Problem 11 - College Algebra Essentials

Question

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$f(x)=4^{x}$$

Step-by-Step Solution

Answer

The function \(f(x) = 4^x\) is an increasing exponential function. By substituting various x-values into the function, we got the points \[(-2, 0.0625), (-1, 0.25), (0, 1), (1, 4), (2, 16)\]. After plotting these and sketching a curve through them, confirmed with a graphing utility, we are able to graph the given function.

Step 1: Creating the Table

Substitute a range of x-values into the function \(f(x) = 4^x\) to obtain corresponding y-values. Since exponential functions can be negative or positive, choose a few positive and negative x-values. For instance, if we substitute \(x = -2, -1, 0, 1, 2\), we will have the following pairs: \[(-2, 0.0625), (-1, 0.25), (0, 1), (1, 4), (2, 16)\].

Step 2: Plot the Coordinates and Sketch the Graph

On an xy-plane, plot each of the points from the table. For an exponential function, the graph generally rises rapidly as x increases and approaches zero as x decreases. Join these points with a smooth curve to create the graph.

Step 3: Confirm with a Graphing Utility

To ensure that the curve is correct, check it against a graphing utility like Desmos or a graphing calculator. Input the function \(f(x) = 4^x\) to see if the curve matches your hand-drawn graph.