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Chapter 3: Problem 125

Find the equation and sketch the graph of each function. A rational function that passes through \((0,5),\) has the line \(y=1\) as a horizontal asymptote, and has the line \(x=2\) as its only vertical asymptote

Short Answer

Expert verified
The function is \( f(x) = \frac{x-10}{x-2} \), with horizontal asymptote \( y = 1 \) and vertical asymptote \( x = 2 \).

Step by step solution

01

General Form of a Rational Function

A rational function can be written in the form \[ f(x) = \frac{P(x)}{Q(x)} \] where both \( P(x) \) and \( Q(x) \) are polynomials.
02

Horizontal Asymptote Identification

The horizontal asymptote for a rational function occurs when the degrees of the polynomials in the numerator \( P(x) \) and denominator \( Q(x) \) satisfy certain conditions. For the horizontal asymptote to be \( y = 1 \), the degrees of \( P(x) \) and \( Q(x) \) must be equal, and the leading coefficients must allow \( \lim_{x \to \infty} \frac{P(x)}{Q(x)} = 1 \). That usually translates to both leading coefficients being 1, assuming \(P(x)\) and \(Q(x)\) are simplified.
03

Vertical Asymptote Identification

The vertical asymptote at \(x = 2\) means that the denominator \( Q(x) \) must have a factor of \( (x-2) \), and there should be no cancellation from the numerator.
04

Using the Point\((0, 5)\)

The function passes through \((0, 5)\), so substituting \(x = 0\) and \(y = 5\) into the function helps to find any remaining constants. If we assume the simplest form \(f(x) = \frac{ax+b}{x-2}\), then \(5 = \frac{a \cdot 0 + b}{0-2} \Rightarrow 5 = \frac{b}{-2} \Rightarrow b = -10\).
05

Simplifying the Equation

With \( b = -10 \), and knowing the numerator and denominator degrees must be equal (single linear term), we can assume \(P(x) = x-10\). Using the horizontal asymptote condition, \(f(x) = \frac{x-10}{x-2}\).
06

Graphing the Function

To sketch the graph, identify the asymptotes first: the horizontal asymptote \( y = 1 \) and vertical asymptote \( x = 2 \). Plot the point \( (0, 5) \) and draw the hyperbolic shape of the function approaching the asymptotes.

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Key Concepts

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

Horizontal Asymptote
A horizontal asymptote of a rational function provides information about the behavior of the function as x approaches infinity or negative infinity.
In this exercise, the horizontal asymptote is given by the line y = 1.
This tells us that no matter how large or small the x-values get, the function value will get closer and closer to 1 but never actually touch it.
To determine the horizontal asymptote, check the degrees of the numerator and the denominator polynomials.
If the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.
Since our function has a horizontal asymptote at y = 1, both leading coefficients of the polynomials in the numerator and denominator must be equal and result in 1.
Vertical Asymptote
Vertical asymptotes occur where the denominator of a rational function equals zero, and the numerator is not zero.
In this exercise, the vertical asymptote is at the line x = 2.
This indicates the function is undefined at x = 2. Therefore, as x approaches 2, the function values will increase or decrease without bound (go to infinity or negative infinity).
For this vertical asymptote, the denominator must have a factor of (x - 2), making the function in a form like \( f(x) = \frac{P(x)}{Q(x)}\), where Q(x) includes (x - 2) as a factor.
Graphing Rational Functions
Graphing a rational function involves a few key steps. Begin with identifying the asymptotes as they form the framework for the graph.
In our exercise, draw the horizontal line y = 1 and the vertical line x = 2 on your graph.
Next, plot any given points. Here, we have the point (0, 5).
Then, consider the behavior of the function around the vertical asymptote. As x approaches 2 from both directions, the function typically diverges to positive or negative infinity.
Finally, sketch the general shape of the rational function, paying attention to these asymptotes and plotted points.
The curve will get closer and closer to the asymptotes but will never intersect or cross them.
Polynomials
Polynomials are essential in constructing rational functions.
A polynomial is an expression that consists of variables and coefficients, combined using addition, subtraction, and multiplication. Examples include \( P(x) = x - 10 \), which is linear.
In rational functions, both the numerator and the denominator are polynomials.
The degree of the polynomial, which is the highest power of the variable, plays a crucial role in determining the function's behavior (e.g., horizontal asymptote criteria).
In our exercise, the polynomial in the numerator \( P(x) = x - 10 \) and the denominator \( Q(x) = x - 2 \) both have the same degree, thus influencing the horizontal asymptote to be a ratio of their leading coefficients.

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