($0.15/MB) most expensive. At 1000 MB: A ≈ $154, C ≈ $168, B ≈ $177.

Table of Contents

5.1 — Recognizing Linear Functions

Race Car Example: y = 210x

How to Test a Table

Non-Linear Examples

5.1 — Why Linear Functions Grow by Equal Differences

The Proof

5.1 — Standard Form & Graphing

Example: Graph 5x + y = 10

Example: −4x + y = 11

5.1 — Discrete vs. Continuous Models

Worked Examples

Solving with a Model

5.2 — Finding Intercepts

Example: 3x − 2y = 6

Practice Answers

5.2 — Interpreting Intercepts

Miners Example

Worked Examples

How to Solve (Tram)

5.2 — Graphing with Intercepts

Example: 1⁄2y = 3 − 3⁄4x

More Examples

Lesson Task: Area of a Sail

5.3 — Rate of Change

Postage Example

Rates That Change: Hikers

5.3 — Slope & the Slope Formula

Example: Points (2, 1) and (3, 4)

Four Kinds of Slope

Practice Answers

5.3 — Interpreting Slope

Applications

Electric Bill: Rates That Change

Lesson Task: Internet Providers

Example: ¹⁄₂y = 3 − ³⁄₄x