Problem 11 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, sketch the graph of the function. $$ h(x)=e^{x-3} $$

Step-by-Step Solution

Answer

The graph of the function is an exponential curve that passes through the y-intercept \(e^{-3}\) and approaches the x-axis (i.e., y=0) as a horizontal asymptote when x goes towards negative infinity.

Step 1: Identify the shape of function

Given the function is of the form \(e^{x}\), we recognize this as an exponential function. For all exponential functions the graph increases as \(x\) increases and decreases as \(x\) decreases. The graph of an exponential function is always concave up.

Step 2: Identify the y-intercept

The y-intercept can be found by subbing \(x=0\) into the equation. Sub in \(x=0\) into \(h(x) = e^{x-3}\) gives \(h(0)= e^{-3}\).

Step 3: Identify the horizontal asymptote

The horizontal asymptote of the function \(h(x) = e^{x-3}\) is \(y = 0\). This is because as \(x\) gets infinitely small, the function approaches zero but never quite reaches it.

Step 4: Plot the function and sketch the graph

On a graph, plot the \(y\)-intercept at \(y = e^{-3}\). Draw a curve starting from the y-axis above the y-intercept and extending towards infinity in the positive x direction. At the same time the function should be sketched to approach the horizontal asymptote of \(y = 0\) as \(x\) goes to negative infinity.

Key Concepts

Graphing Exponential Functions

Exponential functions are a key concept in mathematics, often represented as \( e^{x} \). These functions are characterized by their rapid rate of growth. When graphing exponential functions like \( h(x) = e^{x-3} \), it’s essential to understand the basic shape:

• The graph increases exponentially as \( x \) increases.
• It decreases exponentially as \( x \) decreases.
• The curve of an exponential function is always concave up.

To sketch these graphs correctly, start by identifying key points such as where the curve crosses the axes, and notable features like asymptotes. This ensures the graph is both accurate and functionally representative.

Y-intercept

In the context of exponential functions, the y-intercept is a critical component. It represents the point where the graph crosses the y-axis. For \( h(x) = e^{x-3} \), you can find the y-intercept by setting \( x = 0 \).
Substituting into the equation gives \( h(0) = e^{-3} \), which is the value of the y-intercept.
This point \( (0, e^{-3}) \) serves as the starting reference for plotting the graph, providing insight into the function's behavior at the y-axis. Remember that the y-intercept helps in establishing how the curve begins its ascent or descent from this point.

Horizontal Asymptote

Understanding horizontal asymptotes is crucial when graphing functions. A horizontal asymptote is a horizontal line that the graph of the function approaches but never actually reaches.
For the function \( h(x) = e^{x-3} \), the horizontal asymptote is \( y = 0 \).
This occurs because as \( x \) approaches negative infinity, the value of \( e^{x-3} \) tends to zero, getting infinitely close to the line \( y = 0 \) without ever meeting it. Recognizing the horizontal asymptote helps in accurately sketching the tail behavior of the graph, indicating a limit that the function approaches but never surpasses.

Concavity

Concavity describes the direction in which a graph bends. It is crucial for understanding the shape and behavior of the function's curve. In exponential functions like \( h(x) = e^{x-3} \), the curve is always concave up.

• This means that the function bends upwards as it extends along the x-axis.
• A graph that is concave up has a shape like a cup or a smile.

This upward curvature occurs due to the nature of exponential growth, where the rate of increase itself continues to rise. Knowing this helps mix the graph’s sketch to be visually intuitive and correct in its representation of growth dynamics.