Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. Select integers from \(-2\) to 2 , inclusive, for \(x\). Then describe how the graph of g is related to the graph of \(f .\) If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=2^{x} \text { and } g(x)=2^{x+1}$$

Step-by-Step Solution

Answer

The graph of \(g(x)=2^{x+1}\) is similar to the graph of \(f(x) = 2^{x}\), but it is shifted up by a factor of 2. Each point on g(x) is twice as high as the corresponding point on f(x) at the same x-coordinate.

Step 1: Define the range for x

The problem specifies to use integers from -2 to 2, inclusive for \(x\). This gives us \(x = -2, -1, 0, 1, 2\).

Step 2: Evaluate functions for values of x

Plug the values of \(x\) into both \(f(x) = 2^{x}\) and \(g(x)=2^{x+1}\). This will give us coordinate points for each function.

Step 3: Plot the functions

Next, plot the points calculated in the previous step on the same coordinate grid. Remember that each point is in the form of (x, f(x)) or (x, g(x)), where x is the x-coordinate and f(x) or g(x) is the y-coordinate.

Step 4: Analyze the relation

After plotting, look at the shape and position of the two functions to understand how g(x) is related to f(x).

Step 5: Confirm with a graphing utility

Finally, input both function equations into a graphing utility. This will visualize how g(x) = \(2^{x+1}\) is related to f(x) = \(2^{x}\). Confirm the relationship derived in the previous step.

Key Concepts

Coordinate System

Understanding the coordinate system is essential when graphing any function. A coordinate system provides a framework in which we can plot points, lines, and curves to visualize mathematical relationships. It consists of two number lines: the x-axis (horizontal) and the y-axis (vertical), which intersect at a point called the origin. By using a coordinate system, we can precisely locate any point in a two-dimensional plane by its x (horizontal) and y (vertical) coordinates.

When graphing exponential functions such as in our exercise, we plot points whose coordinates are derived from evaluating the function at different x-values. The graph of an exponential function will pass through these plotted points, and by connecting them smoothly, the continuous increase or decrease of the function is represented visually.

Function Transformation

Function transformation involves altering the basic graph of a function in various ways. When graphing functions, transformations can shift the graph horizontally or vertically, reflect it over one of the axes, or stretch and compress it. For exponential functions, like the ones we are working with, a common transformation is to change the exponent.

In our exercise, function g(x) is linked to function f(x) through a horizontal shift. Specifically, the function g(x) = 2^x+1 is a transformation of f(x) = 2^x that shifts the graph of f(x) to the left by one unit. This transformation is a result of adding one to each x-value before it is evaluated with the exponential expression. As a result, every point on the graph of g(x) will be to the left of the corresponding point on f(x), while having the same vertical position.

Evaluating Functions

Evaluating functions is the process of finding the output of a function for a given input. This is akin to substituting a number into the function and calculating the result. In our exercise, evaluating the function f(x) = 2^x means plugging in the selected values of x and performing the exponentiation to determine the y-coordinate for each point.

To illustrate, when we evaluate f(-2) = 2^-2 and g(-2) = 2^-2+1, we calculate these as f(-2) = 1/4 and g(-2) = 1/2, respectively. These calculations provide us with coordinate pairs that we then plot on the coordinate system. By evaluating the functions for distinct x-values, we can graph the points on the same grid to compare the two functions visually.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a fundamental concept in graphing functions. It enables us to represent every point in a plane by using two perpendicular number lines—the x-axis and the y-axis. When working with exponential functions, we typically see an exponential increase or decrease, which is concisely depicted in the rectangular coordinate system.

In our textbook problem, we are using the rectangular coordinate system to graph two exponential functions. By selecting integers from -2 to 2 for x, and calculating the corresponding y-values (the outputs of the functions), we are plotting points in this system. The grid formed by the x and y axes is well-suited for such graphs since it can accommodate both positive and negative values, as well as rapid increases indicative of exponential growth.