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Compare two functions

Rate of change and start value from an equation, a table or two points · Algebra 1 · 20 problems, easy to harder

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How we do it

  1. Rate of change of A
  2. Rate of change of B
  3. Compare
  4. Check

The steps for Which grows faster: y = 3x + 1 or x 0, 1, 2; y 2, 6, 10. Every solution names its steps, so you always know what comes next.

Worked examples

Which grows faster: y = 3x + 1 or x 0, 1, 2; y 2, 6, 10 Easy

The question

Question: which grows faster: function A, y = 3x + 1, or function B, x: 0, 1, 2; y: 2, 6, 10?

A: y = 3x + 1
= B: x: 0, 1, 2; y: 2, 6, 10

Rate of change means: how much y changes when x goes up by 1.

The bigger one grows faster.

grows faster = bigger rate of change

1 Rate of change of A

In y = m x plus b, m is the number in front of x: 3.

= y = 3x + 1: m = 3

2 Rate of change of B

In the table, y changes by 4 each time x goes up by 1.

So the rate of change is 4.

= changes: + 4, + 4 → m = 4

3 Compare

4 is bigger.

So function B grows faster.

= 3 < 4 → B grows faster

4 Check

Let's check: both lines on one grid.

Line B is steeper: it climbs faster.

So the answer is right!

Let's look back.

First: rate of change of A.

Then: rate of change of B.

Then: compare.

Last: check.

Answer
= 3 < 4 → B grows faster

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Which grows faster: x 0, 1, 2; y 5, 7, 9 or x 0, 1, 2; y 1, 4, 7 Medium

The question

Question: which grows faster: function A, x: 0, 1, 2; y: 5, 7, 9, or function B, x: 0, 1, 2; y: 1, 4, 7?

A: x: 0, 1, 2; y: 5, 7, 9
= B: x: 0, 1, 2; y: 1, 4, 7

Rate of change means: how much y changes when x goes up by 1.

The bigger one grows faster.

grows faster = bigger rate of change

1 Rate of change of A

In the table, y changes by 2 each time x goes up by 1.

So the rate of change is 2.

= changes: + 2, + 2 → m = 2

2 Rate of change of B

In the table, y changes by 3 each time x goes up by 1.

So the rate of change is 3.

= changes: + 3, + 3 → m = 3

3 Compare

3 is bigger.

So function B grows faster.

= 2 < 3 → B grows faster

Careful!

Function A starts higher.

But it grows more slowly.

Careful: the higher start is not the faster one

4 Check

Let's check: both lines on one grid.

Line B is steeper: it climbs faster.

So the answer is right!

Let's look back.

First: rate of change of A.

Then: rate of change of B.

Then: compare.

Last: check.

Answer
Careful: the higher start is not the faster one

▶ Watch it solved

Which grows faster: x 0, 3, 6; y 2, 8, 14 or y = 3x + 1 Harder

The question

Question: which grows faster: function A, x: 0, 3, 6; y: 2, 8, 14, or function B, y = 3x + 1?

A: x: 0, 3, 6; y: 2, 8, 14
= B: y = 3x + 1

Rate of change means: how much y changes when x goes up by 1.

The bigger one grows faster.

grows faster = bigger rate of change

1 Rate of change of A

In the table, y changes by 6 each time x goes up by 3.

So the rate of change is 2.

= changes: + 6, + 6 for 3 of x → m = 2

2 Rate of change of B

In y = m x plus b, m is the number in front of x: 3.

= y = 3x + 1: m = 3

3 Compare

3 is bigger.

So function B grows faster.

= 2 < 3 → B grows faster

Careful!

Function A starts higher.

But it grows more slowly.

Careful: the higher start is not the faster one

4 Check

Let's check: both lines on one grid.

Line B is steeper: it climbs faster.

So the answer is right!

Let's look back.

First: rate of change of A.

Then: rate of change of B.

Then: compare.

Last: check.

Answer
Careful: the higher start is not the faster one

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. which grows faster: y = 3x + 1 or x 0, 1, 2; y 2, 6, 10
    Answer
    = 3 < 4 → B grows faster
  2. which grows faster: y = 2x + 3 or y = 5x - 1
    Answer
    Careful: the higher start is not the faster one
  3. which starts higher: y = 2x + 5 or x 0, 1, 2; y 3, 7, 11
    Answer
    Careful: the faster one is not the higher start
  4. which grows faster: y = 6x or x 0, 1, 2; y 1, 3, 5
    Answer
    Careful: the higher start is not the faster one
  5. which starts higher: y = 4x + 1 or y = x + 6
    Answer
    Careful: the faster one is not the higher start
  6. which grows faster: y = 4x - 2 or the line through (0, 1) and (2, 11)
    Answer
    = 4 < 5 → B grows faster

Medium

  1. which grows faster: x 0, 1, 2; y 5, 7, 9 or x 0, 1, 2; y 1, 4, 7
    Answer
    Careful: the higher start is not the faster one
  2. which grows faster: y = 2x + 8 or x 0, 2, 4; y 1, 7, 13
    Answer
    Careful: the higher start is not the faster one
  3. which starts higher: x 0, 1, 2; y 4, 5, 6 or the line through (0, 2) and (1, 5)
    Answer
    Careful: the faster one is not the higher start
  4. which grows faster: the line through (0, 3) and (2, 7) or y = 3x
    Answer
    Careful: the higher start is not the faster one
  5. which starts higher: y = 3x - 1 or x 0, 1, 2; y 2, 3, 4
    Answer
    Careful: the faster one is not the higher start
  6. which grows faster: y = 10x + 1 or y = 12x - 5
    Answer
    Careful: the higher start is not the faster one
  7. which grows faster: y = 5x + 2 or x 1, 2, 3; y 5, 9, 13
    Answer
    = 5 > 4 → A grows faster

Harder

  1. which grows faster: x 0, 3, 6; y 2, 8, 14 or y = 3x + 1
    Answer
    Careful: the higher start is not the faster one
  2. which starts higher: x 2, 3, 4; y 7, 10, 13 or y = 2x + 2
    Answer
    Careful: the faster one is not the higher start
  3. which grows faster: the line through (1, 2) and (3, 12) or x 0, 1, 2; y 0, 4, 8
    Answer
    Careful: the higher start is not the faster one
  4. which is steeper: y = 7x - 3 or the line through (0, 0) and (2, 12)
    Answer
    Careful: the higher start is not the faster one
  5. which starts higher: the line through (1, 5) and (2, 8) or y = 4x + 3
    Answer
    = 2 < 3 → B starts higher
  6. which grows faster: y = x + 9 or x 0, 2, 4; y 1, 5, 9
    Answer
    Careful: the higher start is not the faster one
  7. which grows faster: y = 2x + 4 or x 0, 1, 2; y 0, 3, 6
    Answer
    Careful: the higher start is not the faster one