Problem 68 - A Graphical Approach to College Algebra

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 \( C \) is \( \frac{1}{e^3} \).

Step 1: Identify the Original Function

Start with the original exponential function, which is given as \( y = e^x \). This is the function to be transformed.

Step 2: Analyze the Transformation

Recognize that the transformation of \( y = e^x \) to \( y = e^{x-3} \) involves a horizontal shift to the right by 3 units. In general, this corresponds to adjusting the x-value by 3.

Step 3: Express the Transformed Function

The function after the transformation becomes \( y = e^{x-3} \). This expression indicates that every x in the original \( e^x \) is replaced by \( x-3 \).

Step 4: Relate to the New Formulation

The problem requires finding a constant \( C \) such that \( y = C e^{x} = e^{x-3} \). To find \( C \), rewrite \( e^{x-3} \) as \( \frac{e^x}{e^3} \).

Step 5: Solve for the Constant C

The expression \( y = \frac{e^x}{e^3} \) can be rewritten as \( y = C e^x \) with \( C = \frac{1}{e^3} \). Thus, \( C \) is found to be \( \frac{1}{e^3} \).

Step 6: Verification through Graphing

To verify, graph both \( y = \frac{1}{e^3} e^x \) and \( y = e^{x-3} \). Both graphs should be identical, confirming that the transformation is correct.

Key Concepts

Graph Transformations

Graph transformations are modifications made to the basic graph of a function to produce a new graph. In the case of exponential functions like \( y = e^x \), transformations can include shifts, stretches, or reflections. The foundation begins with the basic shape and position of the function. Transformations then modify these properties to create variations.

The main types of transformations include:

• Vertical and Horizontal Shifts: Moving the graph up, down, left, or right.
• Reflections: Flipping the graph over a specific axis.
• Stretches and Compressions: Expanding or contracting the graph vertically or horizontally.

Each transformation changes the graph's equation, affecting how the function behaves and is visually represented. For instance, the transformation from \( y = e^x \) to \( y = e^{x-3} \) involves a horizontal movement. Understanding these transformations is key to predicting and verifying graph changes.

Horizontal Shift

A horizontal shift is when the entire graph of a function moves left or right on the Cartesian plane. It's a powerful tool for adjusting the position of a graph without altering its shape.

For exponential functions, a horizontal shift is implemented by adding or subtracting a value from the \( x \)-variable in the function. Specifically, for \( y = e^{x-3} \), this shift is to the right by 3 units, indicating each x-value is replaced by \( x - 3 \). This results in visually moving the graph to the right along the x-axis.

• A negative addition, such as \( x-3 \), shifts the graph to the right.
• A positive addition, such as \( x+3 \), would move it to the left.

Horizontal shifts do not affect the y-values but only the input values of \( x \), hence maintaining the graph's overall shape but changing its position along the axis.

Transformation Verification

Transformation verification is a crucial step in ensuring that the applied changes to a function's graph have been correctly implemented. This involves confirming that all operations and modifications result in the anticipated new form of the function.

A common method of verification is graphical comparison, where the initial equation and its transformed version are plotted to observe if they correspond visually. In our example, you would plot both \( y = \frac{1}{e^3} e^x \) and \( y = e^{x-3} \). If both graphs coincide precisely, the transformation is validated.

• Use graphing tools or software to create accurate plots.
• Ensure the visual alignment of all key points on the graphs.

Transformation verification ensures confidence that the equations used represent the intended transformations. This confirmation step is vital for accurate modeling and understanding of how shifts alter functions.