Problem 95 - Algebra and Trigonometry

Question

Use long division to rewrite the equation for \(g\) in the form $$ \text {quotient}+\frac{\text {remainder}}{\text {divisor}} $$ Then use this form of the function's equation and transformations $$ \text { of } f(x)=\frac{1}{x} \text { to graph } g $$. $$ g(x)=\frac{2 x+7}{x+3} $$

Step-by-Step Solution

Answer

The function \(g(x)\) can be rewritten as \(2+\frac{1}{x+3}\). The graph of \(g(x)\) can be obtained by shifting the graph of \(f(x)=\frac{1}{x}\) 3 units to the left and 2 units upward.

Step 1: Long division of polynomials

First, one needs to rewrite the function \(g\) using long division. When \(2x+7\) is divided by \(x+3\), the quotient is \(2\) and the remainder is \(1\). Therefore, the function \(g\) can be rewritten in the form \(\text{quotient}+\frac{\text{remainder}}{\text{divisor}}\), which is \(2+\frac{1}{x+3}\).

Step 2: Identify transformations

The next step is to identify the transformations that need to be made to the function \(f(x)=\frac{1}{x}\) in order to make it look like our rewritten \(g(x)\). By comparing \(f(x)=\frac{1}{x}\) and \(2+\frac{1}{x+3}\), we can see that the graph of \(f(x)=\frac{1}{x}\) needs to be shifted 3 units to the left and then shifted 2 units upward to obtain the graph of \(g(x)\).

Step 3: Graph the function

To graph our \(g(x)\), we start by drawing \(f(x)=\frac{1}{x}\), which is a hyperbola. Then, we shift the graph 3 units to the left and 2 units upward as we reasoned in Step 2. The result is the graph of \(g(x)=2+\frac{1}{x+3}\).

Key Concepts

Function Transformation

Function transformation involves changing the position or shape of a function's graph in the coordinate plane. It allows us to see how altering an equation affects its graphical representation. In this example, we started with the function \(f(x) = \frac{1}{x}\), which is a basic hyperbolic curve.
To transform \(f(x)\) into \(g(x) = 2 + \frac{1}{x+3}\), we use two steps:

• Horizontal Shift: Sliding the graph left or right. Here, adding \(3\) inside the denominator \((x+3)\) shifts the graph 3 units left.
• Vertical Shift: Adjusting the graph up or down. Adding \(2\) to the entire function moves it 2 units up.

These shifts adjust where the asymptotes (the lines the graph approaches but never touches) occur and change its position on the graph.
Understanding transformations is key to graphing any function just by looking at its equation, without plotting multiple points in advance.

Graphing Rational Functions

Rational functions are functions made up of the ratio of two polynomials. Their graphs are often hyperbolas or other more complicated shapes depending on the polynomials in the numerator and denominator.
To graph the rational function \(g(x) = \frac{2x+7}{x+3}\), first rewrite it using the long division form, \(2+\frac{1}{x+3}\), which makes it more relatable to other simpler rational functions.
The rewritten form \(2 + \frac{1}{x+3}\) tells us:

• There's a horizontal asymptote at \(y=2\) which reflects the highest power term in the quotient.
• The vertical asymptote is at \(x=-3\) because the function is undefined there (division by zero occurs when \(x+3=0\)).

By placing the transformed \(f(x) = \frac{1}{x}\) graph accordingly, respecting these asymptotes and shifts, the function \(g(x)\) is portrayed correctly.

Polynomial Division

Polynomial division is similar to the long division of numbers. It simplifies rational functions by breaking them into manageable parts.
The goal with polynomial division in this context is to rewrite the function \(g(x) = \frac{2x+7}{x+3}\) in a simpler form: \(2 + \frac{1}{x+3}\).
Here's how polynomial long division is carried out:

• Divide: Start with the leading term of the numerator dividing by the leading term of the denominator.
• Multiply: Multiply the entire divisor by this quotient result and subtract from the original polynomial.
• Repeat: Continue this process until what's left (remainder) is of lesser degree than the divisor.

In this problem, dividing \(2x+7\) by \(x+3\) gave us a quotient of \(2\) and a remainder of \(1\). Such simplifications are essential in graphing because they help identify key characteristics of the function's graph including vertical and horizontal asymptotes.