Problem 27 - Precalculus with Calculus Previews

Question

Find the vertical and slant asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph \(f\). $$ f(x)=\frac{x^{2}-2 x-3}{x-1} $$

Step-by-Step Solution

Answer

Vertical asymptote is \(x = 1\), slant asymptote is \(y = x - 1\), x-intercepts are \(x = 3\) and \(x = -1\), y-intercept is \((0, 3)\).

Step 1: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (and the numerator is non-zero at the same values). We solve for where \(x - 1 = 0\). Thus, the vertical asymptote is \(x = 1\).

Step 2: Determine Slant Asymptotes

To find the slant asymptote, perform polynomial long division on \(\frac{x^2 - 2x - 3}{x - 1}\). After dividing, the quotient is \(x - 1\) with a remainder of \(-4\). Therefore, the slant asymptote is \(y = x - 1\).

Step 3: Find x-Intercepts

x-intercepts occur where the numerator is zero. Solve \(x^2 - 2x - 3 = 0\) to find \((x - 3)(x + 1) = 0\). Therefore, the x-intercepts are \(x = 3\) and \(x = -1\).

Step 4: Find y-Intercept

The y-intercept is the value of \(f(x)\) when \(x = 0\). Calculate \(f(0) = \frac{0^2 - 2\cdot0 - 3}{0 - 1} = 3\). The y-intercept is \((0, 3)\).

Step 5: Sketch the Graph

Using the information from the previous steps, sketch the graph. Start by plotting the vertical asymptote at \(x = 1\), the slant asymptote \(y = x - 1\). Plot the x-intercepts at \((3, 0)\) and \((-1, 0)\), and the y-intercept at \((0, 3)\). Draw the curve approaching the asymptotes.

Key Concepts

Vertical Asymptotes

In rational functions, vertical asymptotes occur at the values of the variable that make the denominator zero. This happens because you cannot divide by zero; therefore, the function tends to infinity at these points.
In the given function, \(f(x) = \frac{x^2 - 2x - 3}{x - 1}\), we identify the vertical asymptote by setting the denominator equal to zero: \[ x - 1 = 0 \]

• Solving this gives \(x = 1\).

This means the graph of the function will approach, but never touch, the vertical line \(x = 1\). This asymptote signals a restriction in the domain of the function.

Slant Asymptotes

Slant (or oblique) asymptotes occur when the degree of the polynomial in the numerator is exactly one higher than the degree of the denominator. To determine this, we conduct polynomial long division.
For the function \(f(x) = \frac{x^2 - 2x - 3}{x - 1}\), the division process yields a quotient of \(x - 1\) and a remainder of \(-4\), expressed as:\[\frac{x^2 - 2x - 3}{x - 1} = (x - 1) + \frac{-4}{x - 1}\]

• The equation \(y = x - 1\) is the slant asymptote.

The graph of the function will closely follow this line for large absolute values of \(x\), even crossing it at some points.

Intercepts

Intercepts are the points where the graph crosses the axes. Let's find them for our rational function.

x-intercepts: These happen when the numerator of the rational expression equals zero because that makes the entire fraction zero. We solve:\[x^2 - 2x - 3 = 0\]

• This factors to \((x - 3)(x + 1) = 0\), giving \(x = 3\) and \(x = -1\).

Hence, the x-intercepts are at the points \((3,0)\) and \((-1,0)\).

y-intercept: The y-intercept occurs when \(x = 0\). Substitute zero into the function to find:\[f(0) = \frac{0^2 - 2\cdot0 - 3}{0 - 1} = 3\]

• This gives the y-intercept \((0, 3)\).

Polynomial Long Division

Polynomial long division is a useful tool when dealing with rational functions where the degree of the numerator is greater than the degree of the denominator. This method allows us to divide the polynomials, similar to how we divide numbers.
For \(f(x) = \frac{x^2 - 2x - 3}{x - 1}\), you set it up like long division in arithmetic:

• The divisor is \(x - 1\), and the dividend is \(x^2 - 2x - 3\).

Divide the leading term of the dividend by the leading term of the divisor to get \(x\). Multiply the whole divisor by \(x\), subtract it from the dividend, and bring down the next term. Repeat this process:

• The initial calculation gives \(x - 1\).
• The remainder is \(-4\), as shown with the expression \(-4/(x - 1)\).

Thus, the division concludes here, helping us identify the slant asymptote and analyze the behavior of the graph for rational functions.