Problem 20 - Calculus of a Single Variable

Question

Sketching a Graph In Exercises \(17-22,\) sketch the graph of the function. $$ y=e^{x-1} $$

Step-by-Step Solution

Answer

The graph of y=e^{x-1} is the same as y=e^x, but shifted 1 unit to the right. It crosses the y-axis at (1,1) and increases as x goes to positive infinity, while approaching the x-axis as x goes towards negative infinity.

Step 1: Identify the base function

The base function here is y=e^x. The graph of this base exponential function is a curve that crosses the y-axis at (0,1) and increases as x goes to infinity, and it gets closer to the x-axis but never touches or crosses it as x goes to negative infinity.

Step 2: Identify the transformation

A transformation on the function according to the form y=e^{x-1} implies that there is a horizontal shift of 1 unit to the right, which means the entire graph will move to the right by 1 unit.

Step 3: Sketch the transformed graph

To sketch y=e^{x-1}, start with the graph of the base function y=e^x. Then, move each point on the graph 1 unit to the right. The y-intercept moves to (1,1). As x goes to infinity, the function increases and as x goes to negative infinity, it gets closer to the x-axis but never touches or crosses it.

Key Concepts

Exponential Function

An exponential function is a mathematical expression of the form \( y = a^{x} \), where \( a \) is a constant greater than 0. Such functions have unique characteristics:

• They grow rapidly or decay swiftly depending on the base \( a \).
• In the case of \( y = e^{x} \), it describes continuous growth, with \( e \) (Euler's number) being approximately 2.718.
• This function is defined for all real numbers and crosses the y-axis at the point (0,1), as \( e^{0} = 1 \).
• As \( x \) increases, the function tends to infinity; as \( x \) decreases, it approaches the x-axis but never reaches it.

Understanding these properties is essential for graph sketching, as they help predict the general shape and behavior of the function on the graph.

Horizontal Transformation

Horizontal transformation refers to shifting the graph of a function left or right along the x-axis. If a function is of the form \( y = f(x - c) \), it signifies a horizontal shift. Here:

• A negative \( c \) value indicates a shift to the right by \( c \) units.
• A positive \( c \) value indicates a shift to the left by \( c \) units.

For the function \( y = e^{x-1} \), there is a horizontal shift 1 unit to the right:

• This means every point on the graph of \( y = e^{x} \) moves to the right by one unit.
• The y-intercept of the original function, which is at (0,1), moves to (1,1).

This change keeps the graph's general shape intact while repositioning its alignment along the x-axis.

Y-Intercept

The y-intercept of a graph is the point where it crosses the y-axis. For exponential functions, it plays a vital role in understanding the function's initial value. To find the y-intercept, set \( x = 0 \) in the function.

• For the base function \( y = e^{x} \), the y-intercept is at (0,1), because \( e^{0} = 1 \).
• In the transformed function \( y = e^{x-1} \), the y-intercept shifts due to the horizontal transformation. The new y-intercept is at (1,1).

Knowing the y-intercept allows you to plot this critical point accurately and use it as a reference for other points on the graph, ensuring the graph sketch is correct.

Function Transformation

Function transformation involves shifting, stretching, or reflecting a base graph to obtain a new function. These transformations are fundamental in altering how a graph is positioned and shaped.

• Horizontal shifts, like in \( y = e^{x-1} \), move the graph along the x-axis.
• Vertical shifts move the entire graph up or down the y-axis.
• Stretches and compressions alter the graph's steepness.
• Reflections flip the graph over a given axis.

For any transformed function, identifying these changes allows us to manipulate the base graph to reflect these variations quickly. This understanding is crucial for sketching the graph correctly and interpreting the function's real-world implications.