Before you read

Answer first — see what you already know.

Which function models the decay of a medication in the bloodstream?

A
$f (x) = 3 \cdot 2^{x}$
B
$f (x) = 5 \cdot (0.8)^{x}$
C
$f (x) = - 2 x + 10$
D
$f (x) = x^{2} - 4$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A temperature model is $f (x) = 4 \cdot 3^{x} + 2$ . What is the y-intercept?

A
$(0, 2)$
B
$(0, 4)$
C
$(0, 6)$
D
$(0, 14)$

Check your answer

Answer:

Theory

Exponential Function Basics

The general form: $f(x) = a \cdot b^x + k$

where:
- $a$ : vertical stretch (initial multiplier)
- $b$ : base (growth/decay factor)
- $b > 1$ : exponential growth (graph rises steeply)
- $0 < b < 1$ : exponential decay (graph falls toward zero)
- $k$ : horizontal asymptote ( $y = k$ )

Growth: y = 2ˣ

Decay: y = (½)ˣ

The y-intercept is $f(0) = a \cdot b^0 + k = a + k$ .

Theory

Key Features of Exponential Graphs

Feature	Growth ( $b > 1$ )	Decay ( $0 < b < 1$ )
Direction	Increasing	Decreasing
Shape	Rises steeply to the right	Falls toward asymptote
Asymptote	$y = k$ (approached on the left)	$y = k$ (approached on the right)
Y-intercept	$(0, a + k)$	$(0, a + k)$
Domain	All real numbers	All real numbers
Range	$y > k$ (if $a > 0$ )	$y > k$ (if $a > 0$ )

Critical difference from linear: exponential functions never cross their asymptote. The range is $(k, \infty)$ if $a > 0$ .

Example 1

Describe the graph of $f(x) = 3 \cdot 2^x - 1$ .

$a = 3$ , $b = 2$ (growth), $k = -1$ .

Y-intercept: $f(0) = 3(1) - 1 = 2$ .

Asymptote: $y = -1$ .

Behavior: increases rapidly for large $x$ , approaches $y = -1$ for negative $x$ .

Range: $y > -1$ .

Growth function, y-int

(0, 2)

, asymptote

y = -1

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