/*! 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 5 Draw a graph of each function. D... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 5

Draw a graph of each function. Describe properties of the graph. \(y=\frac{-3}{x}\)

Short Answer

Expert verified
The graph of the function \(y=\frac{-3}{x}\) has point symmetry to the origin, no x or y intercepts, and asymptotes at y=0 and x=0. It decreases in value as x-value increases from negative to positive infinity.

Step by step solution

01

Understanding the function

The function given is \(y=\frac{-3}{x}\). This is a rational function. It can be observed, visually, by plotting values into the function.
02

Plotting values

Begin by deciding on values for x to input into the function. It is suggested to start with -10, -1, 0, 1, 10. Remember that values of x equal to 0 are not permissible in this function.
03

Identifying the function graph's properties

Next, graph the points, observing asymptotic behavior (where the graph approaches but never touches a line), x and y intercepts (where the graph crosses or touches the x and y axes) and end behavior (the y-values as x approaches negative and positive infinity). In this case, there are vertical asymptotes at x=0, and horizontal asymptotes at y=0. The graph decreases from left to right. Also, it can be observed that the function has a point symmetry to the origin.
04

Graphing the function

Using the properties identified in step 3, draw the graph of the function, with additional emphasis on the asymptotes.

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Key Concepts

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

Graphing Functions
To successfully graph a function like the given rational function, it is crucial to understand its general behavior. The function given is \(y = \frac{-3}{x}\), which is a rational function: a quotient of polynomial functions. When graphing this kind of function, you can follow these steps:
  • Choose values for \(x\): Selecting strategic values can simplify the graphing process. Suggested initial values include -10, -1, 1, and 10. Avoid \(x = 0\) because the function is undefined at this point due to division by zero.
  • Calculate corresponding \(y\) values: Substitute the chosen \(x\) values into the equation to find the corresponding \(y\) values.
  • Plot the points: Once you have several (\(x, y\)) points, plot them on a coordinate grid.
The graph of \(y = \frac{-3}{x}\) will form a hyperbola, a curve very characteristic of certain rational functions. This hyperbolic shape reflects the nature of how these functions behave for various \(x\) values.
Asymptotic Behavior
Asymptotic behavior in functions describes how a function behaves as the input values approach certain points. For rational functions like \(y = \frac{-3}{x}\), it鈥檚 essential to identify both horizontal and vertical asymptotes.
  • Vertical Asymptotes: These occur where the function is undefined, typically where the denominator equals zero. For this function, the vertical asymptote is at \(x=0\) because division by zero is undefined.
  • Horizontal Asymptotes: These describe the value \(y\) approaches as \(x\) goes to infinity. For \(y = \frac{-3}{x}\), as \(x\) increases or decreases without bound, \(y\) approaches 0. This is a key insight into the behavior of this function as \(x\) approaches large positive or negative values.
Asymptotes thus guide us to understand that while the graph never actually reaches these lines, it continually tends closer to them as \(x\) moves away from the origin.
Point Symmetry
Point symmetry is a property of certain functions where the graph is symmetric about a single point, often the origin \((0,0)\). For the function \(y = \frac{-3}{x}\), the graph exhibits point symmetry.
  • This means that rotating the graph 180 degrees about the origin leaves the function looking the same.
  • In practical terms, for every point \((x, y)\) on the graph, there is an opposite point \((-x, -y)\) that also lies on the graph. This is a hallmark of hyperbolic shapes and suggests that the function reflects across both axes.
Understanding point symmetry helps in sketching the graph quickly and recognizing patterns in the function鈥檚 behavior. It underscores how this function, particularly, spreads evenly in all quadrants around the origin, showing balance and equivalent depictions in opposite directions.

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Most popular questions from this chapter

A standard number cube is tossed. Find each probability. \(P(\text { greater than } 1 \text { or less than } 5)\)

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