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Chapter 3: Problem 46

Use a graphing utility to graph the exponential function. $$y=1.08 e^{5 x}$$

Short Answer

Expert verified
The graph of the given exponential function \(y = 1.08 e^{5x}\) is a steep, upward-arching curve that gets higher as x gets larger, with a y-intercept of 1.08.

Step by step solution

01

Identify the base, coefficient and exponent

In the given exponential function \(y = 1.08 e^{5x}\), the base is e, which is the base of the natural logarithm (approximately equal to 2.71828). The coefficient is 1.08 and the exponent is 5x. This suggests the graph will increase faster than a simple graph of \(y = ex\). The value 1.08 will slightly alter the y-intercept.
02

Prepare to plot

Set up the graphing utility with an appropriate scale. Given the rapid growth rate of this function, the vertical axis (y) will likely need a far greater range than the horizontal axis (x). Plot this function for a few integral values of x to get a sense of the curve's shape. For example, we can use x values of -2, -1, 0, 1, and 2.
03

Graph the function

Plot the points calculated in Step 2. After plotting these points, sketch a curve to represent the function. Depending upon the graphing utility, it may automatically plot the curve.
04

Interpreting the Graph

The graph should show a steep, upward-arching curve that gets higher as x gets larger. The curve will cross the y-axis at \(y = 1.08\) (since \(e^{5*0}\) equals 1), reflecting the fact that the y-intercept has been slightly altered by the coefficient.

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