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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 10

Compare Linear, Quadratic, and Exponential

Compare the three families using tables (differences and ratios) and graphs.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Linear

Constant first differences in a table. Straight graph. Add the same amount each step.

Quadratic

Constant second differences. Parabola. The amount you add changes by a constant.

Exponential

Constant ratios (multiply by b each time). Graph grows (or decays) by a percent.

Use both a table and a graph

Differences and ratios from the table; shape from the graph.

Level 1

Easy

Read a table’s differences or ratios.

Example
The table is x: 0, 1, 2, 3 and y: 4, 7, 10, 13. Linear, quadratic, or exponential?

First differences 3, 3, 3.

Linear.

Answer: linear

1
x: 0, 1, 2, 3 and y: 2, 4, 8, 16. Which family?
Show Answer

Ratios 2, 2, 2. Exponential.

Answer: exponential

2
x: 1, 2, 3, 4 and y: 2, 5, 10, 17. First differences 3, 5, 7. Which family?
Show Answer

Second differences 2, 2. Quadratic.

Answer: quadratic

3
A graph is a straight line through (0, 1) and (2, 5). Which family?
Show Answer

Linear.

Answer: linear

4
A graph is a U-shape with vertex (1, −2). Which family?
Show Answer

Quadratic.

Answer: quadratic

5
y doubles every time x increases by 1, starting at y = 3 when x = 0. Write a formula.
Show Answer

y = 3 × 2ˣ

Answer: y = 3 × 2ˣ

Level 2

Medium

Write a formula, or reject the wrong family.

Example
The table is x: 0, 1, 2, 3 and y: 5, 8, 13, 20. Write a quadratic.

Second differences 2, 2.

y = x² + 2x + 5

Check: 0+0+5=5, 1+2+5=8, 4+4+5=13, 9+6+5=20.

Answer: y = x² + 2x + 5

1
x: 0, 1, 2 and y: 10, 20, 40. Write an exponential.
Show Answer

y = 10 × 2ˣ

Answer: y = 10 × 2ˣ

2
A graph of y = 2x + 1 and a table 1, 2, 4, 8. Same family?
Show Answer

Line is linear. Table is exponential. No.

Answer: no

3
Which grows faster after many steps: y = 3x + 1 or y = 2ˣ, starting near the same at small x?
Show Answer

Exponential 2ˣ pulls away. At x = 6, 19 vs 64.

Answer: 2ˣ

4
First differences 4, 4, 4. A student says exponential because y is getting larger. Correct them.
Show Answer

Adding 4 is linear, y = 4x + b. Exponential would multiply.

Answer: linear

5
Table: 3, 6, 11, 18. Could it be exponential with integer ratio?
Show Answer

Ratios 2, 11/6, 18/11, not constant. Second differences 2, 2. Quadratic.

Answer: quadratic

Level 3

Hard

A graph and a table together.

Example
A scatterplot looks curved up, and the table ratios are 1.9, 2.1, 2.0. Which family, and why not quadratic?

Ratios near 2: exponential.

Quadratic second differences would be nearly constant; ratios would not stay near one number.

Answer: exponential

1
y = x² and y = 2ˣ at x = 1, 2, 4. Compare.
Show Answer

x=1: both 2. x=2: 4 and 4. x=4: 16 and 16. They meet at those points but are different families.

Second differences vs ratios distinguish them at other x, like x = 3: 9 vs 8.

Answer: different families; 9 vs 8 at x = 3

2
A line on a graph vs a table with second differences 2, 2. Can they be the same function?
Show Answer

No. A line has second differences 0.

Answer: no

3
Write one linear, one quadratic, and one exponential through (0, 2).
Show Answer

y = 2x + 2

y = x² + 2

y = 2 × 3ˣ (or 2 × 2ˣ)

Answer: many; examples 2x+2, x²+2, 2·2ˣ

4
Why does a short table 2, 4, 8 look exponential, but you still want another point?
Show Answer

Three terms with ratio 2 could still bend later. A fourth point tests whether the ratio holds.

Answer: need more data

5
Graph: parabola opening down. Table: first differences −1, −3, −5. Match?
Show Answer

Second differences −2, −2. Quadratic opening down (a < 0). Yes.

Answer: yes

Super Challenge

Super Challenge

1
A student claims every increasing table is exponential. Give a linear counterexample and a quadratic one.
Show Answer

Linear: 3, 5, 7, 9.

Quadratic: 1, 2, 5, 10 (plus 1, plus 3, plus 5).

Answer: 3,5,7,9 and 1,2,5,10

2
Show that y = 2x and y = 2ˣ agree at two points in {0, 1, 2} but not at all three.
Show Answer

x=1: 2 and 2. x=2: 4 and 4. x=0: 0 vs 1.

Answer: miss at x = 0

3
Design a 4-row table that is clearly quadratic, and say what would make you switch to exponential.
Show Answer

Example: 2, 5, 10, 17 (second differences 2).

Switch if ratios settle near one number instead.

Answer: second differences vs ratios

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