/*! 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 50 Using Intercepts and Symmetry to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 50

Using Intercepts and Symmetry to Sketch a Graph In Exercises \(39-56\) , find any intercepts and test for symmetry. Then sketch the graph of the equation. $$ y=\frac{10}{x^{2}+1} $$

Short Answer

Expert verified
The graph of y = \(\frac{10}{x^{2}+1}\) has a y-intercept at (0, 10) and no x-intercepts. The graph is symmetric with respect to the y-axis.

Step by step solution

01

Find the intercepts

The x-intercepts of a graph are the x-coordinates where y=0. For our equation \(y=\frac{10}{x^{2}+1}\), y will only be equal to 0 when the numerator is equal to zero. Since the numerator is 10, a constant, there are no x-intercepts. The y-intercept of a graph is the y-coordinate where x=0. Substituting x=0 into our equation gives y = \(\frac{10}{0^{2}+1} = 10\). So the y-intercept is (0, 10).
02

Test for symmetry

To test for symmetry in respect to the y-axis (even function), replace x with -x in the equation and see if the original equation retains its form. Substituting -x into our equation gives \(y=\frac{10}{(-x)^{2}+1} = \frac{10}{x^{2}+1}\), which is our original equation, therefore, the graph is symmetric about the y-axis. Vertical symmetry (odd function) and symmetry about the origin can also be tested by replacing y with -y and x with -x respectively, but, for this equation, these tests are not relevant.
03

Sketch the graph

Now, armed with the information about intercepts and symmetry, the graph can be sketched. Since there are no x-intercepts and only one y-intercept (0, 10), the graph flirt with the x-axis but never crosses it. The curve is also symmetric about the y-axis. So to sketch the graph properly, one can draw the positive side of the graph and mirror it across the y-axis. Using a graphing tool will also show that the graph is an asymptote towards the x-axis (tends to zero as x gets infinitely large or small) which should be reflected in the sketch.
04

Analyze the graph

The graph represents a function that is always positive, approaching but never reaching 0. It peaks at (0, 10) and is symmetrical about the y-axis. The farther from the y-axis we go, the closer the curve gets to the x-axis without crossing it. This represents an asymptotic behavior.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Intercepts
Intercepts are crucial when sketching graphs, as they provide specific points where the graph crosses the axes. In our example, the function is given by \(y = \frac{10}{x^2+1}\). To find intercepts:
  • x-intercepts: These occur where \(y = 0\). For this particular equation, the numerator must be zero for \(y\) to equal zero. However, since the numerator is the constant 10, the function does not touch the x-axis, resulting in no x-intercepts.
  • y-intercepts: Occur where \(x = 0\). Substitute \(x = 0\) into the equation: \(y = \frac{10}{0^2 + 1} = 10\). This gives a y-intercept at (0, 10).
Thus, the graph has one key intercept at the point \((0, 10)\). This single intercept provides a foundational part of the graph, focusing our attention on the vertical reach of the graph at the y-axis. Remember, intercepts not only show interaction points but often suggest important behavior of graphs.
Symmetry
Symmetry in graphing helps in predicting how functions behave and can simplify graphing. Testing symmetry for our function \(y = \frac{10}{x^2+1}\) is straightforward:
  • y-axis symmetry: Replace \(x\) with \(-x\). This gives us \(y = \frac{10}{(-x)^2 + 1} = \frac{10}{x^2 + 1}\), which matches our original function. Thus, this graph has y-axis symmetry, indicating the graph mirrors itself on either side of the y-axis.
  • Vertical symmetry (origin symmetry): According to this test, replace \(x\) with \(-x\) and \(y\) with \(-y\). Doing so would not retain the equation since \(y\) cannot have a negative value in this function, indicating no origin symmetry.
  • Symmetry about the origin: This would involve switching both x and y coordinates to their negatives, which also doesn't apply here, as reflected by the absence of origin symmetry.
Symmetry simplifies and often doubles the visual understanding of graphing, allowing us to draw half and reflect it across the axis of symmetry. Here, y-axis symmetry tells us the graph shape on one side of the y-axis is mirrored onto the other.
Asymptotes
Asymptotes represent lines that a graph approaches but rarely touches, defining behavior at extreme values. For the function \(y = \frac{10}{x^2+1}\), asymptotic behavior is evident:
  • Horizontal asymptote: As \(x\) becomes very large in magnitude, the term \(x^2+1\) dominates the denominator, thus making \(y\) approach \(0\). Hence, the x-axis \(y = 0\) becomes a horizontal asymptote.
  • Vertical asymptote: Typically seen where the function is undefined, usually when a denominator is zero. However, here \(x^2 + 1 = 0\) has no real solutions, signifying no vertical asymptotes.
  • Behavior Analysis: Observe how the graph behaves near these asymptotes. It gets closer and closer to \(y = 0\) as \(|x|\) increases, making it approach but never actually touching the x-axis.
Understanding asymptotic behavior allows us to expect and visualize the function's end behavior, shaping expectations for how the graph appears at practicable infinities. Asymptotes guide us, especially in determining how graphs behave in real-world applications.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For \(i=1,2,\) let \(T_{i}\) be a triangle with side lengths \(a_{i}, b_{i}, c_{i},\) and area \(A_{i} .\) Suppose that \(a_{1} \leq a_{2}, b_{1} \leq b_{2}, c_{1} \leq c_{2},\) and that \(T_{2}\) is an acute triangle. Does it follow that \(A_{1} \leq A_{2} ?\)

True or False? In Exercises \(93-96,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The equation of any line can be written in general form.

Writing Use a graphing utility to graph the polynomial functions \(p_{1}(x)=x^{3}-x+1\) and \(p_{2}(x)=x^{3}-x .\) How many zeros does each function have? Is there a cubic polynomial that has no zeros? Explain.

Apartment Rental \(A\) real estate office manages an apartment complex with 50 units. When the rent is \(\$ 780\) per month, all 50 units are occupied. However, when the rent is \(\$ 825,\) the average number of occupied units drops to 47 . Assume that the relationship between the monthly rent \(p\) and the demand \(x\) is linear. (Note: The term demand refers to the number of occupied units.) (a) Write a linear equation giving the demand \(x\) in terms of the rent \(p .\) rent \(p .\) (b) Linear extrapolation Use a graphing utility to gredict the demand equation and use the trace feature to predict the number of units occupied when the rent is raised to \(\$ 855\) . (c) Linear interpolation Predict the number of units occupied when the rent is lowered to \(\$ 795 .\) Verify graphically.

Finding the Domain and Range of a Function In Exercises \(11-22,\) find the domain and range of the function. $$ f(x)=|x-3| $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.