Problem 11 Graph each exponential function.... [FREE SOLUTION]

Chapter 13: Problem 11

Graph each exponential function. $$g(x)=2^{x+1}$$

Short Answer

Expert verified

Create a table of values for x and g(x), such as: x | g(x) --|---- -3 | 1/4 -2 | 1/2 -1 | 1 0 | 2 1 | 4 Plot the points (-3, 1/4), (-2, 1/2), (-1, 1), (0, 2), and (1, 4) on a Cartesian plane. Sketch the graph of $g(x)=2^{x+1}$, which looks like y = 2^x shifted 1 unit to the left with a horizontal asymptote at y = 0.

Step by step solution

Create a table of values for x and g(x)

First, we should create a table of values for x and g(x) to get an idea of the graph's shape. Choose at least 5 values of x, including both positive and negative values, and calculate the corresponding values of g(x). x | g(x) --|---- -3 | 2^(-2) = 1/4 -2 | 2^(-1) = 1/2 -1 | 2^(0) = 1 0 | 2^(1) = 2 1 | 2^(2) = 4

Plot the points from the table

Using the table of values, plot the points on a Cartesian plane for each of the x-values we've chosen: (-3, 1/4), (-2, 1/2), (-1, 1), (0, 2), and (1, 4) Remember that the graph of an exponential function will always have an asymptote at y = 0, so as the x-values decrease, the g(x)-values will approach, but never touch, the x-axis.

Sketch the exponential function

Now, connect the points and sketch the graph of g(x) = 2^(x+1). This graph should look like the function y = 2^x shifted 1 unit to the left. The curve should be increasing as it moves to the right. Be sure to add arrows on the ends of the curve to indicate that it continues indefinitely in the positive and negative directions of the x-axis.

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Key Concepts

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

Graphing Exponential Functions

Graphing exponential functions involves understanding how these functions behave and what features define their graphs. Exponential functions, such as $ g(x) = 2^{x+1} $, exhibit specific characteristics:

They have a constant base raised to a variable exponent, which causes a rapid increase (or decrease) in the function's values.
Graphs of exponential functions are curves that increase or decrease swiftly, depending on whether the base is greater than or less than one.
To graph an exponential function, one can start by selecting values for $ x $ to compute corresponding values for $ g(x) $, providing points that can be plotted on a graph.

By understanding these characteristics, one can effectively sketch the curve's path on the Cartesian plane and predict its behavior as $ x $ increases or decreases.

Asymptotes in Graphs

An essential feature of exponential graphs to recognize is the presence of an asymptote. An asymptote is a line that the graph approaches but never touches or crosses. For the function $ g(x) = 2^{x+1} $, the graph has a horizontal asymptote at $ y = 0 $. This tells us several important things about the graph:

As $ x $ approaches negative infinity, the value of $ g(x) $ gets closer and closer to zero, but never actually reaches it.
This asymptotic behavior illustrates that although the function decreases rapidly on the negative side, it doesn't fall below the x-axis.

Understanding asymptotes is crucial as they provide insight into the long-term behavior of graphs, ensuring precise sketching and interpretation.

Exponential Growth

Exponential growth is a key characteristic of exponential functions when the base is greater than one. In our function $ g(x) = 2^{x+1} $, the base is 2, which signifies exponential growth:

The function's values double as $ x $ increases by 1 unit.
This results in a sharp, upward-sloping curve when graphed, such that it expands or "grows" rapidly as $ x $ moves towards positive infinity.
Exponential growth is observed in various real-world contexts, from population growth to interest compounding, making it a key concept in applied mathematics.

Recognizing exponential growth helps in understanding how powerful this rate of increase can be as the function's input becomes larger.

Cartesian Plane Plotting

Plotting points on the Cartesian plane is the first and foundational step in graphing functions. A Cartesian plane is made up of two perpendicular axes: the horizontal x-axis and the vertical y-axis. Plotting the function $ g(x) = 2^{x+1} $ involves the following:

Creating a table of values for $ x $ and $ g(x) $, like $(-3, \frac{1}{4}), (-2, \frac{1}{2}), (-1, 1), (0, 2), (1, 4)$.
Placing each corresponding point on the plane where the x-coordinate climbs along the horizontal axis, and the g(x)-value sets its position on the vertical axis.
Connecting these plotted points with a smooth curve to illustrate the behavior of the function over the x-axis range.

Developing skills in Cartesian plotting is critical for understanding how mathematical relationships unfold visually and is a prerequisite for analyzing any graph's shape and direction.

91影视

Graph each exponential function. $$g(x)=2^{x+1}$$

Short Answer

Step by step solution

Create a table of values for x and g(x)

Plot the points from the table

Sketch the exponential function

Key Concepts

Graphing Exponential Functions

Asymptotes in Graphs

Exponential Growth

Cartesian Plane Plotting

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