Question

What are the characteris of exponential growth

Answer 1

Exponential growth is exhibited when the rate of change—the change per instant or unit of time—of the value of a mathematical function isproportional to the function's current value, resulting in its value at any time being an exponential function of time, i.e., a function in which the time value is the exponent

Answer 2

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College Algebra

Module 11: Exponential and Logarithmic Functions

Characteristics of Graphs of Exponential Functions

LEARNING OBJECTIVES

Determine whether an exponential function and its associated graph represents growth or decay.Sketch a graph of an exponential function.

Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form f(x)=bxf(x)=bx whose base is greater than one. We’ll use the function f(x)=2xf(x)=2x. Observe how the output values in the table below change as the input increases by 1.

x–3–2–10123f(x)=2xf(x)=2x1818141412121248

Each output value is the product of the previous output and the base, 2. We call the base 2 the constant ratio. In fact, for any exponential function with the form f(x)=abxf(x)=abx, b is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of a.

Notice from the table that:

the output values are positive for all values of xas x increases, the output values increase without boundas x decreases, the output values grow smaller, approaching zero

The graph below shows the exponential growth function f(x)=2xf(x)=2x.



Notice that the graph gets close to the x-axis but never touches it.

The domain of f(x)=2xf(x)=2x is all real numbers, the range is (0,∞)(0,∞), and the horizontal asymptote is y=0y=0.

To get a sense of the behavior of exponential decay, we can create a table of values for a function of the form f(x)=bxf(x)=bxwhose base is between zero and one. We’ll use the function g(x)=(12)xg(x)=(12)x. Observe how the output values in the table below change as the input increases by 1.

x–3–2–10123g(x)=(12)xg(x)=(12)x8421121214141818

Again, because the input is increasing by 1, each output value is the product of the previous output and the base or constant ratio 1212.

Notice from the table that:

the output values are positive for all values of xas x increases, the output values grow smaller, approaching zeroas x decreases, the output values grow without bound

The graph below shows the exponential decay function, g(x)=(12)xg(x)=(12)x.



The domain of g(x)=(12)xg(x)=(12)x is all real numbers, the range is (0,∞)(0,∞), and the horizontal asymptote is y=0y=0.

A GENERAL NOTE: CHARACTERISTICS OF THE GRAPH OF THE PARENT FUNCTION f(x)=bxf(x)=bx

An exponential function with the form f(x)=bxf(x)=bx, b>0b>0, b≠1b≠1, has these characteristics:

one-to-one functionhorizontal asymptote: y=0y=0domain: (−∞,∞)(−∞,∞)range: (0,∞)(0,∞)x-intercept: noney-intercept: (0,1)(0,1)increasing if b>1b>1decreasing if b<1b<1

Use the sliders in the graph below to compare the graphs of exponential growth and decay functions. Which one is growth and which one is decay?

HOW TO: GIVEN AN EXPONENTIAL FUNCTION OF THE FORM f(x)=bxf(x)=bx, GRAPH THE FUNCTION

Create a table of points.Plot at least 3 point from the table including the y-intercept (0,1)(0,1).Draw a smooth curve through the points.State the domain, (−∞,∞)(−∞,∞), the range, (0,∞)(0,∞), and the horizontal asymptote, y=0y=0.

EXAMPLE: SKETCHING THE GRAPH OF AN EXPONENTIAL FUNCTION OF THE FORM f(x)=bxf(x)=bx

Sketch a graph of f(x)=0.25xf(x)=0.25x. State the domain, range, and asymptote.

Solution

TRY IT

Sketch the graph of f(x)=4xf(x)=4x. State the domain, range, and asymptote.

Solution

 

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