Equation of Rational Function with Asymptotes and Intercepts

Math Problem Statement

If f has a vertical asymptote given by xequals2, a horizontal asymptote yequals0, y-intercept at minus2, and no x-intercept, then write the equation of a rational function f(x)equalsStartFraction p left parenthesis x right parenthesis Over q left parenthesis x right parenthesis EndFraction having the indicated properties, in which the degrees of p and q are as small as possible. Graph the function using a graphing utility to verify that it has the required properties. Question content area bottom Part 1 Write the equation of the rational function. Choose the correct answer below. A. f(x)equalsStartFraction 4 Over x plus 2 EndFraction B. f(x)equalsStartFraction 4 Over x minus 2 EndFraction C. f(x)equalsStartFraction 2 Over x minus 2 EndFraction D. f(x)equalsStartFraction 2 Over x plus 2 EndFraction Part 2 Use a graphing utility to graph the function obtained in the previous step. Choose the correct graph below. Each graph is shown in a viewing rectangle [minus10,10,2] by [minus10,10,2]. A.

A coordinate system has a horizontal axis with 10 ticks from left to right and a vertical axis with 10 ticks from bottom to top. The xy plane contains two solid branches. From left to right, one branch is a curve falling along the negative x-axis at an increasing rate approaching the line x=2. Another branch is a curve falling from top right along the line x=2 at a decreasing rate approaching the positive x-axis. B.

A coordinate system has a horizontal axis with 10 ticks from left to right and a vertical axis with 10 ticks from bottom to top. The xy plane contains two solid branches. From left to right, one branch is a curve rising along the negative x-axis at an increasing rate approaching the line x= negative 2. The second branch is a curve rising from bottom left along the line x=negative 2 at a decreasing rate approaching the positive x-axis. C.

A coordinate system has a horizontal axis with 10 ticks from left to right and a vertical axis with 10 ticks from bottom to top. The xy plane contains two solid branches. From left to right, one branch is a curve falling along the negative x-axis at an increasing rate approaching the line x= negative 2. The second branch is a curve falling from top left along the line x=negative 2 at a decreasing rate approaching the positive x-axis. D.

A coordinate system has a horizontal axis with 10 ticks from left to right and a vertical axis with 10 ticks from bottom to top. The xy plane contains two solid branches. From left to right, one branch is a curve rising along the negative x-axis at an increasing rate approaching the line x=2. Another branch is a curve rising from bottom right along the line x=2 at a decreasing rate approaching the positive x-axis. Choose the correct answer below.Each graph is shown in a viewing rectangle [10,10,2] by [10,10,2].

Solution

To determine the correct equation of the function $f(x)$ and identify the appropriate graph, let's examine each property:

Vertical Asymptote at $x = 2$ : This indicates that the denominator $q(x)$ has a factor of $(x - 2)$ .
Horizontal Asymptote at $y = 0$ : This suggests that the degree of the numerator $p(x)$ should be less than the degree of the denominator $q(x)$ .
$y$ -intercept at $-2$ : When $x = 0$ , $f(0) = -2$ , which can help determine the constant factor in the numerator.
No $x$ -intercept: This means the numerator $p(x)$ should not have any zeros, so it should be a constant.

Given these conditions, the function that meets all these requirements is: $f(x) = \frac{2}{x - 2}$

This matches option C: $f(x) = \frac{2}{x - 2}$ .

For the graph:

We expect the graph to have a vertical asymptote at $x = 2$ and approach $y = 0$ as $x \to \infty$ or $x \to -\infty$ .
The function should not cross the x-axis and should intersect the y-axis at $y = -2$ when $x = 0$ .

Correct answer for the graph: Option A, which has a curve approaching $x = 2$ vertically and $y = 0$ horizontally, consistent with the given asymptotes.

Would you like more details or have any questions?

Here are some additional questions to explore:

What would happen if the horizontal asymptote were at a nonzero value, like $y = 1$ ?
How would the function change if there was an $x$ -intercept?
How do vertical asymptotes affect the domain of a function?
What is the general rule for determining horizontal asymptotes of rational functions?
How can we determine if a rational function has any holes?

Tip: For rational functions, the degree of the numerator and denominator can help predict horizontal asymptotes.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Asymptotes
Intercepts

Formulas

Rational function form f(x) = p(x) / q(x)
Asymptotes for rational functions

Theorems

Rules for vertical and horizontal asymptotes in rational functions

Suitable Grade Level

Grades 10-12

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