Transcript 171S5.1_p - Cape Fear Community College

MAT 171 Precalculus Algebra
Trigsted - Pilot Test
Dr. Claude Moore - Cape Fear Community College
CHAPTER 5:
Exponential and Logarithmic
Functions and Equations
5.1
5.2
5.3
5.4
5.5
5.6
Exponential Functions
The Natural Exponential Function
Logarithmic Functions
Properties of Logarithms
Exponential and Logarithmic Equations
Applications of Exponential and Logarithmic Functions
5.1 Exponential Functions
·Understand characteristics of exponential functions.
·Sketch graphs of exponential functions using transformations.
·Solve exponential equations by relating bases.
·Solve applications of exponential functions.
Omit Present Value on pages 5.1-21-23;
Example 7.
Exponential Function
The function f(x) = bx, where x is a real number, b > 0 and b ≠ 1, is
called the exponential function, base b.
The base needs to be positive in order to avoid the complex numbers that
would occur by taking even roots of negative numbers.
The following are examples of exponential functions:
Graphing Exponential Functions
To graph an exponential function, follow the steps listed:
1.
Compute some function values and list the results in a table.
2.
Plot the points and connect them with a smooth curve. Be sure to
plot enough points to determine how steeply the curve rises.
Example
Graph the exponential function y = f (x) = 2x.
Example (continued)
As x increases, y increases without bound; as x ∞, y
∞.
As x decreases, y decreases getting close to 0; as x -∞, y
0.
The x-axis, or the line y = 0, is a
horizontal asymptote. As the xinputs decrease, the curve gets
closer and closer to this line, but
does not cross it.
Example
Graph the exponential function
This tells us the graph is
the reflection of the graph
of y = 2x across the yaxis. Selected points are
listed in the table.
Example (continued)
As x increases, the
function values
decrease, getting closer
and closer to 0. The xaxis, y = 0, is the
horizontal asymptote.
As x decreases, the
function values increase
without bound.
Graphs of Exponential Functions
Observe the following graphs of exponential functions and look for
patterns in them.
For the base between 0 and 1, the graph
goes DOWN toward the x-axis to the
right.
For the base between greater than 1, the
graph goes UP to the right.
Example
Graph y = 2x – 2.
The graph is the graph of y = 2x shifted to right 2 units.
Example
Graph y = 5 – 0.5x .
The graph y = 2-x is a reflection of the graph of y = 2x across the y-axis;
y = - 2-x is a reflection across the x-axis;
y = - 2-x + 5 or y = 5 - 2x is a shift up 5 units.
Graph 3
Graph 1
Graph 2
y = 5 H.A.
Graph 4
1
Find the exponential function f(x) = bx whose graph is given as follows.
See the animation for the solutions.
http://media.pearsoncmg.com/ph/esm/esm_trigsted_colalg_1/anim/tca01_anim_0501ex04.html
The amount of money A that a principal P will grow to after t years at interest
rate r (in decimal form), compounded n times per year, is given by the formula
given below.
403/10. Math the function with one of the graphs: f(x) = 1 - ex
403/14. Graph the function by substituting and plotting points. Then check your
work using a graphing calculator: f(x) = 3-x
403/20. Graph the function by substituting and plotting points. Then check your work
using a graphing calculator: f(x) = 0.6 x - 3
403/4. Find each of the following, to four decimal places, using a calculator.
403/34. Sketch the graph of the function and check the graph with a graphing
calculator. Describe how the graph can be obtained from the graph of a basic
exponential function: f(x) = 3 (4 - x)
404/50. Use the compound-interest formula to find the account balance A with the
given conditions:
A = account balance; t = time, in years; n = number of compounding periods per
year; r = interest rate; P = principal
405/58. Growth of Bacteria Escherichia coli. The bacteria Escherichia coli are commonly found in the
human intestines. Suppose that 3000 of the bacteria are present at time t = 0. Then under certain conditions, t
minutes later, the number of bacteria present is N(t) = 3000(2)t/20.
a) How many bacteria will be present after 10 min? 20 min? 30 min? 40 min? 60 min?
b) Graph the function.
c) These bacteria can cause intestinal infections in humans when the number of bacteria reaches 100,000,000.
Find the length of time it takes for an intestinal infection to be possible.
407/76. Use a graphing calculator to match the equation with one of the figures
(a) - (n): y = 2 x + 2 -x
407/82. Use a graphing calculator to match the equation with one of the
figures (a) - (n): f(x) = (e x + e -x) / 2
407/84. Use a graphing calculator to find the point(s) of intersection of the
graphs of each of the following pairs of equations: y = 4 x + 4 -x and y = 8 - 2x
-x2
407/88. Solve graphically: ex = x3