Interpret exponential functions and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

$f(x) = 40 \cdot \left(\tfrac{1}{2}\right)^x$ . What is $f(0)$ ?

$40$
$0$

Key idea

Read $a$ and $b$ from $f(x) = a \cdot b^x$ .

a = f(0)

is the starting value (the

y

-intercept).

b

is the growth factor:

b > 1

growth,

0 < b < 1

decay. Negative inputs go back in time:

f(-1) = a \div b

Worked example

Graph through $(0, 3)$ and $(1, 12)$ .

a = 3

Starting value.

b = 4

12 ÷ 3.

f(-1) = 3 \div 4 = 0.75

One step back.

Your turn, guided

Your turn: $f(x) = 50 \cdot 5^x$ . Find $f(-1)$ .

10

50 ÷ 5.

Practice ★

$f(x) = 24 \cdot 4^x$ . Find $f(-1)$ .
$f(x) = 18 \cdot 3^x$ . Find $f(-1)$ .

Is $f(x) = 4 \cdot \left(\tfrac{1}{ 4 }\right)^x$ exponential growth or decay?
1. Decay: each step multiplies by $\tfrac{1}{ 4 }$
2. Growth, because 4 is greater than 1
3. A linear decrease

Watch out!

Practice ★★

The graph of an exponential function passes through $(0, 8)$ and $(1, 16)$ . Write $f(x)$ .
The graph of an exponential function passes through $(0, 7)$ and $(1, 21)$ . Write $f(x)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 24 \cdot 4^x$ . Find $f(-1)$ .
$f(x) = 18 \cdot 3^x$ . Find $f(-1)$ .
$f(x) = 27 \cdot 3^x$ . Find $f(-1)$ .

Summary card

Reading exponential functions

$a$ : value at $x = 0$ ; $b$ : factor per step; $b^{-1} = 1/b$ .

Watch out: negative exponent, positive value.

End of the lesson

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