Question

The standard form of the rational function \(R(x)=\frac{m x+b}{c x+d}, c \neq 0,\) is \(R(x)=a\left(\frac{1}{x-h}\right)+k\) To write a rational function in standard form requires polynomial division. (a) Write the rational function \(R(x)=\frac{2 x+3}{x-1}\) in standard form by writing \(R\) in the form Quotient \(+\frac{\text { remainder }}{\text { divisor }}\) (b) Graph \(R\) using transformations. (c) Find the vertical asymptote and the horizontal asymptote of \(R\).

Short answer

The standard form of \( R(x) = \frac{2x+3}{x-1} \) is \( R(x) = 5 \left( \frac{1}{x-1} \right) + 2 \). The vertical asymptote is \( x = 1 \) and the horizontal asymptote is \( y = 2 \).

Step-by-step solution

Perform Polynomial Division

To write the rational function in the form Quotient + \( \frac{\text{remainder}}{\text{divisor}} \), perform polynomial division on \( R(x) = \frac{2x+3}{x-1} \). Divide \( 2x+3 \) by \( x-1 \). The quotient is 2 and the remainder is 5. Therefore, we can write: \[ \frac{2x+3}{x-1} = 2 + \frac{5}{x-1} \]

Express in Standard Form

We need to write the function in the form \( a \left( \frac{1}{x-h} \right) + k \). From the result of polynomial division: \[ R(x) = 2 + \frac{5}{x-1} \] We have: \[ a = 5, h = 1, k = 2 \] So, the standard form of \( R(x) \) is: \[ R(x) = 5 \left( \frac{1}{x-1} \right) + 2 \]

Graph the Function Using Transformations

Start with the graph of the basic function \( \frac{1}{x} \), which has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \). Shift the graph 1 unit to the right to account for \( h = 1 \), which results in a vertical asymptote at \( x = 1 \). Then stretch the graph vertically by a factor of 5, and finally shift it up by 2 units to account for \( k = 2 \). Graphing these transformations will give the graph of \( R(x) \).

Identify the Asymptotes

The vertical asymptote can be found by setting the denominator equal to zero: \( x - 1 = 0 \), hence, \( x = 1 \). The horizontal asymptote is given by the value of \( k \) in the standard form. Since \( k = 2 \), the horizontal asymptote is \( y = 2 \).

Key concepts

Polynomial Division

When handling rational functions like \(\frac{2x+3}{x-1}\), polynomial division is an essential technique. The goal is to transform the fraction into a form that's easier to work with. Here's how you do it:

• Divide the numerator by the denominator.
• Identify the quotient and the remainder.

By dividing 2x+3 by x-1, you obtain a quotient of 2 and a remainder of 5. This allows you to express the original function as:
\[ \frac{2x+3}{x-1} = 2 + \frac{5}{x-1} \]. This breakdown helps in identifying transformations and asymptotes.

Vertical Asymptote

A vertical asymptote is a line where the function's value grows unbounded as it approaches a specific x-value. To find the vertical asymptote, set the denominator of the rational function equal to zero. For \(\frac{2x+3}{x-1}\), this means solving x-1 = 0.

Thus, \( x = 1 \) is the vertical asymptote. As x approaches 1, the function's value tends to infinity or negative infinity. In the graph, this manifests as the function skirting infinitely close to the line \( x = 1 \) but never touching it. Identifying vertical asymptotes is crucial for understanding the behavior of rational functions, especially when graphing.

Horizontal Asymptote

Horizontal asymptotes indicate the value that a function approaches as x goes to infinity or negative infinity. For the standard form \(R(x) = 5 \left( \frac{1}{x-1} \right) + 2\), the horizontal asymptote is found by examining the term 'k'.

Here, 'k' is 2, meaning the horizontal asymptote is \( y = 2 \).
No matter how large or small x becomes, the function's value will get closer and closer to 2 without actually reaching it. This is evident in the graph, where the curve levels off as it extends towards infinity and negative infinity.

Graph Transformations

Graphing a rational function involves several steps of transformation, starting from a basic function like \( \frac{1}{x} \).
For \( R(x) = 5 \left( \frac{1}{x-1} \right) + 2 \):

• Start with the graph of \( \frac{1}{x} \), featuring a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \).
• Shift the graph 1 unit to the right since \( h = 1 \), creating a vertical asymptote at \( x = 1 \).
• Next, stretch the graph vertically by a factor of 5 (the value of 'a'). This makes the curves steeper.
• Finally, shift the graph up by 2 units due to 'k', altering the horizontal asymptote to \( y = 2 \).

Each transformation step helps in visualizing how the original function is reshaped, making asymptotes and overall behavior clearer.