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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 6

Compare Linear Functions

Compare linear functions given as equations, graphs, tables, or stories.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Same function, different clothes

An equation, a graph, a table, and a story can all be the same linear function.

Compare slope and intercept

Rewrite everything as y = mx + b, or read m and b from a graph or table.

Which is larger at a point

Evaluate both functions at the same x. The larger output is higher on the graph.

When they meet

Set the two expressions equal to find the x where the graphs cross.

Level 1

Easy

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

Example
Does the table match y = 2x + 1?
x012
y135

When x = 0, y = 1. That matches b = 1.

Each step adds 2, so m = 2.

Yes.

Answer: yes

1
Which is steeper: y = 3x + 1 or y = −4x + 9?
Show Answer

Steeper means larger |m|. |−4| = 4 > 3, so y = −4x + 9.

Answer: y = −4x + 9

2
A story: 5 dollars plus 2 dollars per item. A graph through (0, 5) with slope 2. Same function?
Show Answer

Yes. Both are y = 2x + 5.

Answer: yes

3
Table A increases by 3 each time x increases by 1. Equation B is y = 3x − 2. Same slope?
Show Answer

Yes. Both have m = 3.

Answer: yes

4
y = x + 4 and the graph through (0, 4) and (2, 6). Same line?
Show Answer

Slope of the graph: 22 = 1. Intercept 4. Yes.

Answer: yes

5
Which has the greater y-intercept: y = 2x + 3 or y = 5x − 1?
Show Answer

3 > −1, so y = 2x + 3.

Answer: y = 2x + 3

Level 2

Medium

Rewrite, then compare at a value or find the intersection.

Example
f is y = 2x + 1. g is x − y = −5. Which is larger at x = 3?

g: y = x + 5

f(3) = 7

g(3) = 8

g is larger at x = 3.

Answer: g(3) = 8

1
A table of h:
x012
h(x)468
and k(x) = 3x + 4. Compare slopes.
Show Answer

h has m = 2. k has m = 3. k is steeper.

Answer: k is steeper

2
Plan A: 10 + 3n dollars. Plan B: 4n dollars. For which n is A cheaper?
Show Answer

10 + 3n < 4n

10 < n

A is cheaper when n > 10.

Answer: n > 10

3
2x + 2y = 10 and y = −1x + 5. Same line?
Show Answer

First: y = −1x + 5. Yes, the same line.

Answer: yes

4
f(x) = 12x + 2 and a graph through (0, 2) and (4, 3). Match?
Show Answer

Graph slope 14, not 12. No.

Answer: no

5
Where do y = x + 1 and y = −1x + 7 meet?
Show Answer

x + 1 = −1x + 7

2x = 6

x = 3, y = 4

(3, 4)

Answer: (3, 4)

Level 3

Hard

A story, a table, and an equation at once.

Example
Gym A: 20 dollars plus 8 per class. Table B: (0, 30), (1, 36), (2, 42). Which costs less for 4 classes?

A: 20 + 8(4) = 52

B: m = 6, so 30 + 6(4) = 54

A costs less.

Answer: A; 52 vs 54

1
f(x) = 2x − 3 on a graph, g as the table
x123
g024
. Find x where f(x) = g(x).
Show Answer

g(x) = 2x − 2

2x − 3 = 2x − 2 is never true.

Parallel. They never meet.

Answer: never

2
A tank V = 50 − 4t and a second tank W = 10 + 6t. When do they hold the same amount?
Show Answer

50 − 4t = 10 + 6t

40 = 10t

t = 4

Both hold 34 gallons.

Answer: t = 4; 34 gallons

3
Which representation does not match y = −2x + 6: a graph through (0, 6) and (3, 0), or a table (0, 6), (1, 4), (2, 2)?
Show Answer

Graph: slope 0 − 63 − 0 = −2. Matches.

Table: from 6 to 4 is slope −2. Matches.

Both match.

Answer: both match

4
Line 1: 3x + y = 4. Line 2: slope −3 through (0, 5). Parallel?
Show Answer

Line 1: y = −3x + 4. Same slope, different intercepts. Yes, parallel.

Answer: yes

5
Compare domain in the story: tickets T = 12n with n = 0, 1, 2, … versus the line y = 12x on the whole plane.
Show Answer

The story domain is whole numbers.

The line’s domain is all real numbers.

Same formula, different allowed inputs.

Answer: whole numbers vs all reals

Super Challenge

Super Challenge

1
f is given by a table with slope 2 and f(0) = 5. g(x) = 2x + k. Find k so they are the same function, and k so they are parallel but not the same.
Show Answer

Same: k = 5.

Parallel and different: any k ≠ 5.

Answer: k = 5 same; k ≠ 5 parallel

2
A graph of A through (0, 2) with slope 1 and a story “start at 2, add 1 each hour.” A student says they cannot be the same because one is a picture. Respond.
Show Answer

They are the same function in two representations.

Both are y = x + 2.

Answer: same function, y = x + 2

3
Three representations: y = 13x + 1, a table that goes up 1 every 3 units of x starting at (0, 1), and a graph that looks almost horizontal on −2 ≤ x ≤ 2. Could they all be the same line?
Show Answer

Yes. The slope is small, so a short window looks flat.

The table matches m = 13 and b = 1.

Answer: yes

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