Problem 79 In the following exercises, grap... [FREE SOLUTION]

Chapter 10: Problem 79

In the following exercises, graph each function in the same coordinate system. $$ f(x)=2^{x}, g(x)=2^{x-2} $$

Short Answer

Expert verified

Plot $ f(x) = 2^x $ and $ g(x) = 2^{x-2} $ on the same graph. $ g(x) $ is a horizontal shift of $ f(x) $.

Step by step solution

Understand the Functions

Identify the given functions and their forms: - Function 1: $ f(x) = 2^x $ - Function 2: $ g(x) = 2^{x-2} $

Identify Key Points for f(x)

Determine key points to plot for $ f(x) = 2^x $ by choosing several values for $ x $: - $ x = -2, f(x) = 2^{-2} = \frac{1}{4} $ - $ x = -1, f(x) = 2^{-1} = \frac{1}{2} $ - $ x = 0, f(x) = 1 $ - $ x = 1, f(x) = 2 $ - $ x = 2, f(x) = 4 $

Identify Key Points for g(x)

Determine key points to plot for $ g(x) = 2^{x-2} $ by choosing several values for $ x $: - $ x = -2, g(x) = 2^{-4} = \frac{1}{16} $ - $ x = -1, g(x) = 2^{-3} = \frac{1}{8} $ - $ x = 0, g(x) = 2^{-2} = \frac{1}{4} $ - $ x = 1, g(x) = 2^{-1} = \frac{1}{2} $ - $ x = 2, g(x) = 2^{0} = 1 $ - $ x = 3, g(x) = 2^{1} = 2 $ - $ x = 4, g(x) = 2^{2} = 4 $

Plot the Points on the Coordinate System

Using the key points from Steps 2 and 3, plot $ f(x) = 2^x $ and $ g(x) = 2^{x-2} $ on the same coordinate system. Draw smooth curves through the points for each function.

Analyze the Graphs

Observe the transformations: - The graph of $ g(x) = 2^{x-2} $ is the graph of $ f(x) $ shifted 2 units to the right. - This reflects that $ g(x) $ represents a horizontal shift of $ f(x) $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Transformation

Understanding function transformations is crucial when graphing exponential functions. A transformation alters the position or shape of the graph without changing the function's nature. For instance, in our equation set, we have:

$ f(x) = 2^x $
$ g(x) = 2^{x-2} $

Let's dive into the transformation of $ g(x) $. We can see that $ g(x) $ represents a horizontal shift of the starting function. This shift changes the x-values' positions but retains the same growth rate.Function transformations can include:

Vertical shifts
Horizontal shifts
Reflections
Stretches and compressions

Identifying transformations helps in easily predicting and plotting the function's graph.

Key Points Plotting

Plotting key points is the backbone of graphing functions. To understand how an exponential function behaves, we calculate and plot a series of points.For $ f(x) = 2^x $, consider:

$ x = -2, f(x) = 2^{-2} = \frac{1}{4} $
$ x = -1, f(x) = 2^{-1} = \frac{1}{2} $
$ x = 0, f(x) = 1 $
$ x = 1, f(x) = 2 $
$ x = 2, f(x) = 4 $

For the shifted function $ g(x) = 2^{x-2} $, we use:

$ x = -2, g(x) = 2^{-4} = \frac{1}{16} $
$ x = -1, g(x) = 2^{-3} = \frac{1}{8} $
$ x = 0, g(x) = 2^{-2} = \frac{1}{4} $
$ x = 1, g(x) = 2^{-1} = \frac{1}{2} $
$ x = 2, g(x) = 2^{0} = 1 $
$ x = 3, g(x) = 2^{1} = 2 $
$ x = 4, g(x) = 2^{2} = 4 $

From these points, you can draw smooth curves to obtain the graph of each function. Key points serve as a guide, showing the function's rising or falling trends.

Horizontal Shift

A horizontal shift changes the position of a function's graph along the x-axis. To understand better, examine our functions $ f(x) = 2^x $ and $ g(x) = 2^{x-2} $.The function $ g(x) $ stems from $ f(x) $, but with a shift. This horizontal shift can be seen in the term $ x-2 $, which implies a shift of 2 units to the right.

If the shift had been $ x+2 $, the graph would move 2 units left.
The horizontal shift does not alter the function's shape, only its position.

To transform a function horizontally, you adjust the x-values by the shift's magnitude. This technique helps modify specific characteristics of the original function's graph.

Exponential Growth

Exponential growth describes a rapid increase, where the rate of growth is proportional to the current value. In our example, both functions exhibit exponential growth.

$ f(x) = 2^x $ starts slow, quickly escalating as x increases.
$ g(x) = 2^{x-2} $ follows the same pattern but shifted horizontally.

Exponential functions have common features:

Rapid growth after initial stages
Never touch the x-axis but approach it (asymptotic behavior)
Growth determined by the base (2 in our case)

Understanding exponential growth is essential in real-world applications like population growth, finance, and more. Recognizing these patterns in graphs helps predict future values and trends effectively.

91影视

In the following exercises, graph each function in the same coordinate system. $$ f(x)=2^{x}, g(x)=2^{x-2} $$

Short Answer

Step by step solution

Understand the Functions

Identify Key Points for f(x)

Identify Key Points for g(x)

Plot the Points on the Coordinate System

Analyze the Graphs

Key Concepts

Function Transformation

Key Points Plotting

Horizontal Shift

Exponential Growth

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