Learn on PengienVision, Algebra 1Chapter 6: Exponents and Exponential Functions

Lesson 2: Exponential Functions

In this Grade 11 enVision Algebra 1 lesson, students learn to describe and graph exponential functions of the form f(x) = a · b^x, identifying key features such as the asymptote, domain, range, and constant ratio. Students practice writing exponential functions from tables and graphs, and compare exponential growth to linear functions using real-world applications like virus spread modeling. The lesson covers Chapter 6 on Exponents and 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

Identifying Initial Amount vs Constant Ratio

Property

In exponential functions $f(x) = a \cdot b^x$ , the parameter $a$ is the initial amount (y-intercept) and $b$ is the constant ratio (multiplicative factor).

Examples

Section 3

Writing Exponential Functions from Data

Property

To write an exponential function $f(x) = a \cdot b^x$ from data points or a table:

Find the constant ratio $b$ by dividing consecutive $y$ -values: $b = \frac{y_2}{y_1} = \frac{y_3}{y_2}$
Find the initial value $a$ using any point $(x, y)$ : $a = \frac{y}{b^x}$

Examples

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

Identifying Initial Amount vs Constant Ratio

Property

In exponential functions $f(x) = a \cdot b^x$ , the parameter $a$ is the initial amount (y-intercept) and $b$ is the constant ratio (multiplicative factor).

Examples

Section 3

Writing Exponential Functions from Data

Property

To write an exponential function $f(x) = a \cdot b^x$ from data points or a table:

Find the constant ratio $b$ by dividing consecutive $y$ -values: $b = \frac{y_2}{y_1} = \frac{y_3}{y_2}$
Find the initial value $a$ using any point $(x, y)$ : $a = \frac{y}{b^x}$

Examples

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

Identifying Initial Amount vs Constant Ratio

Property

In exponential functions $f(x) = a \cdot b^x$ , the parameter $a$ is the initial amount (y-intercept) and $b$ is the constant ratio (multiplicative factor).

Examples

Section 3

Writing Exponential Functions from Data

Property

To write an exponential function $f(x) = a \cdot b^x$ from data points or a table:

Find the constant ratio $b$ by dividing consecutive $y$ -values: $b = \frac{y_2}{y_1} = \frac{y_3}{y_2}$
Find the initial value $a$ using any point $(x, y)$ : $a = \frac{y}{b^x}$