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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 2

Compare Linear Functions

Compare linear functions given as equations, graphs, tables, or stories, and check whether two representations match.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Four outfits, one function

The same linear relationship can be an equation, a graph, a table, or a sentence. They must agree on slope, intercepts, and sample points.

Compare slopes

Steeper |m| changes faster. In a story, the larger rate is the faster charge, fill, or drain (watch the sign).

Compare intercepts

The larger y-intercept starts higher. In a story that is a bigger starting fee or a fuller tank.

When they meet

Set the two expressions equal to find the input where the outputs match. Check in both representations.

Level 1

Easy

Match a table, graph, or story to y = mx + b.

Example
Which equation matches this table?
x0123
y14710

Y-intercept 1, slope 3

y = 3x + 1

Answer: y = 3x + 1

1
A graph through (0, 2) and (1, 5) matches which equation: y = 3x + 2 or y = 2x + 3?
Show Answer

Y-intercept is 2, not 3.

Slope = 5 − 21 − 0 = 3

y = 3x + 2

Answer: y = 3x + 2

2
“$7 per hour plus a $10 fee” matches which: C = 7h + 10 or C = 10h + 7?
Show Answer

Rate 7 is the slope. Fee 10 is the intercept.

C = 7h + 10

Answer: C = 7h + 10

3
Table A has y-values 2, 4, 6, 8. Table B has 5, 5, 5, 5 for x = 0, 1, 2, 3. Which is steeper?
Show Answer

Table A has slope 2. Table B has slope 0.

Table A is steeper.

Answer: Table A

4
y = x − 4 and y = x + 1 are shown. What is the same, and what is different?
Show Answer

Same slope 1, so they are parallel.

Different intercepts: (0, −4) vs (0, 1).

Answer: same slope; different intercepts

5
The story “starts at 20 gallons and drains 4 gallons per minute” matches V = 20 − 4t or V = 4t − 20?
Show Answer

Start 20 is the intercept. Drain is a negative slope.

V = 20 − 4t

Answer: V = 20 − 4t

Level 2

Medium

Two representations of one function, or two functions to compare.

Example
Show that the table and the equation describe the same line.
x024
y51−3

From the table, intercept 5 and slope 1 − 52 − 0 = −2.

Equation would be y = −2x + 5.

Check x = 4: −2(4) + 5 = −3.

Answer: y = −2x + 5

1
Gym A: $30 plus $8 per class. Gym B: M = 12c + 10. Which has the larger starting fee? Which has the larger rate?
Show Answer

A: intercept 30, slope 8

B: intercept 10, slope 12

A has the larger fee. B has the larger rate.

Answer: A fee; B rate

2
A graph through (0, 0) and (4, 2) and the table x = 0, 2, 4 with y = 0, 1, 2: same function?
Show Answer

Graph slope 24 = 12, through origin: y = 12x

Table: 0, 1, 2 is also slope 12 through the origin.

Yes, same function.

Answer: yes; y = 12x

3
f(x) = 2x + 3 and a table for g: (0, 4), (1, 6), (2, 8). Which function has the larger output at x = 0? At x = 5?
Show Answer

f(0) = 3, g(0) = 4, so g starts higher.

Both slopes are 2, so they stay 1 apart.

f(5) = 13, g(5) = 4 + 10 = 14. g is still larger.

Answer: g is larger at both

4
Equation y = 4x − 1 vs story “4 more than a number.” Are they the same?
Show Answer

“4 more than a number” is y = x + 4, slope 1, intercept 4.

y = 4x − 1 has slope 4.

Different functions.

Answer: no

5
Plan A: 15 + 2t. Plan B’s graph goes through (0, 9) and (3, 18). Which plan costs more at t = 3? At t = 0?
Show Answer

A(0) = 15, B(0) = 9. A costs more at the start.

A(3) = 21. B’s slope = 18 − 93 = 3, so B(3) = 18.

A still costs more at t = 3 (21 vs 18).

Answer: A at both times

Level 3

Hard

Find where two representations agree, and catch a mismatch.

Example
C₁ = 6h + 20 and C₂ = 4h + 32. For which h are the costs equal? Which is cheaper for a 2-hour job?

6h + 20 = 4h + 32

2h = 12

h = 6

At 2 hours: C₁ = 32, C₂ = 40, so C₁ is cheaper.

After 6 hours they match at $56. After that, C₂ (smaller rate) is cheaper.

Answer: equal at 6 hours; C₁ cheaper at 2 hours

1
A table shows (0, 7), (2, 11), (4, 15). A student writes y = 2x + 7. A graph of that equation is drawn through (0, 7) with slope 2. Do all three representations match?
Show Answer

Table slope 11 − 72 = 2, intercept 7. Yes, y = 2x + 7.

Check (4, 15): 2(4) + 7 = 15.

The graph with intercept 7 and slope 2 matches too.

Answer: yes, all three match

2
Line A is x + 2y = 10. Line B’s table is x = 0, 2, 4 with y = 5, 4, 3. Compare slopes and intercepts.
Show Answer

A: 2y = −1x + 10, y = −12x + 5

B: intercept 5, slope 4 − 52 = −12

Same slope and intercept, so A and B are the same line.

Answer: same line

3
Story: “20 plus 3 per guest.” Graph: through (0, 20) and (4, 28). Equation: C = 3g + 16. Which one does not belong?
Show Answer

Story: C = 3g + 20

Graph: slope 84 = 2, intercept 20, so C = 2g + 20

Equation: intercept 16, slope 3

All three are different. None matches another.

If the graph point (4, 28) was a plotting error and should have been (4, 32), then story and graph would match C = 3g + 20, and the equation would be the odd one out.

Answer: all three disagree as written

4
f is given by y = 32x − 1. g is given by a graph through (0, −1) and (2, 2). Are f and g the same function?
Show Answer

g has intercept −1 and slope 2 − (−1)2 = 32

g(x) = 32x − 1

Yes, same function.

Answer: yes

5
Company A: table hours 1, 2, 3 and pay 18, 26, 34. Company B: P = 10h + 4. Who pays more for 5 hours? Who has the larger hourly rate?
Show Answer

A: slope 81 = 8. When h = 1, pay is 18, so 8(1) + b = 18 and b = 10.

P_A = 8h + 10

A(5) = 40 + 10 = 50. B(5) = 10(5) + 4 = 54.

B has the larger hourly rate (10 vs 8) and pays more for 5 hours ($54 vs $50).

Answer: B; $54 vs $50

Super Challenge

Super Challenge

1
Two representations claim to be the same job: a table with (0, 12), (3, 18) and a story “$4 per hour plus $12.” A graph is drawn through (0, 12) with slope 1. Which representation is the mismatch, and how would you fix it?
Show Answer

Story: C = 4h + 12, so at 3 hours C = 24, not 18.

Table slope 63 = 2, so C = 2h + 12.

Graph slope 1: C = h + 12.

All three disagree. The story does not match the table. The graph does not match either.

To match the table, use $2 per hour plus $12, and draw slope 2 from (0, 12).

Answer: none match as written; table is C = 2h + 12

2
f(x) = 2x + 1 is given as an equation. Invent a table, a graph description, and a story that all match f, then change only the story’s starting fee so the story no longer matches.
Show Answer

Table: (0, 1), (1, 3), (2, 5)

Graph: through (0, 1), slope 2

Story: $2 per item plus $1 tax, C = 2n + 1

If the tax becomes $3, the story is C = 2n + 3, which no longer matches the table or graph.

Answer: matching set uses intercept 1; changing the fee breaks it

3
Why can two tables look different (different x-values) and still represent the same linear function? Give an example.
Show Answer

A linear function is the whole line, not a finite list of points.

Table A: (0, 4), (2, 8). Table B: (1, 6), (3, 10). Both fit y = 2x + 4.

Different sample points, same slope and intercept, same function.

Answer: same rule, different sample points

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