Problem 46 - Precalculus with Limits

Question

Graphing a Natural Exponential Function In Exercises \(45-50\) , use a graphing utility to graph the exponential function. $$y=1.08 e^{5 x}$$

Step-by-Step Solution

Answer

To graph the function, evaluate it for chosen x-values from a particular range, plot these values and join them to form the curve. The function \(y=1.08e^{5x}\) grows rapidly for positive x and approaches 0 for negative x.

Step 1: Understand the Function

First, comprehend the exponential function given, which is \(y=1.08e^{5x}\). This is a variant of the basic exponential function \(y=e^x\), where 1.08 is a multiplier (amplitude change), and 5 is the power of x which influences the rate of growth or decay.

Step 2: Populate x-values

Chose a range of x-values to compute the corresponding y-values. A common choice is an interval from -2 and 2 with a 1 unit increment, giving the range as {-2,-1,0,1,2}.

Step 3: Compute y-values

Use the function \(y=1.08e^{5x}\) and compute the y-values corresponding to the chosen x-values. The base of natural logarithm, e, equals approximately 2.71. For example, when x=1, \(y=1.08e^{5 \times 1} = 1.08 \times e^5\).

Step 4: Plot the Points

Use a graphing calculator or another graphing utility to plot the calculated points. Each point on the graph represents an ordered pair (x,y). Plot each of these pairs to get the points.

Step 5: Draw the Curve

Join the points plotted on the graph to illustrate the shape and trend of the function. The curve of the function \(y=1.08e^{5x}\) should be increasing, due to the positive exponential. For \(x<0\), the function approaches 0, and for \(x>0\), it grows rapidly due to the 5x exponent.

Key Concepts

Natural Exponential Function

The natural exponential function is simply an exponential function with a base of the mathematical constant 'e', approximately equal to 2.71. Represented by the formula \(y=e^x\), it's essential to understand its unique properties. For instance, the rate of change of the natural exponential function is proportional to its value at any point.

This quality is particularly important in contexts like population growth, compound interest, and radioactive decay, to name a few real-world applications. In the given exercise, \(y=1.08e^{5x}\) is a variation of the natural exponential function, where the constant 1.08 scales the function and the term 5x dictates how rapidly the function's value increases for positive x or decreases for negative x.

Graphing Calculator

A graphing calculator is an invaluable tool for visualizing mathematical functions, especially exponential ones. When dealing with the natural exponential function, graphing calculators help students to transform abstract equations into concrete visual representations.

By inputting the function \(y=1.08e^{5x}\) into a graphing calculator and choosing a set of x-values, students can obtain the corresponding y-values and plot these to form a curve. These visual aids greatly enhance understanding of the function's behavior. It’s essential to utilize the graphing calculator's features like zooming and adjusting axes to capture the function's rapid growth or decay adequately.

Rate of Growth and Decay

The terms 'growth' and 'decay' in the context of exponential functions refer to how quickly the value of the function increases (growth) or decreases (decay) as x changes. In the function \(y=1.08e^{5x}\), the coefficient 5 before the x powers the rate at which the function grows for positive x-values and decays as x becomes negative.

This behavior is central to understanding exponential functions: the greater the absolute value of the coefficient of x in the exponent, the steeper the slope of the graph. In applications, this concept can be used to predict populations, model investments, and describe natural phenomena like radioactive decay.

Exponential Function Properties

Exponential functions have distinct properties that set them apart from other types of functions. For the function \(y=1.08e^{5x}\), crucial properties include:

• The function is always positive, no matter the value of x.
• The rate of growth is proportional to the function's current value, resulting in a 'J-shaped' curve.
• As x approaches negative infinity, the function value approaches zero, but never touches the x-axis (asymptote).
• The function's slope increases as x increases, depicting exponential growth.

By understanding these properties and observing them on a graphing calculator, a student can gain deeper insights into the nature of exponential functions and their implications.