Problem 44 - Algebra and Trigonometry

Question

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(t(x)=\frac{x-2}{x^{2}-4 x}\)

Step-by-Step Solution

Answer

X-intercept: (2, 0); vertical asymptotes: x=0, x=4; horizontal asymptote: y=0.

Step 1: Simplify the Function

The given function is \(t(x)=\frac{x-2}{x^2-4x}\). First, simplify the denominator: \(x^2-4x=x(x-4)\). Therefore, the function simplifies to \(t(x)=\frac{x-2}{x(x-4)}\).

Step 2: Find the Intercepts

To find the x-intercepts, set the numerator equal to zero: \(x-2=0\), solving gives \(x=2\). Thus, the x-intercept is \((2, 0)\). There is no y-intercept because \(t(x)\) is undefined at \(x=0\).

Step 3: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero unless canceled by the numerator. Since \(x(x-4)=0\), vertical asymptotes are at \(x=0\) and \(x=4\).

Step 4: Determine Horizontal Asymptotes

Since the degree of the polynomial in the denominator (2) is greater than that of the numerator (1), the horizontal asymptote is \(y=0\).

Step 5: Confirm with Graphing

Sketch the simplified function \(t(x)=\frac{x-2}{x(x-4)}\), ensuring to reflect the calculated intercepts and asymptotes. Use a graphing calculator to confirm that the function has an x-intercept at \(x=2\), vertical asymptotes at \(x=0\) and \(x=4\), and a horizontal asymptote at \(y=0\).

Key Concepts

Intercepts in Algebra

Intercepts in algebra are essential points where the graph of a function crosses the axes. These include:

• X-intercepts: Points where the graph touches or crosses the x-axis. To find these, set the numerator of the rational function equal to zero and solve for \(x\). For example, with the function \(t(x) = \frac{x-2}{x(x-4)}\), setting \(x-2=0\) solves to \(x=2\). Thus, the x-intercept is \((2, 0)\).


• Y-intercepts: Points where the graph touches or crosses the y-axis. You find these by evaluating the function at \(x=0\). However, in rational functions like \(t(x) = \frac{x-2}{x(x-4)}\), if the function is undefined at \(x=0\), there is no y-intercept.

Understanding intercepts helps to sketch the initial framework of the graph, showing where it interacts with the axes.

Vertical Asymptotes

Vertical asymptotes are lines that the graph approaches but never touches or crosses. They occur where the denominator of the rational function is zero, leading to undefined values.

• To locate vertical asymptotes, solve for values of \(x\) that make the denominator zero. For the function \(t(x) = \frac{x-2}{x(x-4)}\), the denominator \(x(x-4)\) equals zero when \(x=0\) or \(x=4\), indicating vertical asymptotes at these points.
• Remember, if the zero in the denominator cancels with the numerator, it might not necessarily imply a vertical asymptote.

Vertical asymptotes indicate a form of discontinuity and suggest that the function's values increase or decrease without bound as it approaches the asymptote line.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a function, showing how it behaves as \(x\) goes to positive or negative infinity.

• To determine horizontal asymptotes, compare the degrees of the polynomials in the numerator and the denominator. If the degree of the denominator is greater than the numerator, the horizontal asymptote is at \(y=0\). This is the case for \(t(x)=\frac{x-2}{x(x-4)}\), since the denominator's degree (2) is greater than the numerator's (1).


• If the degrees are equal, divide the leading coefficients to find the horizontal asymptote's equation. If the numerator's degree is greater, there is typically no horizontal asymptote.

Horizontal asymptotes help predict the behavior of the graph at extreme \(x\) values, stabilizing the sketch of the rational function as it sets bounds on how far up or down the values can go.