State the domain and range of each function. 3.1 Graphs of Exponential Functions.

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State the domain and range of each function

3.1 Graphs of Exponential Functions

Exponential Function Notice that now the variable is the exponent. Where base (b), b > 0, b  1, and x is any real number.

Graphs of Exponential Equations To see the basic shape of an exponential function such as make a table of values and plot points.

xy Graph f(x) = 2 x

Notice the end behavior of the graph. As x → + ∞, f(x) → +∞, which means that the graph moves up to the right. As x → - ∞, f(x) → 0, which means that the graph has the line y=0 as an asymptote. An asymptote is a line that a graph approaches (gets close to) as you move away from the origin.

Exponential Growth Function: The graph passes through the point (0,a) [the y-intercept]. Have these characteristics… Where a > 0 and b > 1 The x-axis is an asymptote of the graph. The domain is all real numbers. The range is y > 0 if a > 0

Identify the y-intercept and asymptote State the domain and range

What do you think would happen to the shift of the graph? h determines the horizontal shift k determines the vertical shift

Domain: All Real Numbers Range: All Positive Real Numbers Domain: All Real Numbers Range: All Positive Real Numbers Asymptote: y = 0

Domain: All Real Numbers Range: All Positive Real Numbers Domain: All Real Numbers Range: y > -3 Asymptote: y = -3 Asymptote: y = 0

Domain: All Real Numbers Range: y > 2 Domain: All Real Numbers Range: y > 1 Asymptote: y = 2 Asymptote: y = 1

Exponential Decay Function Where a > 0, base b is 0 < b < 1 (b is a fraction), and x is any real number.

Decide whether f(x) is an exponential growth or exponential decay function. Because 0 < b < 1 f is an exponential decay function Because b > 1 f is an exponential growth function 1) 2)

Graphs of Exponential Equations To see the basic shape of an exponential function such as make a table of values and plot points.

xy ½ ¼ Graph f(x) = (½) x -3

State the domain, range, make a table of values and graph the following

where e is an irrational constant whose value is The exponential function with base e is called “The Natural Exponential Function” The Number e

The Natural Exponential Function

Homework: Page 185 #7-13 odd, 15-18, odd