Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function.\(h(x)=e^{x-2}\)

Step-by-Step Solution

Answer

The graph of the function \( h(x)= e^{x-2} \) is similar to the graph of the base function \( e^x \) but it starts at the point (2,1) instead of (0,1), due to the 2-unit shift to the right.

Step 1: Understand the Function

We start by understanding the function \(h(x)=e^{x-2}\). It is an exponential function with base \(e\) which is approximately equal to 2.718. When \(x\) is decreased by 2 in the function, it results in a shift of the graph of the function 2 units to the right along the x-axis.

Step 2: Recognize the base function

Recognize that the base function here is \( e^x \). The graph of the base function \( e^x \) will start at (0,1) on the y-axis, since any number to the power of 0 is 1, and will increase rapidly with a slope that gets steeper as x grows.

Step 3: Shift the base function

Apply the shift to the base function. The function \(h(x)=e^{x-2}\) is equivalent to the function \(e^x\) shifted 2 units to the right. This means that instead of starting at (0,1) our function will start at (2,1) and will increase in the same way as \( e^x \).

Key Concepts

Graphing Exponential Functions

Graphing exponential functions might seem tricky at first, but with a few guidelines, it's easier than you think! An exponential function is generally in the form of \(f(x) = a \, b^{x}\), where \(b\) is the base and it has a power of \(x\). In the given function, \(h(x) = e^{x-2}\), our base is \(e\), which is approximately 2.718. This special base is often involved in natural processes because of its unique mathematical properties.

The general shape of an exponential graph is a curve that either increases or decreases. For \(e^x\), the curve rapidly increases; as \(x\) becomes larger, the function values grow very quickly. This growth pattern is useful in fields like finance and biology, where exponential relationships frequently appear.

To sketch such a function, we start with the basic graph of \(e^x\), which passes through the point (0,1) because \(e^0 = 1\). For a more accurate graph, calculate a few points by substituting values for \(x\), then plot them to see the curve's steep rise for small values of \(x\).

Remember: the larger the base of the exponential function, the steeper the graph.

Horizontal Shifts

A horizontal shift moves a graph left or right on the Cartesian plane. This happens when a constant is either added to or subtracted from \(x\) in a function. In \(h(x) = e^{x-2}\), we see a horizontal shift in action.

The "\(x-2\)" inside the exponent indicates that we should move the graph of \(e^x\) 2 units to the right. Simplifying this:

• Addition to \(x\) shifts the graph to the left, while subtraction shifts it to the right.
• In this function, the graph will pass through (2,1), since \(e^0 = 1\) happens when \(x - 2 = 0\), or \(x = 2\).

The shape and steepness of the curve remain unchanged; it’s just its position that shifts on the x-axis. Visualizing horizontal shifts helps you manipulate graphs without recalculating the entire function.

Base of Natural Logarithms

The natural base \(e\) plays a crucial role in mathematics and science due to its unique properties. It's known as the "base of natural logarithms," a constant approximately equal to 2.718. This euler's number" is used extensively in calculus and complex analysis.

In the context of exponential functions, \(e\) serves as the base for the function \(e^x\), giving the exponential function the form \(a \, e^{bx}\), where \(b\) is the exponent's coefficient.

Why \(e\)? Because using \(e\) as a base results in a function that has a constant rate of change proportional to its current value; this is vital in modeling growth processes such as populations or compound interest.

• The derivative of an exponential function with base \(e\) is straightforward: the derivative of \(e^x\) is just \(e^x\).
• This feature simplifies the computation of rates of change and makes modeling dynamic systems more intuitive.

Understanding the significance of \(e\) leads you to appreciate its pervasive influence across both theoretical and applied mathematics.