Problem 96 - 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 \( \text { of } f(x)=\frac{1}{x} \text { to graph } g \). $$ g(x)=\frac{3 x+7}{x+2} $$

Step-by-Step Solution

Answer

The rewritten equation using long division is \(g(x)=3+\frac{1}{x+2}\). The function's graph is a translation of the standard graph \(f(x)=\frac{1}{x}\), shifted up by 3 units and to the left by 2 units.

Step 1: Perform Long Division

Start by dividing the numerator \(3x+7\) by the divisor \(x+2\) using long division method. The result will be a quotient of \(3\) with a remainder of \(1\). Therefore, the equation for \(g(x)\) can be rewritten as \(3 + \frac{1}{x+2}\).

Step 2: Identify the Parent Function and its Transformations

The parent function in this case is \(f(x) = \frac{1}{x}\). Our transformed function is \(g(x) = 3 + \frac{1}{x+2}\). The transformations include vertical shift up by 3 units and horizontal shift to the left by 2 units.

Step 3: Graph the Parent and Transformed Functions

To plot the function \(g(x) = 3 + \frac{1}{x+2}\), first sketch the parent function \(f(x) = \frac{1}{x}\). Then, apply transformations to the parent function. That is, shift the graph of \(f(x)\) up by 3 units and to the left by 2 units. Now, you have the graph of \(g(x) = 3 + \frac{1}{x+2}\).

Key Concepts

Rational Functions

Rational functions are expressions that involve fractions with polynomials in both the numerator and the denominator. They take the form \( \frac{p(x)}{q(x)} \), where \(p(x)\) and \(q(x)\) are polynomials, and \(q(x) eq 0\). One interesting aspect of rational functions is how they can exhibit vertical asymptotes, which occur when the denominator equals zero, and horizontal asymptotes, which relate to the degree or the leading coefficients of the polynomials.

• The rational function in the exercise is \( g(x) = \frac{3x+7}{x+2} \).
• When we divide, often called **long division**, it's possible to decompose the function into a simpler form.
• This involves dividing the numerator by the denominator, leading us to a quotient with a remainder, as seen with \( 3 + \frac{1}{x+2} \).

Long division in rational functions helps to split the function into easily recognizable components, making further mathematical operations, such as transformations or graphing, more straightforward.

Function Transformation

Function transformation involves shifting, reflecting, stretching, or compressing the graph of the original function to obtain a new function. From the given exercise, the goal is to apply transformations to the parent rational function \( f(x) = \frac{1}{x} \) to obtain \( g(x) = 3 + \frac{1}{x+2} \).

• **Vertical transformations** can shift graphs up or down. In the exercise, we see a vertical shift upwards by 3 units due to the constant term 3 in \( 3 + \frac{1}{x+2} \).
• **Horizontal transformations** can shift graphs left or right. A horizontal shift to the left by 2 units is noted because of the \(+2\) in the denominator \((x+2)\).

These transformations can significantly affect the overall shape and position of the function graph, changing the location of asymptotes and intercepts—helping to piece together where the function's graph sits on the coordinate plane.

Graphing Functions

Graphing functions is a crucial part of understanding their behavior visually. By graphing, we can interpret a function's asymptotes, intercepts, and general shape. Let's break down the process:

• We begin with the parent function \( f(x) = \frac{1}{x} \), which is a hyperbolic curve with vertical and horizontal asymptotes on the axes.
• For \( g(x) = 3 + \frac{1}{x+2} \), we apply the transformations identified in the previous section.
• Start by plotting **a vertical shift**, moving the graph up by 3 units. You will adjust the horizontal asymptote from \(y = 0\) to \(y = 3\).
• Next, implement **a horizontal shift** by shifting the graph's entirety to the left by 2 units. This moves our vertical asymptote from \(x = 0\) to \(x = -2\).

When each of these transformations is applied, using some anchor points and asymptotes will help sketch an accurate plot of the transformed function. A critical understanding of graphing allows one to see how the mathematical operation stretches beyond numbers to real-world applications of these transformations.