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Chapter 10: Problem 11

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \(h(0)= e^{-3}\).
03

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.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 sketch to be visually intuitive and correct in its representation of growth dynamics.

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Most popular questions from this chapter

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