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Notes for 4.2 Exponential Functions (pp. 402 – 414)
I. Exponents and Their Properties (pp. 402 – 404)
Def. An exponential function has its variable as an exponent.
It is written as f ( ) x = ax , where a > 0, a ≠ 1 and x is any real number.
Ex. Recall: x^3^ = 8 vs. 2 x^ = 8 is an exponential equation.
*Calculator reminders: To put an exponent into the calculator, use the ^ key. If the exponent is a fraction, you must put parentheses around the entire fraction so that the calculator will “see” the entire exponent.
Ex. 2 3.14 = 2 ^ 3.14 ENTER and
3 22 = 2 ^ (3 / 2) ENTER
Ex. If f ( ) x = 2 x , find f(-1), f(3), f ( 5/2), f(4.92) Use calculator.
Recall Properties of Exponents:
- am i an = a m^ + n
m n mn a = a
a^0^ = 1
m^1 a (^) am
- 1 anything^ = 1
II. Exponential Functions (pp. 404 – 407)
A. For f ( ) x = ax ,where a > 1. (p. 405) Ex. f ( ) x = 2 x. Graph is an upwards swoop.
Characteristics of the exponential when a (the base) > 1:
1. Domain: ( −∞ ∞, ) and Range: ( 0, ∞ ) graph lies on top of the x-axis.
- Graph has no x intercept because it doesn’t cross the x axis… only approaches it.
- Graph has a y intercept at (0, 1), since 2 0 = 1 (in fact, a^0^ = 1 , so that point is on all exponential functions!!)
- Graph is increasing.
- Graph is continuous.
- Any graph contains the points (0, 1), (1,a), and (-1,
a
- The larger the value of the base, the steeper the graph.
B. For f ( ) x = ax , where a < 1 (fraction base) Ex.
x f x = ;
x g x = Graph is a
downwards swoop. Characteristics of an exponential, where the base ( a ) is < 1:
1. Domain: ( −∞ ∞, ) and Range: ( 0, ∞ ) graph lies on top of the x-axis.
- Graph has no x intercept because it doesn’t cross the x axis… only approaches it.
- Graph has a y intercept at (0, 1).
- Graph is decreasing.
- Graph is continuous.
- Any graph contains (0, 1), (1, a) and (-1,
a
- The smaller the value of a , the steeper the graph.
Translations still work. p. 407, Ex. 3 x 2 x^ − 2 x 2 x +^3 2 x^ + 3
III. Solving Exponential Equations (pp. 407 - 408)
- This type of exponential uses the property that if a m^ = an (the bases are the same), then m = n (the exponents will be the same.
Ex. 3 x +^1 = 81
Since 81 = 3^4 , we can write the equation as 3 x +^1 = 34 Now that the bases (3) are the same, then x + 1 = 4 And x = 3
Ex. 2 7
x − (^) =
5 2 x −^7 = 5 −^1 (recall that a negative exponent will take care of a fraction) 2 x – 7 = − 1 2 x = 6 x = 3 Omit p. 408, Example 6.
Continuously compounded interest runs the compounding times faster and faster and faster, until it get to be “instantaneous”. The formula is: A = Pert , where A = Future Value, P = Present Value or Principal, r = rate as a decimal, and t = time in years. The e is that number 2.71828 that the ( ) always settles to…(demo in video).
Ex. For an investment of $1000 at 4% for 10 years, compare the different amounts earned when it is compounded in the chart below:
Compoundings $1 $ Annually Semiannually Quarterly Monthly Daily Continuously (use different Formula)
Assignments: Text: pp. 414 – 417#1 – 21 every other odd, 33, 38, 39 – 55 odd, 71, 73, 75ab, 77 - 80
