/*! 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 7 Draw a graph of each function. D... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 7

Draw a graph of each function. Describe properties of the graph. \(y=\frac{-5}{x}\)

Short Answer

Expert verified
The graph of \(y=-\frac{5}{x}\) is a hyperbola with no x-intercept and y-intercept, with vertical and horizontal asymptotes at x=0 and y=0 respectively. The graph is in the second and fourth quadrants, and never touches the axes.

Step by step solution

01

Determine the asymptotes

As \(x\) approaches 0 from either the positive or negative side, the value of \(y\) will approach negative or positive infinity. Therefore, the x-axis (y=0) is a horizontal asymptote. Since the function is undefined when \(x=0\), the y-axis (x=0) is a vertical asymptote.
02

Compute the intercepts

The x-intercept of the graph of a function is the value of \(x\) for which \(y=0\). But this function has no \(x\) values that result in \(y=0\), so the graph has no x-intercept. The y-intercept is the value of \(y\) when \(x=0\). As \(x\) can't be zero, therefore there is no y-intercept.
03

Draw the shape of the curve

After taking the asymptotes and intercepts into account, we may sketch the curve. The negative sign in the function indicates that the graph will be in the second and fourth quadrants. When \(x\) is positive, \(y\) will be negative and when \(x\) is negative, \(y\) will be positive. So the graph will approach but never reach the asymptotes.

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Key Concepts

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

Graphing Rational Functions
Rational functions are expressions obtained from the ratio of two polynomials. To graph a rational function, it's important to recognize certain characteristics that define its shape and behavior. The function given in the original step-by-step solution is \( y = \frac{-5}{x} \), which is a simple rational function. It鈥檚 characterized by having a fractional form with \(-5\) as a constant numerator and \(x\) as a linear denominator.
To draw this function:
  • Analyze the function's symmetry鈥攖here is no symmetry about the y-axis or x-axis, but it does present origin symmetry. This means the graph reflects evenly when rotated 180 degrees about the origin.
  • Identify the end behavior by observing how \( y \) reacts as \( x \) approaches very large positive or negative numbers. \( y \) approaches zero, implying a horizontal asymptote.
  • Because of the negative constant in the numerator, when \( x \) is positive, \( y \) will be in the negative region of the second quadrant. Conversely, when \( x \) is negative, \( y \) will be positive, appearing in the fourth quadrant.
Utilizing these traits helps in forming an accurate sketch of the function on a coordinate plane, ensuring you correctly mirror the nature of rational functions.
Asymptotes
Asymptotes are lines that a graph of a function approaches but never actually touches. They serve as guidelines that help shape the behavior of the function at its extremes or near undefined values.
For the function \( y = \frac{-5}{x} \), there are two types of asymptotes:
  • Vertical Asymptote: This occurs where the function is undefined because the denominator is zero, which in this case is at \( x = 0 \). This line acts as a barrier that the graph cannot cross.
  • Horizontal Asymptote: As \( x \) approaches positive or negative infinity, the value of \( y \) gets closer to zero, indicating \( y = 0 \) is the horizontal asymptote.
The presence of these asymptotes implies that as you examine the graph near these regions, the function will tend toward these lines but will not intersect them. Having both vertical and horizontal asymptotes is a common trait in rational functions, often guiding their plotting and improving our understanding of how to depict their vast reaches.
Function Intercepts
Function intercepts are points where the graph crosses the x-axis or y-axis, crucial for defining the location and intersections of a graph.
For the given function \( y = \frac{-5}{x} \):
  • X-Intercepts: This where \( y = 0 \), but for this function, the numerator is constant and non-zero, implying no real x-intercepts exist. The graph of this function will not cross the x-axis, as this condition cannot logically fulfill \( y = 0 \).
  • Y-Intercepts: These occur when \( x = 0 \). However, since plugging \( x = 0 \) into the denominator makes the function undefined, there is also no y-intercept. Therefore, the graph won't have a point intersecting the y-axis.
When intercepts are absent, as in this case, it reflects specific behaviors of rational functions, such as avoiding particular axes upon graphing. This absence is another significant factor when mapping the graph and understanding the exceptions often observed in these mathematical scenarios.

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Most popular questions from this chapter

Industrial Design A storage tank will have a circular base of radius \(r\) and a height of \(r .\) The tank can be either cylindrical or hemispherical (half a sphere). a. Write and simplify an expression for the ratio of the volume of the hemispherical tank to its surface area (including the base). For a sphere, \(V=\frac{4}{3} \pi r^{3}\) and \(S .\) A. \(=4 \pi r^{2} .\) b. Write and simplify an expression for the ratio of the volume of the cylindrical tank to its surface area (including the bases). c. Compare the ratios of volume to surface area for the two tanks. d. Compare the volumes of the two tanks.

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A month is selected at random; a day of that month is selected at random.

Use the fact that \(\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b} \div \frac{c}{d}\) to simplify each rational expression. State any restrictions on the variables. $$ \frac{\frac{8 x^{2} y}{x+1}}{\frac{6 x y^{2}}{x+1}} $$
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