exponential functions algebra iii, sec. 3.1 objective recognize, evaluate, and graph exponential...
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Exponential FunctionsAlgebra III, Sec. 3.1
Objective
Recognize, evaluate, and graph exponential functions.
Important Vocabulary
Algebraic functions – linear, polynomial, and rational functions
Transcendental functions – nonalgebraic functions (exponential & logarithmic functions)
Natural base e – 2.718281828…
Continuous compounding – the number of compoundings increase without bound
Exponential Functions
The exponential function f with base a is denoted by ______________ where a > 0, a ≠ 1, and x is any real number.
Example (on your handout)
Use a calculator to evaluate the expression
2.6265
Example 1
Use a calculator to evaluate each function at the indicated value of x.
a.) at x = π
687.2913
Example 1
Use a calculator to evaluate each function at the indicated value of x.
b.) at x = ½
0.3536
Example 1
Use a calculator to evaluate each function at the indicated value of x.
c.) at x = -2.5
1.7469
Graphs of Exponential Fns
For a > 1, the graph of is ___________________ over its domain.
For a > 1, the graph of is ___________________ over its domain.
increasing
decreasing
Graphs of Exponential Fns (cont.)
For the graph of or , a > 1,
the domain is ___________,
the range is __________,
and the y-intercept is __________.
Also both graphs have _________ as a horizontal asymptote.
(-∞, ∞)
(0, ∞)
(0, 1)
y = 0
Example 2In the same coordinate plane, sketch the graph of each function.
a.)
b.) Create a table.
x f(x) g(x)
-2
-1
0
1
2
Plot the points. Sketch the curve.
Example 3In the same coordinate plane, sketch the graph of each function.
a.)
b.) Create a table.
x f(x) g(x)
-2
-1
0
1
2
Plot the points. Sketch the curve.
Example (on your handout)
Sketch the graph of the function.
Create a table. x f(x)
-2
-1
0
1
2
Plot the points. Sketch the curve.
Exponents
x x2 x3 x4 x5
1
2
3
4
5
6
Example 4
Solve using the one-to-one property.
a.) Write as common bases.
Exponents must equal.
Solve for x.
Example 4
Solve using the one-to-one property.
b.)Write as common bases.
Exponents must equal.
Solve for x.
Example 5
Describe the graph as a transformation of the graph of
a.)
b.)
c.)
Horizontal shift: 3 units right
Reflection over x-axis
Reflection over y-axis, Vertical shift: 3 units up
The Natural Base e
The natural exponential function is given by the function _______________.
e = 2.718281828… is the constant x is the variable
Example 6Use a calculator to evaluate the function given by
at each individual value of x to four decimal places.
a.) x = -0.4
b.) x = -7.1
c.) x = 0.72
0.6703
0.0008
2.0544
Example (on your handout)
Use a calculator to evaluate the expression
1.8221
Example 7Sketch the graph of the natural exponential function.
Create a table. x f(x)
-2
-1
0
1
2
Plot the points. Sketch the curve.
ApplicationsAfter t years, the balance A in an account with principle P and annual interest rate r (in decimal form) is given by the formulas:
For n compoundings per year:
For continuous compounding:
Example (on your handout)
Find the amount in an account after 10 years if $6000 is invested at an interest rate of 7%.
a.) compounded monthly
b.) compounded continuously
Example 8On the day of a child’s birth, a deposit of $25,000 is made in a trust fund that pays 8.25% interest. Determine the balance in this account on the child’s 26th birthday if the interest is compounded…
a.) quarterly.
b.) monthly.
c.) continuously.