7.2 Graph Exponential Decay Functions

 

Exponential decay function

A function of the form y = abx where a > 0 and 0 < b< 1

 

Decay factor

In a function of the form y = abx, the base b is the decay factor.

 

PARENT FUNCTION FOR EXPONENTIAL DECAY FUNCTIONS

The function y = bx, where 0 < b < 1, is the parent function for the family of exponential decay functions with base b. The general shape of the graph of y = bx is shown below.

                                                               The graph falls
from left to right,
passing through the
points (0, 1) and (1, b)

The x- axis is an
asymptote of the
graph.

The domain of y = bx is _all real numbers_. The range is _y > 0_.

 

Example 1

Graph y = abx for 0 < b < 1

Graph the function

 


Plot (0, _-2_) and                   Then, from right to left, draw a curve that begins just __below__ the x-axis, passes through the two points, and moves __down__ to the left.

 

 Example 2

Graph y = abx - h + k for 0 <b< 1

 


Graph                    + 1. State the domain and range.

 

 

 

 

Solution

Begin by sketching the graph of                      , which passes through (0, _2_ ) and 1,        . Then translate the graph _right 1 unit_ and _up 1 unit_ . Notice that the graph passes through (1, _3_) and                          .

 


The graph's asymptote is the line _y = 1_. The domain is _all real numbers_, and the range is y > 1.

 

Example 3

Televisions A new television costs $1200. The value of the television decreases by 21% each year. Write an exponential decay model giving the television's value y (in dollars) after t years. Estimate the value after 2 years. Graph the model. Use the graph to estimate when the value of the television will be $300.

 

Solution

a.   The initial amount is a = _1200_ and the percent decrease is r = _0.21_. So, the model is:

y = a(l - r)t                       Write exponential decay formula.

= _1200(1 - 0.21)t_      Substitute for a and r

= 1200(0.79)t_               Simplify.

When t = 2, the television's value is y = 1200(0.79)2 = _$748.92_.

b.    The graph passes through the points (0, _1200_) and (1, _948_). It has the _t-axis_ as its asymptote. Plot a few other points. Then draw a smooth curve through the points.

 

 

 


c.    Using the graph, you can estimate that the value of the television will be $300 after about _6_ years.