Exponential Graphs

Understanding the characteristics and behavior of exponential graphs

Lesson Overview

Instructor Presentation

Introduction

Exponential functions follow the form y=(a^x) with the following definitions:

a = base of the function, a is a real number.

x is defined everywhere

The function is strictly monotonic in cases where a is not 1.

The function is continuous everywhere

Plot[(2^x),{x,-5,5}, PlotRange->{-2,20}]

Goals

Students will be able to plot and analyze graphs of exponential functions of the form f(x)=b·(a^(hx-c))+d using Mathematica.

Students will make correct conjectures about the appearance and behavior of exponential graphs after transformations are applied.

In exponential functions, in general, the further the base is from 0 on the positive side, the "steeper" the exponential curve.

Because the function y=(2^x) contains no varying denominator, the graph has no vertical asymptotes, and therefore is continuous everywhere.  In addition, this lack of an exclusion means that the domain consists of all real numbers.

Due to the fact that no exponent will allow the expression (2^x) become negative, there exists a horizontal asymptote at y=d.

Horizontal and vertical shifting occur when c and d are non-zero numbers.  This shifting follows the same rules as in polynomial functions, studied earlier.

Reflections in the x-axis and y-axis occur when b or h is negative.  When b<0, the graph is reflected in the y-axis.  When h<0, the graph shows an x-axis reflection.

Concept Exploration

Bases of exponential functions

Manipulate[Plot[a^x,{x,-5,5}, PlotRange->{-4,40}, PlotLabel->a^x],{a,0.001,5}]

For values of the base between 0 and 1, the closer the base is to zero, the steeper the graph appears.

For values of the base greater than 1, as the base increases, the graph gets steeper.

Exponential functions and horizontal & vertical shifting

Plot[{2^x, 2^x-4, (2^(x-4))}, {x,-10,10}, PlotRange->{-10,10}]

Vertical and horizontal shifting behave in the same way for exponential graphs as with other "algebraic" graphs previously studied.

Exponential functions and reflection

Manipulate[Show[Plot[{(a^x),-(a^x)}, {x,-10,10}, PlotRange->{-10,10}],PlotLabel->{(a^x),-(a^x)}],{a,0.1,5}]

For b = -1, the exponential graph is reflected in the x-axis.

Manipulate[Show[Plot[{(a^x),(a^(-x))}, {x,-10,10}, PlotRange->{-10,10}],PlotLabel->{(a^x),(a^(-x))}],{a,0.1,5}]

For h = -1, the exponential graph is reflected in the y-axis.

Examples

Example 1

Run the following demonstration and give an explanation for what each variable does to the graph when changed.

Manipulate[Show[Plot[d*(a^(x+b))+c,{x,-10,10},PlotRange->{-10,10}],PlotLabel->d*(a^(x+b))+c],{a,0.1,5},{b,-5,5},{c,-5,5},{d,{-1,1}}]

a :

b :

c :

d :

Example 2

Create and plot the following exponential graphs choose an appropriate window.

An exponential graph with a base of 6, reflected in the y-axis.

An exponential graph with a base of (1/2), shifted up 3 units and left 5 units

An exponential graph with a base of 7.3, shifted up 1 unit, then reflected in the x-axis.

Example 3

Describe the transformations, in the correct order, that occured to create the given exponential graph.

y=-(3^x)-8

y=(4^(-x))+3

y=2-(9^(x+5))

Answer Key

Example 1

a : As a gets very close to zero from the positive side, the graph gets steeper. 
   For a > 1, as a increases, the graph gets steeper.  As a approaches 1 from the 
right or left, the graph gets less steep.

b : Indicates the horizontal shift.  As b gets smaller, the graph shifts 
right b units.  As b gets larger, the graph shifts left b units.

c : Indicates the vertical shift.  As c increases, the graph shift up c units.  
  As c decreases, the graph shifts down c units.

d : When d = -1, the graph is decreasing.  When d = 1, the graph is increasing.  
  Comparing the same graph, except the value of d, when changing from 1 to - 1, the
graph is reflected about a horizontal line, namely, y = c.

Example 2

An exponential graph with a base of 6, reflected in the y-axis.

Plot[-(6^x), {x,-5,5},PlotRange->{-800,10}]

An exponential graph with a base of (1/2), shifted up 3 units and left 5 units

Plot[3+((((1/2)))^(x+5)), {x,-10,5},PlotRange->{-1,5}]

An exponential graph with a base of 7.3, shifted up 1 unit, then reflected in the x - axis.

Plot[-((7.3^x)+1),{x,-5,5},PlotRange->{-10,10}]

Example 3

Describe the transformations, in the correct order, that occured to create the given exponential graph.

y=-(3^x)-8

Exponential graph with a base of 3 reflected in the x - axis, then shifted down 8 units.  Or Exponential graph with a base of 3 shifted up 8 units, then reflected about the x - axis.

y=(4^(-x))+3

Exponential graph with a baes of 4, reflected in the y - axis, then shifted up 3 units.

y=2-(9^(x+5))

Exponential graph with a base of 9, shifted left 5 units, reflected in the x - axis, then shifted up 2 units.