Problem 33 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Rewrite the function \(g\) 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 \(f(x)=\frac{a}{x}\). \(g(x)=\frac{12 x}{x-5}\)

Step-by-Step Solution

Answer

The given function \(g(x)=\frac{12x}{x-5}\) can be simplified into the form \(g(x)=\frac{12}{x-5}\), when \(x\ne5\). The graph of this function has vertical and horizontal asymptotes at \(x = 5\) and \(y = 0\) respectively. Compared to \(f(x)=\frac{12}{x}\), the graph of \(g(x)\) is a shift to the right by \(5\) units and 12 times higher.

Step 1: Simplifying \(g(x)\) into the form \(g(x)=\frac{a}{x-h}+k\)

To simplify the function \(g(x)=\frac{12 x}{x-5}\) into the form \(g(x)=\frac{a}{x-h}+k\), we break up the fraction as follows: \(g(x)=\frac{12x}{x-5} = \frac{12}{1}\cdot\frac{x}{x-5} = 12\cdot\frac{1}{\frac{x}{5}-1}.\) So the function can be rewritten as \(g(x)=\frac{12}{\frac{x}{5}-1} = \frac{12}{x - 5}\) when \(x\ne5\). Here \(a = 12, h = 5, k = 0\).

Step 2: Graphing g(x)

To graph the function \(g(x)=\frac{12}{x-5}\), we note that it has a vertical asymptote at \(x = 5\) and a horizontal asymptote at \(y = 0\). We plot several points for \(x\) on either side of \(x = 5\), and draw the graph of \(g(x)\) approaching the asymptotes as \(x\) approaches \(5\) from either direction, and as \(x\) approaches positive or negative infinity.

Step 3: Describing the Transformation of the Graph

The graph of the function \(g(x)=\frac{12}{x-5}\) is a transformation of the graph of the function \(f(x)=\frac{12}{x}\). Specifically, it is a shift to the right by \(5\) units compared to \(f(x)\). Additionally, due to the value of a, the graph of g(x) will be 12 times higher than the graph of f(x).

Key Concepts

Function Transformation

Function transformation refers to the shifting, stretching, compressing, or reflecting of a function's graph. For the function in our problem, we are focusing on rewriting and interpreting it as a transformation of the parent function. The given function is initially expressed as \( g(x)=\frac{12x}{x-5} \). After conversion, it takes the form \( g(x)=\frac{12}{x-5} \), which is a variant of \( f(x)=\frac{a}{x-h}+k \).

• "\( a \)" in the function denotes vertical scaling. Here, \( a = 12 \), indicating the graph is stretched vertically by a factor of 12 compared to the simple reciprocal function \( f(x)=\frac{1}{x} \).
• "\( h \)" affects horizontal translation. The term \( x-h \) means the graph is shifted "h" units right. In this function, \( h = 5 \), thus \( g(x) \) moves 5 units to the right compared to \( f(x)=\frac{a}{x} \).
• "\( k \)" describes vertical translation. Since \( k = 0 \), there is no vertical shift.

These transformations result in the graph of \( g(x) \) being scaled, shifted, and thereby differing significantly from its parent form.

Vertical Asymptote

Vertical asymptotes are lines that the graph approaches but never actually reaches or crosses. In rational functions, vertical asymptotes typically occur where the denominator of the function equals zero. For the function \( g(x)=\frac{12}{x-5} \), the denominator becomes zero when \( x = 5 \). Therefore, \( x = 5 \) is a vertical asymptote.

• Approaching the asymptote from the left, \( g(x) \) decreases towards negative infinity.
• Approaching from the right, \( g(x) \) rises towards positive infinity.

The behavior near the vertical asymptote reflects a dramatic change in the function’s output, showcasing the critical points where the function is undefined due to division by zero. This phenomenon is crucial in sketching the accuracy of the graph and understanding the discontinuities inherent in rational functions.

Horizontal Asymptote

Horizontal asymptotes are lines that the graph of a function approaches as \( x \) moves towards positive or negative infinity. In this rational function, the horizontal asymptote is at \( y = 0 \). This occurs because the degree of the polynomial in the denominator is greater than the one in the numerator, leading the terms to disappear as \( x \) approaches infinity.

• As \( x \) tends to infinity, \( g(x) \) approaches the horizontal line \( y = 0 \).

Horizontal asymptotes provide valuable information about the end behavior of the function. They indicate how the function behaves as \( x \) becomes very large (positively or negatively). In this exercise, no matter how large \( x \) gets, the values of \( g(x) \) continue to move closer and closer to zero. Understanding horizontal asymptotes is essential for grasping the overall behavior and trend of rational functions.