Linear or exponential: equal differences or equal factors

Warm-up

Three quick questions from earlier lessons.

Discover

Offer A: 1,000 dollars a day for 30 days. Offer B: 1 cent the first day, then double every day. Which pays more on day 30?

Offer B
Offer A

Key idea

Linear adds, exponential multiplies.

Over equal steps in

x

, a linear function changes by equal differences (add

m

); an exponential function changes by equal factors (multiply by

b

). A growing exponential eventually overtakes any linear function.

Worked example

$y$ : 3, 6, 12, 24.

Not linear

Differences 3, 6, 12: not equal.

Exponential, factor 2

Ratios 2, 2, 2.

24 \cdot 2 = 48

Next value.

Your turn, guided

Your turn: an exponential function has $g(0) = 5$ , $g(1) = 15$ . Find $g(2)$ .

45

Factor 3.

Practice ★

A table gives $x = 0, 1, 2, 3$ and $y = 4, 12, 36, 108$ . What kind of relationship is it?
1. Exponential: $y$ is multiplied by 3 at each step
2. Linear: $y$ goes up by 8 at each step
3. Neither
A table gives $x = 0, 1, 2, 3$ and $y = 8, 24, 72, 216$ . What kind of relationship is it?
1. Exponential: $y$ is multiplied by 3 at each step
2. Linear: $y$ goes up by 16 at each step
3. Neither

An exponential function has $g(0) = 9$ and $g(1) = 18$ . Find $g(3)$ .

Watch out!

Practice ★★

$f(x) = 57x + 380$ and $g(x) = 1.04^x$ . For very large $x$ , which is greater?
1. $g$ : exponential growth eventually overtakes linear growth
2. $f$ : it is greater now and grows faster
3. They stay about equal
$f(x) = 73x + 397$ and $g(x) = 1.04^x$ . For very large $x$ , which is greater?
1. $g$ : exponential growth eventually overtakes linear growth
2. $f$ : it is greater now and grows faster
3. They stay about equal

Exit ticket

Three questions to check you have got it.

An exponential function has $g(0) = 9$ and $g(1) = 18$ . Find $g(3)$ .
An exponential function has $g(0) = 4$ and $g(1) = 20$ . Find $g(3)$ .
An exponential function has $g(0) = 5$ and $g(1) = 15$ . Find $g(3)$ .

Summary card

Linear or exponential?

Equal differences: linear. Equal factors: exponential.

Watch out: exponential growth wins in the end.

End of the lesson

Next lesson: Exponential growth and decay: multiplying by a constant factor Back to the chapter

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