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Chapter 1: Problem 20

Graph each of the functions without using a grapher. Then support your answer with a grapher. $$ y=5^{-x^{2}} $$

Short Answer

Expert verified
The graph of \(y = 5^{-x^2}\) is a symmetric curve with key point (0,1) and asymptote at \(y = 0\).

Step by step solution

01

Identify the General Form

The function given is \( y = 5^{-x^2} \). This is an exponential function where the base is 5 and the exponent is \(-x^2\). The negative sign in the exponent suggests that this is a decreasing function as \(x\) moves away from zero in either direction.
02

Determine Key Points

To graph the function, identify key points. When \( x = 0 \), \( y = 5^{0} = 1 \). As \(x\) increases or decreases from zero, the value of \(-x^2\) becomes more negative, making \(5^{-x^2}\) approach zero. Calculate additional points such as \(x = 1\) and \(x = -1\) to find that \(y = 5^{-1} \approx 0.2\).
03

Analyze Symmetry and Asymptotes

Since \(x^2\) is even, the function is symmetric about the y-axis. The exponential factor means there is a horizontal asymptote at \(y = 0\), as the function never touches the x-axis.
04

Sketch the Graph

Plot the determined key points: \((0, 1)\), \((1, 0.2)\), \((-1, 0.2)\), and note the symmetry. Draw a smooth, continuous curve, decreasing towards the asymptote \(y = 0\) as \(|x|\) increases.
05

Verify with Graphing Tool

Use a graphing calculator or software to graph \(y = 5^{-x^2}\) and verify the manually-drawn graph matches. Check that key points and the shape of the graph align with your sketch.

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Key Concepts

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

Graphing Techniques
To graph exponential functions like \( y = 5^{-x^2} \), start by understanding its general behavior. This function represents exponential decay since the exponent is negative. Follow these basic steps to depict the graph accurately.
  • Identify the general form and base. Here, the base is 5, and it is raised to the power of \(-x^2\).
  • Notice the direction. As the power is negative, the graph will show a decay pattern, decreasing as \( |x| \) increases.
  • Plot key points. Begin with the essential point where \( x = 0 \), resulting in \( y = 1 \).
By identifying these elements, you can sketch a rough draft of the graph's behavior without using technological aids. Keep practicing these techniques with different functions to improve your graphing prowess.
Symmetry in Functions
Symmetry is a crucial aspect when graphing functions. In the function \( y = 5^{-x^2} \), the exponent \( -x^2 \) is even. This indicates that the function is symmetric about the y-axis.

Here's why that symmetry occurs:
  • For any positive value of \( x \), the same value of \( -x \) will produce the same \( y \).
  • The even power \( x^2 \) ensures equality in the absolute value of \( x \) on both sides of the y-axis.
Recognizing symmetry helps significantly in graphing as it reduces the amount of computation for different \( x \) values. Just compute a few key points for positive \( x \) and mirror them on the negative side.
Asymptotes
In exponential functions, especially those approaching zero, asymptotes play a crucial role. For \( y = 5^{-x^2} \), a horizontal asymptote exists at \( y = 0 \).
  • As \( x \) becomes larger in positive or negative direction, \( y = 5^{-x^2} \) approaches zero but will never reach it.
  • The asymptote is a guideline that shows the direction of the graph as it stretches towards infinity.
Understanding the concept of asymptotes is important, as it describes the behavior of the graph at its extremes. The graph will always approach but never touch this horizontal reference line.
Key Points in Graphing
Identifying key points is essential for accurately sketching a function. For \( y = 5^{-x^2} \), pinpoint certain values of \( x \) to see how \( y \) changes.
  • Start with \( x = 0 \). At this point, \( y = 1 \).
  • Try \( x = 1 \) and \( x = -1 \). Both give \( y \approx 0.2 \), demonstrating symmetry.
  • Test further points like \( x = 2 \) and \( x = -2 \), where \( y \) becomes even smaller (around \( y \approx 0.04 \)).
By plotting these points, you lay the groundwork for a reliable sketch. This strategy enables you to capture both the core structure of the graph and ensure accuracy in its representation.

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Most popular questions from this chapter

In a report of the Federal Trade Commission \((\mathrm{FTC})^{41}\) an example is given in which the Portland, Oregon, mill price of 50,000 board square feet of plywood is \(\$ 3525\) and the rail freight is \(\$ 0.3056\) per mile. a. If a customer is located \(x\) rail miles from this mill, write an equation that gives the total freight \(f\) charged to this customer in terms of \(x\) for delivery of 50,000 board square feet of plywood. b. Write a (linear) equation that gives the total \(c\) charged to a customer \(x\) rail miles from the mill for delivery of 50,000 board square feet of plywood. Graph this equation. c. In the FTC report, a delivery of 50,000 board square feet of plywood from this mill is made to New Orleans, Louisiana, 2500 miles from the mill. What is the total charge?

Suppose you were one of the people who in 1955 purchased a 40 -year Treasury bond with a coupon rate of \(3 \%\) per year. Presumably, you thought that inflation over the next 40 years would be low. (The inflation rate for the 30 years before 1955 was just a little over \(1 \%\) a year.) In fact, the inflation rate for the 40 -year period of this bond was \(4.4 \%\) a year. Thus, in real terms you would have lost \(1.4 \%\) a year. If you bought \(\$ 1000\) of such a bond at issuance and held it to maturity, how much would the \(\$ 1000\) bond have been worth in real terms?

You are given a pair of functions, \(f\) and \(g .\) In each case, use your grapher to estimate the domain of \((g \circ f)(x)\). Confirm analytically. $$ f(x)=x^{2}, g(x)=\sqrt{x} $$

Chakravorty and Roumasset \(^{45}\) showed that the revenue \(R\) in dollars for cotton in California is approximated by the function \(R(w)=-0.2224+\) \(1.0944 w-0.5984 w^{2},\) where \(w\) is the amount of irrigation water in appropriate units paid for and used. What happens to the revenue if only a small amount of water is paid for and used? A large amount? What is the optimal amount of water to use?

Cotterill and Haller \(^{54}\) recently estimated a demand curve for Kellogg's Special \(\mathrm{K}\) cereal to be approximated by \(x=A p^{-2.0598},\) where \(p\) is the price of a unit of cereal, \(x\) is the quantity sold, and \(A\) is a constant. Find the percentage decrease in quantity sold if the price increases by \(1 \% .\) By \(2 \%\).
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