Question

In Exercises 33-40, rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{x+2}{x-8} $$

Short answer

The given function in the form \(g(x)=\frac{a}{x-h}+k\) is \(g(x) = \frac{1}{x-8} + \frac{2}{x-8}\). When graphed, this function is a hyperbola shifted 8 units to the right and \(\frac{2}{x-8}\) units up from the graph of \(\frac{a}{x}\).

Step-by-step solution

Rewrite the function

The function can be rewritten as \(g(x)= \frac{x}{x-8} + \frac{2}{x-8}\). Given that \(f(x) = \frac{a}{x}\), this can be identified as \(g(x)=f(x)+k\) with \(h = 8\) for \(f(x)\), and \(k = \frac{2}{x-8}\).

Graph the function

To plot this using a standard graphing tool or calculator, use the x-values to get the corresponding y-values. Select various x-values including those for x>8 and x<8, excluding x=8 because it will make the denominator zero, hence undefined. The function \(g(x)\) is a hyperbola shifted 8 units to the right and \(\frac{2}{x-8}\) units up.

Describe transformations

The graph of the function \(g(x)\) is obtained by translating the graph of \(f(x)=\frac{1}{x}\) 8 units to the right and \(\frac{2}{x-8}\) units up. This is due to the addition of the \(h\) and \(k\) values in the \(g(x)\) equation compared to the \(f(x)\) equation. The transformation can also lead to a shape change in the graph, making it appear stretched or compressed.

Key concepts

Graphing Hyperbolas

Graphing a hyperbola involves understanding its shifts and how each part of the equation contributes to its graph. A hyperbola is a type of rational function characterized by its two disconnected curves called branches. These branches symmetrically open in opposite directions.

When graphing a hyperbola like \(g(x)=\frac{x+2}{x-8}\), you must be aware of asymptotes. Asymptotes are lines that the graph approaches but never touches or crosses, guiding the hyperbola's shape. For the given function, there is a vertical asymptote at \(x = 8\) because the denominator cannot be zero. There is also a horizontal asymptote determined by the behavior of the function as \(x\) becomes very large or very small, which in this case runs towards \(y = 1\) because \(\frac{x}{x-8}\) approaches 1.

• Vertical Asymptote: \(x = 8\).
• Horizontal Asymptote: As \(x\rightarrow \pm \infty, y \rightarrow 1\).

The branches of the hyperbola will hug these asymptotes closely without touching them and will reflect symmetrically relative to the point where they intersect, which is sometimes called the hyperbola’s center.

Rational Functions

To understand what rational functions are, we need to recognize that they are ratios of two polynomial functions. The general form of a rational function is \(f(x) = \frac{p(x)}{q(x)}\) where \(p(x)\) and \(q(x)\) are polynomials and \(q(x)\) is not zero.

The function \(g(x) = \frac{x+2}{x-8}\) fits the description of a rational function. Its behavior is influenced heavily by its denominator, which can affect the domain of the function, specifically creating points at which the function is undefined. Here, this occurs at \(x = 8\).

• The numerator's degree (1, for \(x+2\)) and the denominator's degree (1, for \(x-8\)) are the same, indicating that the horizontal asymptote can be determined by the ratio of the leading coefficients (both are 1 here), resulting in an asymptote at \(y = 1\).

Rational functions are essential in modeling real-world situations that involve rates or ratios.

Translation of Functions

Translation is a kind of transformation that shifts a graph horizontally and/or vertically from its original position. When working with functions, translation does not alter the shape or orientation of the graph.

For the function \(g(x)=\frac{x+2}{x-8}\), the graph is born from translating the base function \(f(x)=\frac{1}{x}\). The term \(-8\) in the denominator induces a horizontal shift, moving the graph 8 units to the right. Meanwhile, the numerator \(x+2\) is restructured from the base function and affects the graph's position as part of the transformation.

• Horizontal Translation: Right by 8 units due to \(h=8\) in the function setup.
• Vertical Translation: Adjusted by the impact of \(\frac{2}{x-8}\).

This movement underscores how a function's graph shifts without altering its fundamental shape, allowing you to discern connections and similarities between transformed and parent functions efficiently.