Question

Graph each exponential function, manually or by calculator, for the given values of \(x\) Take \(e=2.718\).$$y=4 e^{x / 2} \quad(x=0 \text { to } 4)$$

Short answer

To graph \(y=4e^{x/2}\), calculate the \(y\) values for \(x=0\) to \(x=4\), plot the points, and then draw a smooth curve that represents the exponential growth.

Step-by-step solution

Identify the Function and Constants

Recognize that the function provided is an exponential function in the form of \(y=ae^{bx}\), where \(a=4\), \(b=\frac{1}{2}\), and \(e\) is the base of the natural logarithm, approximately equal to 2.718. The range of \(x\) values that will be used to plot the graph is from 0 to 4.

Calculate y Values for the Given x Values

Calculate the corresponding \(y\) values for \(x\) values from 0 to 4 using the function \(y=4e^{x/2}\). Plug in each \(x\) value into the function to find each \(y\) value.

Plot the Points

Plot the calculated \((x,y)\) points on a graph. Be sure to accurately scale your axes to accommodate the range of \(x\) and the potential range of \(y\) values.

Draw the Graph

Connect the plotted points with a smooth curve to complete the graph of the exponential function. The graph should continuously rise from left to right since it's an exponential growth function.

Key concepts

Natural Base e

In mathematics, the natural base e is a unique and important number, approximately equal to 2.718. It arises naturally in the study of exponential growth and decay, and it serves as the base for natural logarithms. To appreciate its role in exponential functions, think of e as the rate of growth if we continuously compound 100% interest on a unit amount of money. It is crucial to recognize this number because the formulas involving e show up in many disciplines, such as economics, biology, and physics.

When dealing with exponential functions, base e is often preferred over other bases because of its unique mathematical properties. For instance, the derivative of e^x, is itself, making calculus computations more straightforward. Understanding the significance of e is essential for mastering topics in calculus and higher-level math.

Calculating Exponential Functions

Exponential functions like y = ae^{bx} can intimidate students, but with the right approach, they are quite manageable. First, identify the constants: a, which is the initial amount, and b, the rate of growth or decay. When b is positive, the function will show exponential growth, and if it's negative, it will depict exponential decay.

To calculate y for specific x values, simply plug the x into the equation and perform the calculation step by step. If x is negative, remember that you're essentially looking at the reciprocal of e to a power. A calculator with an e button or using a software tool can make this process easier, especially for non-integer values of x. Keep in mind that for large values of x, the y values will quickly become very large due to the nature of exponential growth.

Plotting Points on a Graph

Plotting points is the foundation of understanding any mathematical function visually. With an exponential function like y = 4e^{x/2}, first calculate the y values for a range of x, as seen in the step-by-step solution. Now, you're ready to plot.

Start with a pair of axes: the horizontal axis (x-axis) for the input values and the vertical axis (y-axis) for the output. Mark your calculated x values on the x-axis, and find the corresponding y values on the y-axis. Place a dot for each pair of x and y. When all points are plotted, draw a smooth curve through them that best fits the pattern of the points. Important: because we're working with an exponential function that represents growth, the curve should rise from left to right. The graph helps visualize the behavior of the function and is essential for understanding how the function behaves across different values of x.