PreCal Notes 4-1

Exponential Functions CLICK

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Definition: Exponential Function

The exponential function with base The exponential function with base aa is defined by: is defined by:

ff((xx) = ) = aaxx

where where a a > 0 and > 0 and aa ≠ ≠ 1.1.

Ex 1—Evaluating Exponential

FunctionsLet Let ff((xx) = 3) = 3xx and evaluate the following: and evaluate the following:

(a)(a) ff(2)(2)

(b)(b) ff(–(–⅔)⅔)

(c)(c) ff((ππ) )

(d)(d) ff( √2 ) ( √2 )

Hint: you will need a calculator!!Hint: you will need a calculator!!

= 32 = 3 ^ 2 ENTER = 9= 3-⅔ =3 ^((–) 2÷3 ) ENTER ≈ 0.4807

= 3π = 3^π ENTER ≈ 31.544

= 3√2 = 3^√ 2) ENTER ≈ 4.7288

Graphs of Exponential Functions

F(x) = ax

Domain: All real numbersRange: (0, ∞)

a< 1 a>1decreases increases

ex 2: Identifying Graphs of Exponential

FunctionsFind the exponential function Find the exponential function ff((xx) = ) =

aaxx whose graph is given. whose graph is given.

Since f(2) = a2 = 25, we see that the base is 5.

SO… f (x) = 5x

Find the exponential function Find the exponential function ff((xx) = ) = aaxx whose graph is given. whose graph is given.

Since f(3) = a3 = 1/8 , we see that the base is ½ .

SO... f (x) = (½)x

ex 3: Identifying Graphs of Exponential

Functions

Brain Break!!!!

Speaking of Infinity…Make an infinity symbol with your right hand out in Make an infinity symbol with your right hand out in front of you. and stop your finger on the far right side of front of you. and stop your finger on the far right side of the infinity sign. the infinity sign.

Lift your left hand to be at the far left side of the infinity Lift your left hand to be at the far left side of the infinity sign. Now move your hands at the same time and the sign. Now move your hands at the same time and the same pace in the same direction to continue your same pace in the same direction to continue your infinity sign. Your hands should cross the middle at the infinity sign. Your hands should cross the middle at the same time. same time.

This one seems easy at first. Then you try to do it when This one seems easy at first. Then you try to do it when your hands are doing the infinity signs in different your hands are doing the infinity signs in different directions. You should look like a choir director if you directions. You should look like a choir director if you are doing it correctly. are doing it correctly.

We can graph Exponential Functions using transformations.

Find the parent function f(x) = 2x in the front of your book.

Ex4—TransformationsUse the graph of Use the graph of ff((xx) = 2) = 2xx to sketch the to sketch the graph of the function.graph of the function.

gg((xx) = 1 + 2) = 1 + 2xx

Notice that the line y = 1 is now a horizontal asymptote.

h(x) = –2x

Ex5 - Use the graph of f(x) = 2x to sketch the graph of the function.

EX 6: Use the graph of EX 6: Use the graph of ff((xx) = 2) = 2xx to sketch to sketch the graph of the function.the graph of the function.

k(x) = 2x –1

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