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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 6

Graph Linear Functions

Graph a linear function from slope-intercept or standard form, including real-world restrictions.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Slope-intercept

From y = mx + b, plot (0, b), then use rise over run.

Two intercepts

From Ax + By = C, plot the x-intercept and the y-intercept, then connect them.

A story may cut the graph

If t is time, only t ≥ 0 belongs on the graph in the story.

No figure in the question

When you are asked to graph, the sketch is in the answer.

Level 1

Easy

Integer slope. Graph in the answer.

Example
Graph. y = x + 2

y-intercept (0, 2)

Slope 1: up 1, right 1.

Answer: see graph

1
Graph. y = −x + 4
Show Answer

(0, 4), slope −1

Answer: see graph

2
Graph. y = 3
Show Answer

Horizontal through (0, 3).

Answer: see graph

3
Graph. x = −2
Show Answer

Vertical through (−2, 0).

Answer: see graph

4
Graph. y = 2x − 1
Show Answer

(0, −1), slope 2

Answer: see graph

5
Graph. y = −3
Show Answer

y = −3

Answer: see graph

Level 2

Medium

Standard form, or a fractional slope.

Example
Graph. 2x + 4y = 8

x-intercept (4, 0), y-intercept (0, 2)

Answer: see graph

1
Graph. y = 12x − 1
Show Answer

(0, −1), up 1, right 2

Answer: see graph

2
Graph. x − y = 3
Show Answer

y = x − 3

Answer: see graph

3
A tank has V = 8 − 2t gallons after t minutes. Graph the part that makes sense in the story.
Show Answer

t ≥ 0 and V ≥ 0, so t from 0 to 4.

Plot (0, 8) to (4, 0).

Answer: segment from (0, 8) to (4, 0)

4
Graph. y = −23x + 2
Show Answer

(0, 2), down 2, right 3

Answer: see graph

5
Graph. 5x + y = 0
Show Answer

y = −5x

Answer: see graph

Level 3

Hard

A restricted domain, or technology to check a point.

Example
Graph C = 8n + 4 for n = 0, 1, 2, 3, 4 (tickets). Should the graph be a solid line?

n is a whole number, so plot points, not a solid connecting line, in the story.

Points: (0, 4), (1, 12), (2, 20), (3, 28), (4, 36).

A calculator graph of the line can check that those points sit on y = 8x + 4.

Answer: discrete points on y = 8x + 4

1
Graph the part of y = −2x + 8 with x ≥ 0 and y ≥ 0.
Show Answer

First-quadrant segment from (0, 8) to (4, 0).

Answer: segment from (0, 8) to (4, 0)

2
Use a table to graph y = 3x − 2 for x = −1, 0, 1, 2.
x−1012
y−5−214
Show Answer

Plot those four points and connect them.

Answer: see graph

3
Graph x + 2y = 6 and label both intercepts.
Show Answer

(6, 0) and (0, 3)

Answer: (6, 0) and (0, 3)

4
A calculator shows a line through (2, 5) with slope −1. Write the equation and sketch it.
Show Answer

5 = −1(2) + b

b = 7

y = −1x + 7

Answer: y = −x + 7

5
Why does graphing y = 15x + 1 with a window from −6 to 6 make the line look almost horizontal?
Show Answer

The slope is small, so in 6 units of x the rise is only about 1.

A window with a larger x-span makes a gentle slope look flat. Zoom or change the scale to see the tilt.

Answer: small slope; window hides the tilt

Super Challenge

Super Challenge

1
Sketch y = 2x − 1 and 2x − y = 5 on the same plane. What do you notice?
Show Answer

Second: y = 2x − 5

Same slope, different intercepts: parallel, never meet.

Answer: parallel

2
A graphing tool plots y = (1/3)x + 2. Name two points you would use to check the calculator by hand.
Show Answer

(0, 2) and (3, 3)

Plug x = 3: 13(3) + 2 = 3.

Answer: (0, 2) and (3, 3)

3
Graph the constraint x ≥ 0, y = 4 − x in a ticket story. What part of the line is extra?
Show Answer

The line y = 4 − x continues into x < 0 and y < 0.

The story keeps the segment from (0, 4) to (4, 0).

Answer: only the first-quadrant segment

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