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Amy Ferguson Moncure

Algebra 1 · Unit 6

Key Features of Linear Functions

Name domain, range, zeros, slope, and intercepts from an equation or a graph, and interpret them.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Slope and intercepts

Slope is the rate of change: m = y₂ − y₁x₂ − x₁. The y-intercept is (0, b). A zero is where y = 0.

Domain and range

A non-vertical line has domain all real numbers unless a story restricts x. Range is all reals unless the line is horizontal or a story restricts y.

What they mean

Slope is a rate (dollars per hour). The y-intercept is a start. A zero is when the output hits 0.

From an equation or a graph

From y = mx + b you can see m and b at once. From Ax + By = C, plug in x = 0 and y = 0.

Level 1

Easy

Read m, intercepts, or a zero from y = mx + b.

Example
For y = 3x − 9, name the slope, the y-intercept, and the zero.

m = 3

y-intercept (0, −9)

0 = 3x − 9

x = 3

Zero (3, 0)

Answer: slope 3; (0, −9); zero (3, 0)

1
For y = 5x + 2, name the slope and the y-intercept.
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m = 5

(0, 2)

Answer: slope 5; (0, 2)

2
Find the zero of y = −2x + 10.
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0 = −2x + 10

x = 5

(5, 0)

Answer: (5, 0)

3
A line is horizontal through (0, 4). Name slope, domain, and range.
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slope 0

domain all reals

range only 4

Answer: slope 0; domain all reals; range 4

4
The table is linear. Find the slope and y-intercept.
x0123
y581114
Show Answer

Each step +1 in x adds 3 to y, so m = 3.

When x = 0, y = 5.

Answer: slope 3; (0, 5)

5
A cab costs C = 4t + 7 dollars after t miles. What do 4 and 7 mean?
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4 dollars per mile

7 dollar starting fee

Answer: 4 per mile; 7 start

Level 2

Medium

Standard form, a graph, or a story.

Example
Find the intercepts and slope of 3x + 6y = 12.

y = 0: 3x = 12, x = 4, so (4, 0)

x = 0: 6y = 12, y = 2, so (0, 2)

m = 2 − 00 − 4 = 2−4 = −12

Answer: intercepts (4, 0), (0, 2); slope −1/2

1
For y = 34x − 3, name slope, y-intercept, and zero.
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m = 34

(0, −3)

0 = 34x − 3

x = 4

Zero (4, 0)

Answer: slope 3/4; (0, −3); (4, 0)

2
A tank has V = 30 − 6t gallons after t minutes. Interpret the intercepts.
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Starts with 30 gallons.

Empty at t = 5 minutes.

Answer: start 30 gal; empty at 5 min

3
Find the intercepts of 4x − 5y = 20.
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(5, 0) and (0, −4)

Answer: (5, 0) and (0, −4)

4
Name domain and range of y = −4x + 2 as a graph (no story).
Show Answer

Both all real numbers.

Answer: all reals; all reals

5
The graph through (0, 6) and (3, 0). Name slope and the zero.
Show Answer

m = 0 − 63 − 0 = −2

Zero (3, 0)

Answer: slope −2; zero (3, 0)

Level 3

Hard

Context can restrict domain and range.

Example
Tickets cost C = 9n + 4 dollars for n tickets, n = 0, 1, 2, … . Name slope, intercept, domain, and range in the story.

Slope 9: 9 dollars per ticket.

Start 4 dollars.

Domain: whole numbers 0, 1, 2, …

Range: 4, 13, 22, … dollars.

Answer: m = 9; start 4; n = 0, 1, 2, …

1
A = 80 − 5w dollars after w weeks. Find the zero and say whether negative w makes sense.
Show Answer

0 = 80 − 5w, w = 16

Empty after 16 weeks.

w ≥ 0 in the story.

Answer: w = 16; negative weeks do not make sense

2
Find k so y = kx − 4 has zero x = 2.
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0 = 2k − 4

k = 2

Answer: k = 2

3
A vertical line through (3, 0). What is the slope, domain, and range?
Show Answer

Slope is undefined.

Domain: only 3.

Range: all reals.

Answer: undefined; domain 3; range all reals

4
f(x) = −25x + 6. Find the zero.
Show Answer

0 = −25x + 6

25x = 6

x = 15

(15, 0)

Answer: (15, 0)

5
On 0 ≤ t ≤ 8, V = 40 − 5t. Name the domain and range in the story.
Show Answer

Domain: 0 ≤ t ≤ 8

V goes from 40 down to 0.

Range: 0 ≤ V ≤ 40

Answer: t in [0, 8]; V in [0, 40]

Super Challenge

Super Challenge

1
Two lines: y = 2x + 1 and y = 2x − 5. Compare slope, intercepts, and range.
Show Answer

Same slope 2, so parallel.

y-intercepts (0, 1) and (0, −5).

Both have range all reals.

Neither is a vertical line, so both have domain all reals.

Answer: same slope; different intercepts; both all reals

2
A story uses y = 3x + 2 with x = the number of boxes. Why is the mathematical range of the line not the story’s range?
Show Answer

The line takes every real y.

Boxes are whole numbers, so y = 2, 5, 8, 11, … only.

Answer: story range is discrete

3
Find b so that y = −3x + b has x-intercept 4. Then name the y-intercept.
Show Answer

0 = −3(4) + b

b = 12

y-intercept (0, 12)

Answer: b = 12; (0, 12)

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