Category Archives: 6.3 Exponential Functions

5.1 Exponential Functions, 6.3 Exponential Functions

Exponents: Simplify

Example:  Rewrite the expression in the form 5^p where p is an algebraic expression.

25/root{3}{5}

Solution:

The original expression
25/root{3}{5}
Rewrite the numerator with a base of 5. 
25=5^2
25/root{3}{5}=5^2/root{3}{5}
 
Rewrite the radical as a fractional exponent.  In general root{n}{x^m}=x^{m/n}.  For this example root{3}{5}= 5^{1/3}
5^2/root{3}{5}=5^2/5^{1/3}
 
Use the quotient rule for exponents to subtract the exponents.
Quotient Rule for Exponents
b^m/b^n=b^{m-n}
5^2/5^{1/3}=5^{2-1/3}
 
Simplify by getting a common denominator and combining the fractions.
5^{2-1/3}=5^{6/3-1/3}=5^{5/3}
 

 

3.5 Graphing Techiques: Transformations, 5.1 Exponential Functions, 6.3 Exponential Functions, Graphing Exponential Functions

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)

Here is a youtube video with examples.