/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Sketch the graph of each functio... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 26

Sketch the graph of each function. $$f(x)=5+2 e^{x}$$

Short Answer

Expert verified
The graph of the function has a y-intercept at \(y = 7\), and it rises indefinitely as \(x\) increases, and approaches a horizontal line as \(x\) decreases. Actual y-values depend on specific x-values taken, and should be accurately calculated for precise sketching.

Step by step solution

01

Determine the y-intercept

The y-intercept of the graph will occur when \(x = 0\). Therefore, substituting \(x = 0\) into the function, the y-intercept can be found as \(f(0) = 5 + 2 * e^0 = 5 + 2 = 7\).
02

Choose a set of x-values

Now, choose some x-values to determine the shape of the function. Note that since it is an exponential function, it is useful to choose both negative and positive x-values. Let's choose -2, -1, 0, 1, 2.
03

Compute corresponding y-values

The y-values will be found by substituting each chosen x-value into the function, the y-values will be -2: \(f(-2) = 5 + 2 * e^{-2}\), -1: \(f(-1) = 5 + 2 * e^{-1}\), 0: \(f(0) = 7\) (from the previous step), 1: \(f(1) = 5+2 * e^{1}\), and 2: \(f(2) = 5+2 * e^{2}\) approximately.
04

Sketch the graph

Now use the computed points to sketch the graph, keeping in mind that as an exponential function, the graph will approach a horizontal asymptote as \(x\) approaches negative infinity, and it will rise indefinitely as \(x\) approaches positive infinity.

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Key Concepts

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

Graphing Functions
Graphing functions can be a fun and intuitive process, especially with exponential functions. To graph a function, follow these steps:
  • First, understand the general shape of the function you are dealing with. For exponential functions like \(f(x) = 5 + 2e^x\), the graph tends to rise steeply as \(x\) increases and flattens towards a horizontal asymptote as \(x\) decreases.
  • Next, determine critical points such as the y-intercept by setting \(x = 0\).
  • Then, evaluate the function at several x-values, both positive and negative, to get a sense of the graph's behavior and shape.
  • Finally, plot these points and sketch a smooth curve through them, keeping the growth pattern of the function in mind.
By visualizing each part of this process, graphing functions becomes a powerful tool for understanding exponential relationships.
Y-intercept
The y-intercept of a graph is crucial because it tells us where the function crosses the y-axis. For the function \(f(x) = 5 + 2e^x\), the y-intercept is found by substituting \(x = 0\). This means you look at \(f(0) = 5 + 2e^0\).
  • Since \(e^0 = 1\), you simply calculate \(f(0) = 5 + 2 \times 1 = 7\).
  • This tells us that when \(x = 0\), \(f(x) = 7\), so the y-intercept is the point (0, 7).
The y-intercept is useful for anchoring the graph and providing a reference point for plotting other values.
Horizontal Asymptote
A horizontal asymptote refers to a line that a graph approaches but never quite reaches as \(x\) moves towards positive or negative infinity. For an exponential function like \(f(x) = 5 + 2e^x\), the constant term 5 identifies the horizontal asymptote. This is because as \(x\) approaches negative infinity, \(2e^x\) approaches zero.
  • Thus, the horizontal asymptote is the line \(y = 5\).
  • This means that as \(x\) becomes very negative, the function approaches a value of 5, but never dips below.
Recognizing the horizontal asymptote helps to frame the graph and gives an understanding of the function's long-term behavior.
Exponential Growth
Exponential growth is a hallmark of functions like \(f(x) = 5 + 2e^x\). These functions are characterized by rapid increases in value as \(x\) becomes larger. This happens because the base of the exponential function, in this case \(e\), is greater than one.
  • As \(x\) increases, \(e^x\) grows quickly, causing \(f(x)\) to increase without bound.
  • This is evident in the graph, where the curve steeply rises as it moves rightward.
Understanding exponential growth is crucial in real-world applications where populations, investments, or other quantities grow at a constant rate, highlighting the significance of exponential functions in various fields.

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Most popular questions from this chapter

The following table gives the temperature, in degrees Celsius, of a cup of hot water sitting in a room with constant temperature. The data was collected over a period of 30 minutes. (Source: www.phys. unt.edu, Dr. James A. Roberts)$$\begin{array}{|c|c|} \hline\text { Time } & \text { Temperature } \\\\(\mathrm{min}) & (\text { degrees Celsius }) \\ \hline0 & 95 \\\1 & 90.4 \\\5 & 84.6 \\\10 & 73 \\\15 & 64.7 \\\20 & 59 \\\25 & 54.5 \\\29 & 51.4\\\\\hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form \(f(t)=C a^{2}\) that best fits the data. Let \(t\) be the number of minutes the water has been cooling. (b) Using your modicl, what is the projected temperature of the water after 1 hour?

Evaluate the expression to four decimal places using a calculator. $$\ln \sqrt{2}$$

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Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{2} 20$$

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$. \log _{7} 150$$
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