Problem 68 - A Graphical Approach to Precalculus with Limits

Question

The graph of \(y=e^{x-3}\) can be obtained by translating the graph of \(y=e^{x}\) to the right 3 units. Find a constant \(C\) such that the graph of \(y=C e^{x}\) is the same as the graph of \(y=e^{x-3}\). Verify your result by graphing both functions.

Step-by-Step Solution

Answer

The constant is \(C = e^{-3}\).

Step 1: Understanding the Translation

The transformation from the graph of \( y = e^x \) to \( y = e^{x-3} \) is a horizontal translation. Specifically, the original graph \( y = e^x \) is shifted 3 units to the right. This changes the exponent from \( x \) to \( x - 3 \).

Step 2: Substituting to Find C

To find a constant \( C \) such that \( y = C e^x \) is identical to \( y = e^{x-3} \), we equate the two expressions: \( C e^x = e^{x-3} \). We can rewrite the right side using the property of exponents to solve for \( C \): \( e^{x-3} = e^x \times e^{-3} \). Therefore, \( C = e^{-3} \).

Step 3: Expressing C in Terms of e

The constant \( C \) is expressed as \( e^{-3} \). This means that to obtain the graph \( y = e^{x-3} \) from \( y = e^x \), we multiply \( e^x \) by \( e^{-3} \).

Step 4: Verification by Sketching Graphs

For verification, you can sketch both graphs \( y = e^{x-3} \) and \( y = e^{-3}e^x \). Both functions will produce identical graphs by confirming the application of the transformations and multiplications involved.

Key Concepts

Graph Transformations

Graph transformations in mathematics describe the way a graph changes in response to modifications in its equation. This exercise deals with a specific transformation of the exponential graph of the function \( y = e^x \).

The transformation from \( y = e^x \) to \( y = e^{x-3} \) is a simple shift. The modification happens in the exponent part of the exponential expression. By changing \( x \) to \( x-3 \), we are introducing a horizontal movement of the entire graph.

Let's look at some of the common graph transformations:

• Vertical translations involve adding or subtracting a constant directly to the function, shifting the graph up or down.
• Horizontal translations modify the input variable (x) by a constant, which moves the graph left or right.
• Vertical stretching/shrinking involves multiplying the output by a constant.
• Reflection flips the graph across an axis.

Understanding these basic transformations is crucial because they show how changing small parts of the function equation can have significant graphical effects.

Horizontal Translation

Horizontal translation specifically involves moving a graph left or right. It is a straightforward transformation that affects the graph's position along the x-axis.

In our exercise, we started with the graph of \( y = e^x \). The transformation involved a horizontal translation of 3 units to the right, resulting in \( y = e^{x-3} \). This adjustment in the equation's exponent subtracts 3 from every x-value on the original graph.

Here's how it impacts the function:

• The original point \( (0, e^0) \) of the graph of \( y = e^x \) becomes \( (3, e^{0}) \) in the graph of \( y = e^{x-3} \).
• For any point on the graph \( (a, e^a) \) in \( y = e^x \), it becomes \( (a+3, e^a) \) in \( y = e^{x-3} \).

This translation is an effective method to modify a graph's position without altering its shape. The function values (outputs) remain the same; only their corresponding inputs (x-values) change.

Properties of Exponents

The properties of exponents form the backbone of manipulating exponential functions and their transformations. Understanding these rules makes it easier to manage changes in equations as seen in our exercise.

Key properties include:

• Product of powers: \( a^m \times a^n = a^{m+n} \)
• Power of a power: \( (a^m)^n = a^{mn} \)
• Quotient of powers: \( \frac{a^m}{a^n} = a^{m-n} \) if \( a eq 0 \)
• Negative exponent: \( a^{-n} = \frac{1}{a^n} \)

This exercise uses the property that \( e^{x-3} \) can be rewritten as \( e^x \times e^{-3} \). This helps in verifying the transformation by deducing the constant \( C = e^{-3} \) to match the between the two expressions \( y = C e^x \) and \( y = e^{x-3} \).

When learning about exponential functions, these properties allow one to easily simplify expressions and recognize equivalent forms. They help anchor complex transformations with simple arithmetic.