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March 27, 2009
Honors Algebra 2 notes and problems
“More on exponential functions” page 1
More on exponential functions
Recognizing exponential functions from their tables



In the table of an exponential function, if the x-values go up by 1’s, the y-values are
repeatedly multiplied by b.
A table can be recognized as representing an exponential function if whenever a fixed number
is repeatedly added to x, the y-values are repeatedly multiplied by some number (example:
a table where every time that 2 is added to x, y is multiplied by 9).
Compare this “add-multiply” property of exponential function tables with the “add-add”
property of linear function tables. (For linear functions, when 1 is added to x, the slope m is
added to y.)
Finding an exponential function that exactly fits a table (given perfect
correlation)
Here is how to find a function formula f(x) = a · bx when given a table (or some points or
function values):
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At a minimum, two points are needed to be able to find the function formula.
a is always equal to f(0). [Reason: f(0) = a · b0 = a · 1 = a.]
The “base” b can be found as a ratio using any two points whose x-values are 1 apart; for
example:
b = ff ((10)) , or b = ff ((12)) , or b = ff ((23)) , etc. [Why is this true?]
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If the above facts aren’t enough to find a and b, here is a process that will work when given
any two points:
 Write the equation f(x) = a · bx twice.
 Substitute the values of x and f(x) for one point into the first equation; do the same for the
second point and the second equation.
 Now you have two equations with two unknowns a and b. Solve for a and b using
substitution. [Other equation solving methods we’ve studied, such as elimination or
matrix row operations, cannot be used here because they work only for linear equations.]
 Finally, write f(x) = a · bx and put in the a and b values you’ve found.
Finding an exponential function that approximately fits a table
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Finding the “best fitting” exponential function for a table is called exponential regression.
Your calculator can perform exponential regression. The method is almost identical to linear
regression. (The only difference is: on the CALC menu choose ExpReg instead of LinReg.)
If you need a review of the calculator steps, see page 419 example 3.
Name
March 27, 2009
Honors Algebra 2 notes and problems
“More on exponential functions” page 2
Problem set
1. For all parts of this problem, let f(x) = 2x.
a. Show that f(x + 1) = 2 f(x).
b. Complete this sentence with a transformation description, based on the equation from
part a: “Translating the graph of f(x) to the left by 1 is equivalent to
_____________________________________________.”
c. Make a carefully drawn graph of f(x) and 2 f ( x) on a your TI. Use the table to confirm
the transformation description you wrote in part b.
2. Answer the questions that follow about these tables:
x
0
1
2
3
y
8
4
2
1
x
0
2
4
6
y
¼
1
4
16
x
2
4
6
8
y
3
18
33
48
x
2
4
6
8
y
3
18
54
162
a. Decide whether each table above represents an exponential function, or a linear function,
or neither. Explain how you make each decision.
b. For each function above that you identified as exponential, find an a · bx formula
(don’t use a calculator, and try to do any arithmetic in your head).
c. For each function above that you identified as linear, find a linear function formula
(don’t use a calculator, and try to do any arithmetic in your head).
3. a. Solve the equation 2x = 8 in your head.
b. Solve the equation 2x = 7 using a graphical method on your calculator. (For example,
make two graphs and find their intersection.)
4. Solve each equation by factoring.
a. 4 2 x  10  4 x  16  0
b. 2 2 x  3  2 x 1  8  0
2
1
c. x 3  7 x 3  12  0
5. Suppose f(x) is an exponential function with f(2) = 100 and f(5) = 60.
a. Without finding a general function formula for f(x), find the values of f(8) and f(11).
Hint: Use the add-multiply property.
b. Find a function formula for f(x) using a method where you set up two f(x) = a · bx
equations, then solve the system of equations to find a and b. (Ugly Numbers!! Round to
the nearest thousandth)
6. Here are some values of a function g(x) that is approximately but not exactly exponential:
g(0) = 4, g(1) = 7, g(2) = 15, g(3) = 40. Using your calculator, find the exponential function
that best fits function g(x). Graph both the data points and the exponential function together
on your calculator screen.
Name
March 27, 2009
Honors Algebra 2 notes and problems
“More on exponential functions” page 3
7. The standard form for an exponential function formula is a · bx. Here are some exponential
functions that are not given in standard form. Rewrite the formulas in standard form.
b. g(x) = 4 · 10(x – 1).
a. f(x) = 5 · 3(2x).
c. h(x) = 2(1 – 3x)
8. A bank account pays interest at some fixed rate, compounded monthly. Suppose that
someone puts $1,000 in the account. After 10 years, the balance has grown to $1,500.
Determine the monthly interest rate and annual interest rate, as a percentage.
Hint: Use a compound interest equation (as covered last Friday or in section 6.2), but in this
problem the rates will be the unknown while all the other quantities are known. Solve the
equation either using substitution or a time conversion adjustment.
SOLUTIONS:
1. b, Vertical stretch by a factor of 2.
2. a. exp, exp, lin, neither
b. y  8(.5) x , y  .25(2) x
c. y 
15
x  12
2
3. a. x=3; b. x=2.8073549
4. a. x 
1
3
, x
2
2
b. x=1, x=2
c. x=27, x=64
5. a. f(8)=36, f(11)=21.6
b. y  140.933.843
6.
y  3.603(2.153) x
7.
a. f ( x)  5  9
8.
Monthly: .338%
x
x
b. g ( x)  .4(10)
x
Annually: 4.138%
1
c. h( x)  2   
8
x