Problem 37 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

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 \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{x+18}{x-6} $$

Step-by-Step Solution

Answer

The function \(g(x)=\frac{x+18}{x-6}\) can be rewritten in the form \(g(x)=1+\frac{18}{x-6}\) which represents a shift to the right by 6 units and an upward shift by 1 unit from the graph of the original function \(f(x)=\frac{a}{x}\).

Step 1: Rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\)

Dividing every term by \(x\), the function \(g(x)=\frac{x+18}{x-6}\) can be rewritten as: \(g(x)=1+\frac{18}{x-6}\). So, comparing this with the standard function form \(g(x)=\frac{a}{x-h}+k\), we get \(a=18\), \(h=6\), and \(k=1\).

Step 2: Graph the function

For the given function \(g(x)=1+\frac{18}{x-6}\), the graph can be constructed by shifting the hyperbola \(\frac{a}{x}\) horizontally by 6 units to the right and then vertically upwards by 1 unit. The horizontal asymptote is at \(y=k=1\) and the vertical asymptote is at \(x=h=6\).

Step 3: Describe the transformation

From the graph of the function \(f(x)=\frac{a}{x}\), the graph of the function \(g(x)=1+\frac{18}{x-6}\) can be obtained by a transformation that involves a shift to the right by 6 units and an upward shift by 1 unit. This is because the parameter \(h=6\) and \(k=1\).

Key Concepts

Graphing Rational Functions

Rational functions, which are ratios of polynomials, can be a challenging topic. When graphing the rational function like the one in the exercise, \(g(x)=\frac{x+18}{x-6}\), understanding its structure is crucial. First, rewriting it to the standard form \(g(x)=\frac{a}{x-h}+k\) helps us see the transformations more clearly.

To graph \(g(x)\), we start with the basic function \(f(x)=\frac{a}{x}\) which is a hyperbola. For \(g(x)=1+\frac{18}{x-6}\), we apply a horizontal shift to the right by looking at \(h\), and a vertical shift by considering \(k\). Here, \(h=6\) and \(k=1\), indicating a 6-unit right and 1-unit upward shift, respectively, from the basic hyperbola graph. This means for every point on the graph of \(f(x)\), you move it 6 units right and 1 unit up to get to the corresponding point on \(g(x)\).

Using graph paper or technology, we can visualize these shifts and plot our transformed graph accordingly, ensuring we mark our new asymptotes and observe the behavior approaching them.

Horizontal and Vertical Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. They can be horizontal, vertical, or oblique, and understanding them is essential when graphing rational functions. The given function, \(g(x)=1+\frac{18}{x-6}\), has both horizontal and vertical asymptotes.

The vertical asymptote occurs where the function's denominator is zero, which is at \(x=h\). In our example, setting the denominator \(x-6=0\) gives us \(x=6\), indicating a vertical line at \(x=6\) that our graph will never cross.

The horizontal asymptote represents the value that \(y\) approaches as \(x\) goes to infinity. It's found by looking at \(k\) in our standard function form. Here, \(k=1\), giving us a horizontal line at \(y=1\). As \(x\) becomes very large or very small, the term \(\frac{18}{x-6}\) approaches zero, and the graph approaches the line \(y=1\), but will never actually reach it. These asymptotes frame the graph and help us understand its limits and behavior at extremes.

Shifts in Graphs of Functions

Shifts are one of the primary transformations we can apply to the graph of a basic function to get another function’s graph. They include horizontal shifts (left or right) and vertical shifts (up or down). To correct a common confusion, remember that for horizontal shifts, the direction is opposite the sign inside the function’s argument. So, \(x-h\) results in a shift \(\text{h}\) units to the right.

In our example \(g(x)=1+\frac{18}{x-6}\), we identified a horizontal shift \(\text{h}=6\) units to the right and a vertical shift \(\text{k}=1\) unit upward. To visualize shifts:

• Start with the basic graph of \(f(x)\).
• Shift every point \(\text{h}\) units horizontally in the correct direction based on the sign.
• Then shift it \(\text{k}\) units vertically.

Visualizing these shifts is essential for graphing functions accurately, and doing so systematically can greatly simplify the graphing process. With practice, identifying and applying these transformations becomes more intuitive, allowing for quick sketches of complex functions’ graphs.