Exponential Functions by Gf7D6vn

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									7.1 Graphing Exponential & Growth Functions

I.      Exponential Functions         y  ab x , where 0  b  1or b  1

Graph the family of parent functions given below on the same Cartesian plane using different colors.
Make a table of values if necessary to find accurate points or do the mental math.
For    b>1                                    y  2x                  y  3x             y  10 x




                                             What do you notice as the exponential base changes?




7.2 Graphing Exponential Decay Functions

For 0 <b <1 (a fraction)
                    x                          x
              1                        1
Graph      y                       y  
              2                        4

 For 0 <b <1, notice that the only change in the
characteristics is that the function is now
________________.

Rewrite the functions above using negative
exponents.

y                             y

If we consider the parent functions to have a base of 2
and 4 respectively, what type of transformation
occurs due to the negative in the exponent? _______________


________________ Asymptote at _______________.            Write the end behavior of the functions above

in sentence form. _______________________________________________________________

________________________________________________________________________________

Why do all five graphs have a y intercept of (0,1)?
II.       Transformations of the Exponential function

                                f ( x)  a  b x  h  k


WITHIN the function (exponent)                                      OUTSIDE the
                                                               function(exponent)
Horizontal translation, r y                                    r x , Vertical Translation
      affect x-coordinate                                      affect y-coordinate

Horizontal Translation FIRST                                         Vertical Translation LAST
x     y

0     1        (this is the pivot point)
1     base

Steps to graph Exponential Functions
1. Determine the transformations
2. Determine the horizontal asymptote and graph.
3. Using the T-chart apply the transformations
4. Graph the two points
 Each reflection changes a function from increasing to decreasing or from decreasing to increasing.
 5. Sketch the curve.
 6. Identify the y-intercept. ( Let x equal to zero and calculate)

Examples: Sketch. Label the H.A. and y-intercept and pivot point. Then give the end behavior below.
1) y  2 x1                      2) y  3 x  2                       3) y  2 5 x  1




                                                           x
                                                  1
                                         4) y      2                   5) f  x   23x6
                                                  2
III. Simplifying Exponential Expressions

                                                                                  5x
1. 23  2 4           2.    e 2  e 6                 3. e x  5e x 3   4.
                                                                                 5 2 x4



IV. Solving Exponential Equations

The PROPERTY OF EQUALITY for exponential functions:                        If au = av , then u = v for a > 1

To solve exponential equations:
Step 1) Get the same base on both sides.
Step 2) Set the exponents equal to each other. Solve for the variable.

Solve for the variable in each equation below.


                                                                                       3x  9 x
                                                                                           3
5. 3 x+1
           = 81               6.              4   x2
                                                        2   x
                                                                           7.




                                                                                                            1
                                                                                       e  x  (e x ) 2 
                                                                                               2
                                                   x
8.         4 2 0
            x     x
                              9.    2 8  x
                                                        4       x
                                                                           10.
                                                                                                            e3

								
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