Problem 6 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function \(g\) and describe how the graph is related to the graph of \(f(x)=1 / x\) $$g(x)=\frac{1}{x-6}$$

Step-by-Step Solution

Answer

The graph of \(g(x) = 1/(x - 6)\) is a hyperbola with a vertical asymptote at \(x = 6\) and a horizontal asymptote at \(y = 0\), which is the graph of \(f(x) = 1/x\) shifted to the right by 6 units.

Step 1: Understand the Parent Function

Consider the parent function \(f(x) = 1/x\). The graph of \(f(x) = 1/x\) is a hyperbola with a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\). When \(x\) is positive, the function is positive and when \(x\) is negative, the function is negative.

Step 2: Identify the Transformation

Now consider the function \(g(x) = 1/(x - 6)\). Notice that compared to the parent function \(f(x) = 1/x\), the denominator in the function \(g\) is \(x - 6\) instead of \(x\). This means that every \(x\) in the parent function is replaced with \(x - 6\) in the function \(g\). Therefore, the graph of \(g\) is just the graph of \(f\) shifted to the right 6 units. There are no other transformations.

Step 3: Sketch the Graph

The graph of \(g\) will have the same shape as the graph of \(f\), but it will be moved 6 units to the right. Therefore, the vertical asymptote will be at \(x = 6\) instead of \(x = 0\), and the horizontal asymptote will still be at \(y = 0\). Choose several key points on the graph of \(f\), shift them 6 units to the right to get the corresponding points on the graph of \(g\), and plot the points. Then connect the points to draw the graph.

Key Concepts

Parent Function

In mathematics, a parent function is the simplest form of a set of functions that form a family. These parent functions provide the basic shape and characteristics from which variations of the function are derived through transformations. For rational functions like f(x) = 1/x, the parent function is a hyperbola symmetrical about the origin. Hyperbolas have two separate curves or 'branches', and for f(x) = 1/x, as x approaches zero from the right, the y-values become increasingly positive; as x approaches zero from the left, the y-values become increasingly negative.

Horizontal Asymptote

A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as x goes to positive or negative infinity. It represents a value that the output of the function (the y-value) gets closer and closer to, but never actually reaches. For the parent function f(x) = 1/x, the horizontal asymptote is the x-axis, or the line y = 0. This is because as x becomes very large or very negative, the value of 1/x gets closer and closer to zero.

Vertical Asymptote

A vertical asymptote is a vertical line that the graph of a function approaches as the independent variable x approaches a specific value. Vertical asymptotes occur in rational functions when the denominator is zero and the function is undefined. For the parent function f(x) = 1/x, the function is undefined at x = 0, making x = 0 the vertical asymptote. As x gets very close to zero, the function values increase or decrease without bound, causing the graph to approach this vertical line asymptotically.

Transformation of Functions

A transformation of functions involves changing the position, shape, or size of the graph of the function. In the case of the function g(x) = 1/(x - 6), a horizontal shift is applied to the parent function f(x) = 1/x. The entire graph of the parent function is moved 6 units to the right because every x value in the parent function is replaced with x - 6 in the transformed function. Transformations can include shifts, reflections, stretches, and compressions. Understanding how these transformations modify the original graph is essential for graphing the new function accurately.