Learn on PengiBig Ideas Math, Algebra 1Chapter 6: Exponential Functions and Sequences

Lesson 3: Exponential Functions

Property An exponential function is a function of the form $f(x) = b^x$, where $b$ is a positive real number $(b 0)$ and $b \neq 1$. The domain of an exponential function is the set of all real numbers.

Section 1

Exponential Functions

Property

An exponential function is a function of the form $f(x) = b^x$ , where $b$ is a positive real number $(b > 0)$ and $b \neq 1$ . The domain of an exponential function is the set of all real numbers.

Examples

$f(x) = 5^x$ is an exponential function since the base, 5, is a positive number not equal to 1.

$g(x) = (\frac{1}{3})^x$ is an exponential function because its base, $\frac{1}{3}$ , is positive and not equal to 1.

Section 2

Properties of Exponential Graphs

Property

Properties of the Graph of $f(x) = a^x$

when $a > 1$	when $0 < a < 1$
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$(0, \infty)$	$(0, \infty)$
x-intercept	none	none
y-intercept	$(0, 1)$	$(0, 1)$
Contains	$(1, a), (-1, \frac{1}{a})$	$(1, a), (-1, \frac{1}{a})$
Asymptote	x-axis, the line $y=0$	x-axis, the line $y=0$
Basic shape	increasing	decreasing

Examples

The graph of $f(x) = 4^x$ is an increasing curve because $a > 1$ . It passes through the points $(0, 1)$ and $(1, 4)$ .
The graph of $g(x) = (\frac{1}{4})^x$ is a decreasing curve because $0 < a < 1$ . It passes through the points $(0, 1)$ and $(-1, 4)$ .
Both graphs $f(x) = 4^x$ and $g(x) = (\frac{1}{4})^x$ have a domain of all real numbers, $(-\infty, \infty)$ , and a range of all positive numbers, $(0, \infty)$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Exponential Functions

Property

An exponential function is a function of the form $f(x) = b^x$ , where $b$ is a positive real number $(b > 0)$ and $b \neq 1$ . The domain of an exponential function is the set of all real numbers.

Examples

$f(x) = 5^x$ is an exponential function since the base, 5, is a positive number not equal to 1.

$g(x) = (\frac{1}{3})^x$ is an exponential function because its base, $\frac{1}{3}$ , is positive and not equal to 1.

Section 2

Properties of Exponential Graphs

Property

Properties of the Graph of $f(x) = a^x$

when $a > 1$	when $0 < a < 1$
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$(0, \infty)$	$(0, \infty)$
x-intercept	none	none
y-intercept	$(0, 1)$	$(0, 1)$
Contains	$(1, a), (-1, \frac{1}{a})$	$(1, a), (-1, \frac{1}{a})$
Asymptote	x-axis, the line $y=0$	x-axis, the line $y=0$
Basic shape	increasing	decreasing

Examples

The graph of $f(x) = 4^x$ is an increasing curve because $a > 1$ . It passes through the points $(0, 1)$ and $(1, 4)$ .
The graph of $g(x) = (\frac{1}{4})^x$ is a decreasing curve because $0 < a < 1$ . It passes through the points $(0, 1)$ and $(-1, 4)$ .
Both graphs $f(x) = 4^x$ and $g(x) = (\frac{1}{4})^x$ have a domain of all real numbers, $(-\infty, \infty)$ , and a range of all positive numbers, $(0, \infty)$ .

Overview

Lesson overview

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

Overview

Section 1

Exponential Functions

Property

An exponential function is a function of the form $f(x) = b^x$ , where $b$ is a positive real number $(b > 0)$ and $b \neq 1$ . The domain of an exponential function is the set of all real numbers.

Examples

$f(x) = 5^x$ is an exponential function since the base, 5, is a positive number not equal to 1.

$g(x) = (\frac{1}{3})^x$ is an exponential function because its base, $\frac{1}{3}$ , is positive and not equal to 1.

Section 2

Properties of Exponential Graphs

Property

Properties of the Graph of $f(x) = a^x$

when $a > 1$	when $0 < a < 1$
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$(0, \infty)$	$(0, \infty)$
x-intercept	none	none
y-intercept	$(0, 1)$	$(0, 1)$
Contains	$(1, a), (-1, \frac{1}{a})$	$(1, a), (-1, \frac{1}{a})$
Asymptote	x-axis, the line $y=0$	x-axis, the line $y=0$
Basic shape	increasing	decreasing

Examples

The graph of $f(x) = 4^x$ is an increasing curve because $a > 1$ . It passes through the points $(0, 1)$ and $(1, 4)$ .
The graph of $g(x) = (\frac{1}{4})^x$ is a decreasing curve because $0 < a < 1$ . It passes through the points $(0, 1)$ and $(-1, 4)$ .
Both graphs $f(x) = 4^x$ and $g(x) = (\frac{1}{4})^x$ have a domain of all real numbers, $(-\infty, \infty)$ , and a range of all positive numbers, $(0, \infty)$ .