Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. \(\mathbf{I}\) \(f(x)=\frac{x^{2}+3 x+4}{x-5}\) \(\mathbf{II}\) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote.

Step-by-Step Solution

Answer

The graph has an oblique asymptote, matching choice F.

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function is zero and the numerator is non-zero. For the function \( f(x) = \frac{x^2 + 3x + 4}{x-5} \), set the denominator equal to zero: \( x - 5 = 0 \). Solving this, we find \( x = 5 \). Therefore, the vertical asymptote is \( x = 5 \), which is not in Column II.

Step 2: Identify Horizontal Asymptotes

For rational functions of the form \( \frac{p(x)}{q(x)} \), if the degree of \( p(x) \) (numerator) is equal to the degree of \( q(x) \) (denominator), the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients. Here, both the numerator and denominator have leading terms with degree 2 and 1, respectively, so no horizontal asymptote matches here.

Step 3: Identify Oblique Asymptotes

An oblique asymptote exists when the degree of the numerator is one more than the degree of the denominator. In \( f(x) = \frac{x^2 + 3x + 4}{x-5} \), the numerator degree (2) is higher than the denominator degree (1), confirming an oblique asymptote. Thus, \( f(x) \) has an oblique asymptote, matching choice F.

Step 4: Determine the X-Intercepts

The \( x \)-intercepts are found by setting the numerator equal to zero and solving for \( x \). For \( x^2 + 3x + 4 = 0 \), using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \( a = 1, b = 3, c = 4 \): \( x = \frac{-3 \pm \sqrt{9 - 16}}{2} \), which results in no real roots, hence no \( x \)-intercept.

Step 5: Determine the Y-Intercept

The \( y \)-intercept is found by setting \( x = 0 \) in \( f(x) = \frac{x^2 + 3x + 4}{x-5} \). Substituting, \( f(0) = \frac{0^2 + 3 \, \times \, 0 + 4}{0 - 5} = \frac{4}{-5} = -\frac{4}{5} \), no match for intercept \((0,5)\) in Column II.

Key Concepts

Vertical Asymptotes

When you encounter a rational function, identifying vertical asymptotes helps you understand where the graph will shoot off to infinity. Vertical asymptotes are found at values of \( x \) that make the denominator zero.

Consider a function \( f(x) = \frac{p(x)}{q(x)} \). To find vertical asymptotes, solve \( q(x) = 0 \). In the example function \( f(x) = \frac{x^2 + 3x + 4}{x-5} \), set \( x - 5 = 0 \), leading to \( x = 5 \). Here, the graph of the function heads towards positive or negative infinity as \( x \) approaches 5.

Vertical asymptotes remind us of regions where the function is undefined. However, not all zeroes of the denominator lead to vertical asymptotes; sometimes, a hole might occur if a factor is shared with the numerator, but more on that another time.<<

Oblique Asymptotes

Oblique asymptotes appear in rational functions when the degree of the numerator is exactly one more than that of the denominator. This means the function's graph will slant rather than level out over time.

To identify oblique asymptotes in \( f(x) = \frac{x^2 + 3x + 4}{x-5} \), observe that the numerator’s degree (2) is one more than the denominator’s degree (1), suggesting an oblique asymptote exists. We find this asymptote by performing polynomial long division on \( x^2 + 3x + 4 \) by \( x - 5 \).

The result of this division, typically in the form of \( mx + b \), will represent your oblique asymptote, illustrating how the function behaves asymptotically as \( x \) moves towards infinity or negative infinity. In our exercise, this feature matches with choice F. Remember, finding oblique asymptotes accurately describes how your function stretches into space.<<

Intercepts

Intercepts are vital for plotting and understanding a rational function's graph. They describe where the graph intersects the axes, giving us key insight into a graph's position and behavior.

• X-Intercepts: These occur where the numerator equals zero, solving \( p(x) = 0 \). Thus, the value(s) of \( x \) make the entire fraction equal zero. For \( f(x) = \frac{x^2 + 3x + 4}{x-5} \), using the quadratic formula on \( x^2 + 3x + 4 = 0 \) to solve for \( x \) reveals no real roots, meaning no \( x \)-intercepts exist.
• Y-Intercepts: Simply substitute \( x = 0 \) into the function, which tells you the point at which the graph crosses the \( y \)-axis. Here, \( f(0) = \frac{0^2 + 3(0) + 4}{0 - 5} = -\frac{4}{5} \), placing the \( y \)-intercept at \( (0, -\frac{4}{5}) \).

These intercepts provide valuable starting points for sketching graphs without needing a calculator, relying solely on our understanding of algebra. Remember, knowing intercepts deepens our grasp of graphing rational functions effectively.<<