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Chapter 3: Problem 22

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. $$f(x)=4^{x-3}+3$$

Short Answer

Expert verified
The graph of the function \(f(x) = 4^{x-3}+3\) is a shifted and scaled version of the graph of the base function \(f(x) = 4^x\). It represents an exponential growth, starting slightly above 3 (at y=3.015625 when x=0) and rises increasingly rapid as x increases. Exploring with a table of values helps to see the trend within the graph.

Step by step solution

01

Construct a Table of Values

Using a graphing utility, input the function \(f(x) = 4^{x-3}+3\). Choose a range of x-values, for the table accordingly, for example from -2 to 5. Substitute each chosen x-value into the function and record the corresponding y-values. For instance, substituting x=0 into the function, we get \(f(0) = 4^{0-3}+3 = 4^{-3}+3 = 1/64+3 = 3.015625\). Hence, we now have the coordinate point (0,3.015625). Repeat this process for all chosen x-values to complete the table.
02

Plot the Coordinate Points

With the completed table of values, start plotting the coordinate points on a graph. Make sure that the x-axis and the y-axis are appropriately labeled and scaled. Also ensure that you have accurately plotted each coordinate point from the table.
03

Sketch the Graph of the Function

After plotting the coordinate points, connect them to form a smooth curve. This will represent the graph of the function \(f(x) = 4^{x-3}+3\). Always remember that the graph of an exponential function is a smooth curve that rises or falls increasingly rapidly as x increases or decreases, and does not have any breaks or sharp corners.

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Most popular questions from this chapter

The management at a plastics factory has found that the maximum number of units a worker can produce in a day is \(30 .\) The learning curve for the number \(N\) of units produced per day after a new employee has worked \(t\) days is modeled by \(N=30\left(1-e^{k t}\right) .\) After 20 days on the job, a new employee produces 19 units. (a) Find the learning curve for this employee (first, find the value of \(k\) ). (b) How many days should pass before this employee is producing 25 units per day?

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