7.1 Graph Exponential Growth Functions

.

Exponential function

A function of the form y = ab^x where a ¹ 0 and the base b is a positive number other than one

Exponential growth function

A function of the form y = ab^x where a > 0 and b > 1

Growth factor

In a function of the form y = ab^x, b is the growth factor.

Asymptote

A line that a graph approaches more and more closely

PARENT FUNCTION FOR EXPONENTIAL GROWTH FUNCTIONS

The function y = b^x, where b _>_ 1, is the parent function for the family of exponential growth functions with base _b_. The general shape of the graph of y = b^x is shown below.

The x-axis is an asymptote of the graph. An asymptote is a line that a graph approaches more and more closely.

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

The domain of y = b^x is _all real numbers_. The range is _y > 0_.

Example 1

Graph y = ab^x for b > 1

Graph the function y = · 6^x.

Solution

Plot and . Then, from left to right, draw a curve that begins just _above_ the x-axis, passes through the two points, and moves _up to the right_.

Example 2

Graph y = ab^{x - h} + k for b > 1

Graph y = 2 · 3^{x - 2} - 2. State the domain and range.

Solution

Begin by sketching the graph of y = 2 · 3^x, which passes through (0, _2_) and (1, _6_). Then translate the graph _right 2 units_ and _down 2 units_.

The graph's asymptote is the line _y = -2_. The domain is all real _numbers_, and the range is _y > -2_.

Example 3

Buffalo In the last 12 years, an initial population of 38 buffalo in a state park grew by about 7% per year.

a. Write an exponential growth model giving the number n of buffalo after t years. About how many buffalo were in the park after 7 years?

b. Graph the model. Use the graph to estimate the year when there were about 53 buffalo.

Solution

a. The initial amount is a = _38_ and the percent increase is r = _0.07_ . So, the exponential growth model is:

n = a(l + r)^t Write exponential growth model.

= _38(1 + 0.07)^t_ Substitute for a and r.

= _38 · 1.07^t_ Simplify.

Using this model, you can estimate the number of buffalo after 7 years (t = 7) to be n = _38 · 1.07⁷_ » _61_ buffalo.

b. The graph passes through the points (0,_38_) and (1,_40.66_). Plot a few other points. Then draw a smooth curve through the points. Using the graph, you can estimate that the number of buffalo was 53 after about _5_ years.

COMPOUND INTEREST

Consider an initial principal P deposited in an account that pays interest at an annual rate r (expressed as a decimal), compounded n times per year. The amount A in the account after t years is given by this equation:

A =

Example 4

You deposit $2900 in an account that pays 3.5% annual interest. Find the balance after 1 year if the interest is compounded monthly and annually.

a. With interest compounded monthly, the balance after 1 year is:

A = 2900 Substitute for P, r, n, and t.

= 2900 _(1.0029)¹²_ Simplify.

» _3002.55_ Use a calculator.

The balance at the end of 1 year is _$3002.55_.

b. With interest compounded annually, the balance after 1 year is:

A = 2900

= 2900 _(1.035)¹_

= _3001.50_

The balance at the end of 1 year is _$3001.50_.