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

8-2 Transformations of Exponential Functions

In this Grade 9 lesson from California Reveal Math, Algebra 1, students learn how to apply transformations — including vertical and horizontal translations, vertical and horizontal dilations, and reflections — to exponential functions of the form g(x) = a·b^(x−h) + k. Students practice identifying how the parameters a, h, and k affect the graph of the exponential parent function f(x) = b^x, including shifts in the asymptote. They also write equations of transformed exponential functions by analyzing key features of graphs.

Section 1

Vertical Translations of Exponential Functions

Property

A vertical translation of an exponential function shifts the graph up or down by adding or subtracting a constant $k$ outside the base:

f(x) = b^x + k

Section 2

Horizontal Translations of Exponential Functions

Property

A horizontal translation of an exponential function shifts the graph left or right. Starting from the parent function $f(x) = b^x$ , a horizontal translation produces:

g(x) = b^{(x-h)}

Section 3

Vertical Dilation and Reflection: f(x) = ab^x

Property

For an exponential function of the form $f(x) = ab^x$ , the value of $a$ determines a vertical dilation (stretch or compression) and, when $a < 0$ , a reflection across the $x$ -axis.

If $|a| > 1$ , the graph is vertically stretched by a factor of $|a|$ .
If $0 < |a| < 1$ , the graph is vertically compressed by a factor of $|a|$ .
If $a < 0$ , the graph is reflected across the $x$ -axis.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Vertical Translations of Exponential Functions

Property

A vertical translation of an exponential function shifts the graph up or down by adding or subtracting a constant $k$ outside the base:

f(x) = b^x + k

Section 2

Horizontal Translations of Exponential Functions

Property

A horizontal translation of an exponential function shifts the graph left or right. Starting from the parent function $f(x) = b^x$ , a horizontal translation produces:

g(x) = b^{(x-h)}

Section 3

Vertical Dilation and Reflection: f(x) = ab^x

Property

For an exponential function of the form $f(x) = ab^x$ , the value of $a$ determines a vertical dilation (stretch or compression) and, when $a < 0$ , a reflection across the $x$ -axis.

If $|a| > 1$ , the graph is vertically stretched by a factor of $|a|$ .
If $0 < |a| < 1$ , the graph is vertically compressed by a factor of $|a|$ .
If $a < 0$ , the graph is reflected across the $x$ -axis.

Overview

Lesson overview

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

Overview

Section 1

Vertical Translations of Exponential Functions

Property

A vertical translation of an exponential function shifts the graph up or down by adding or subtracting a constant $k$ outside the base:

f(x) = b^x + k

Section 2

Horizontal Translations of Exponential Functions

Property

A horizontal translation of an exponential function shifts the graph left or right. Starting from the parent function $f(x) = b^x$ , a horizontal translation produces:

g(x) = b^{(x-h)}

Section 3

Vertical Dilation and Reflection: f(x) = ab^x

Property

For an exponential function of the form $f(x) = ab^x$ , the value of $a$ determines a vertical dilation (stretch or compression) and, when $a < 0$ , a reflection across the $x$ -axis.

If $|a| > 1$ , the graph is vertically stretched by a factor of $|a|$ .
If $0 < |a| < 1$ , the graph is vertically compressed by a factor of $|a|$ .
If $a < 0$ , the graph is reflected across the $x$ -axis.