Learn on PengiCalifornia Reveal Math, Algebra 1Unit 8: Exponential Functions

8-1 Exponential Functions

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn to identify and graph exponential functions of the form f(x) = ab^x, distinguishing between exponential growth and exponential decay functions based on the value of b. The lesson covers key vocabulary including asymptotes and uses real-world contexts like the Richter scale and paper folding to recognize exponential behavior through constant multiplicative rates of change. Students also compare exponential and linear behavior and practice finding y-intercepts, domain, range, and asymptotes when graphing exponential functions.

Section 1

Exponential Function

Property

An exponential function has the form

f(x) = ab^x, \quad \text{where } b > 0 \text{ and } b \neq 1, \quad a \neq 0

The constant $a$ is the $y$ -intercept of the graph because $f(0) = a \cdot b^0 = a \cdot 1 = a$ .
The positive constant $b$ is called the base. We do not allow $b$ to be negative, because if $b < 0$ , then $b^x$ is not a real number for some values of $x$ . We also exclude $b = 1$ because $1^x = 1$ for all values of $x$ , which is a constant function.

Examples

The function $f(x) = 5(2)^x$ is an exponential function where the initial value is $a=5$ and the growth factor is the base $b=2$ .

The function $P(t) = 100(0.75)^t$ represents exponential decay with an initial amount of $100$ and a decay factor of $0.75$ .

Section 2

Exponential Growth vs. Decay: Identifying Base b

Property

For an exponential function $f(x) = ab^x$ with $a > 0$ , the base $b$ determines whether the function represents growth or decay:

If $b > 1$ , the function models exponential growth (output increases as $x$ increases).
If $0 < b < 1$ , the function models exponential decay (output decreases as $x$ increases).

Section 3

Graphing Exponential Functions: Table of Values and Smooth Curve

Property

To graph $f(x) = ab^x$ , build a table using at least two negative $x$ -values, $x = 0$ , and at least two positive $x$ -values. Plot each point $(x, f(x))$ , then draw a smooth, continuous curve that:

passes through the y-intercept $(0,\, a)$
rises steeply (growth) or falls steeply (decay) on one side
approaches but never crosses the horizontal asymptote $y = 0$ on the other side

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Exponential Function

Property

An exponential function has the form

f(x) = ab^x, \quad \text{where } b > 0 \text{ and } b \neq 1, \quad a \neq 0

Examples

The function $f(x) = 5(2)^x$ is an exponential function where the initial value is $a=5$ and the growth factor is the base $b=2$ .

The function $P(t) = 100(0.75)^t$ represents exponential decay with an initial amount of $100$ and a decay factor of $0.75$ .

Section 2

Exponential Growth vs. Decay: Identifying Base b

Property

For an exponential function $f(x) = ab^x$ with $a > 0$ , the base $b$ determines whether the function represents growth or decay:

If $b > 1$ , the function models exponential growth (output increases as $x$ increases).
If $0 < b < 1$ , the function models exponential decay (output decreases as $x$ increases).

Section 3

Graphing Exponential Functions: Table of Values and Smooth Curve

Property

passes through the y-intercept $(0,\, a)$
rises steeply (growth) or falls steeply (decay) on one side
approaches but never crosses the horizontal asymptote $y = 0$ on the other side

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Exponential Function

Property

An exponential function has the form

f(x) = ab^x, \quad \text{where } b > 0 \text{ and } b \neq 1, \quad a \neq 0

Examples

The function $f(x) = 5(2)^x$ is an exponential function where the initial value is $a=5$ and the growth factor is the base $b=2$ .

The function $P(t) = 100(0.75)^t$ represents exponential decay with an initial amount of $100$ and a decay factor of $0.75$ .

Section 2

Exponential Growth vs. Decay: Identifying Base b

Property

For an exponential function $f(x) = ab^x$ with $a > 0$ , the base $b$ determines whether the function represents growth or decay:

If $b > 1$ , the function models exponential growth (output increases as $x$ increases).
If $0 < b < 1$ , the function models exponential decay (output decreases as $x$ increases).

Section 3

Graphing Exponential Functions: Table of Values and Smooth Curve

Property

passes through the y-intercept $(0,\, a)$
rises steeply (growth) or falls steeply (decay) on one side
approaches but never crosses the horizontal asymptote $y = 0$ on the other side