Category Archives: 5.1 Exponential Functions

Solving an Exponential Equation: Relating the Bases

Example:  Solve the exponential equation.

4^{x-9}=1/1024

Solution:

The exponential equation
4^{x-9}=1/1024
Try to write both sides of the equation with the same base.  Try 4 since there is a base of 4 on the left 4^{x-9}=1/4^5
Using a property of negative exponents move the base to the numerator  4^{x-9}=4^{-5}
Now that that the bases are the same the exponents must be equal  4^{x-9}=4^{-5}

x-9=-5

Solve for x
 x-9=-5

x-9+9=-5+9

x=4

The solution the the exponential equation is 4.

 

Graphing an Exponential Equation by Transformations

Example: For the function below.  Graph using transformations.  Find the y-intercept.  State the horizontal asymptote and the domain and range.

f(x)=-2^{x+2}+2

First we must examine the base function y=2^x

Graph using plotting points.  We can use the standard set of x-values to find ordered pairs.

xy
-22^(-2)=1/4
-12^(-1)=1/2
02^0=1
12^1=2
22^2=4

The graph below shows the points plotted and the line that connects them.  This graph has a horizontal asymptote at y=0.  The domain is (- infty, infty)  and the range is (0, infty)

 

Analyze the transformations.

f(x)=-2^{x+2}+2

The +2 in the exponent shifts the graph left 2 units.

The – in the front of the base reflects the graph over the x-axis.

The +2 next to the base shifts the graph and the horizontal asymptote up two units.

You can see the graph after the transformations.

The horizontal asymptote is y=2.  The domain is (- infty, infty)  and the range is (- infty, 2)

To find the y-intercept we let x=0.

f(0)=-2^{0+2}+2

f(0)=-(2^{2})+2

f(0)=-(4)+2

f(0)=-4+2

f(0)=-2

Thus the y-intercept is (0,-2)