Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{20+6 x-2 x^{2}}{8+6 x-2 x^{2}}$$

Step-by-Step Solution

Answer

The rational function has vertical asymptotes at \(x=4\) and \(x=-1\) and a horizontal asymptote at \(y=1\).

Step 1: Identify the Type of Function

The function given is a rational function, which is a ratio of two polynomials: \(f(x)=\frac{20+6x-2x^{2}}{8+6x-2x^{2}}\). Here both the numerator and denominator are quadratic polynomials.

Step 2: Simplify the Function

Simplify the function by factoring both the numerator and the denominator. Notice both have a term \(-2x^2\). Factor them:\(f(x) = \frac{-2(x^2 - 3x - 10)}{-2(x^2 - 3x - 4)}\). Cancel the common factor \(-2\) in the numerator and the denominator: \(f(x) = \frac{x^2 - 3x - 10}{x^2 - 3x - 4}\).

Step 3: Factorize the Polynomials

Further factorize the quadratic expressions: \(x^2 - 3x - 10 = (x-5)(x+2)\) and \(x^2 - 3x - 4 = (x-4)(x+1)\). The function simplifies to: \(f(x) = \frac{(x-5)(x+2)}{(x-4)(x+1)}\).

Step 4: Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero and the numerator does not. Solve \(x-4=0\) and \(x+1=0\): the vertical asymptotes are at \(x=4\) and \(x=-1\).

Step 5: Determine Horizontal Asymptotes

Since the degrees of the numerator and the denominator are the same (degree 2), the horizontal asymptote is given by the ratio of the leading coefficients: \(\frac{-2}{-2} = 1\). Thus, the horizontal asymptote is at \(y=1\).

Step 6: Sketch the Graph

Using the asymptotes as a guide, sketch the rational function. Plot the polynomial zeros of numerator \((x-5=0 \Rightarrow x=5\) and \(x+2=0 \Rightarrow x=-2\)) to find the x-intercepts where the graph touches or crosses the x-axis (at \(x=5\) and \(x=-2\)). Consider the behavior of the function around the asymptotes.

Step 7: Analyze Behavior Near Asymptotes

As \(x\) approaches \(4\) and \(-1\), the value of the function tends toward infinity or negative infinity, confirming the vertical asymptotes. The function starts near one region of the horizontal asymptote \(y=1\) at extreme \(x\) values.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are crucial in understanding the behavior of rational functions. They occur at values of \( x \) that make the denominator zero, thereby causing the function to be undefined at those points. To determine the vertical asymptotes for the given rational function \( f(x) = \frac{(x-5)(x+2)}{(x-4)(x+1)} \), we look for solutions to the denominator's equation:

• Set each factor in the denominator equal to zero: \( x-4 = 0 \) and \( x+1 = 0 \).
• Solving these equations, we find vertical asymptotes at \( x = 4 \) and \( x = -1 \).

These points are where the graph of the function will shoot up or drop down towards infinity or negative infinity. Importantly, the numerator should not be zero at these points, confirming that they are indeed vertical asymptotes, not removable discontinuities. In this context, watch how the graph behaves by approaching these values from either side. It typically reveals much about the function's overall characteristics.

Horizontal Asymptotes

Horizontal asymptotes describe how a function behaves as \( x \) approaches infinity or negative infinity. They specifically indicate the line which the graph of the function gets closer to as \( x \) increases or decreases beyond all bounds.For rational functions, horizontal asymptotes depend on the degree of the polynomials in the numerator and denominator:

• If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

In this example, the degree of both the numerator and denominator is 2. So, we look at the ratio of the leading coefficients: \( \frac{-2}{-2} = 1 \).Thus, the horizontal asymptote of the function is \( y = 1 \). The graph will approach this line as \( x \) moves towards positive or negative infinity. It provides a visual boundary that the function will not cross but merely approach very closely, giving insight into the function's end behavior.

Polynomial Factorization

Polynomial factorization is a helpful tool to simplify rational functions and identify key features like asymptotes and intercepts. Factoring involves breaking down a polynomial into a product of simpler terms to make the function easier to analyze.In the given function \( f(x) = \frac{20+6x-2x^{2}}{8+6x-2x^{2}} \), both the numerator and the denominator can be factored:

• Start with factoring out the common term \(-2x^2\): \( f(x) = \frac{-2(x^2 - 3x - 10)}{-2(x^2 - 3x - 4)} \). This simplifies the expression.
• Next, further factor the quadratic equations: \( x^2 - 3x - 10 = (x-5)(x+2) \) and \( x^2 - 3x - 4 = (x-4)(x+1) \).

With factorization, we transform \( f(x) \) into \( \frac{(x-5)(x+2)}{(x-4)(x+1)} \). Factoring aids in identifying the function's roots (\( x-5=0 \to x=5 \) and \( x+2=0 \to x=-2 \)), which are the x-intercepts, and allows us to precisely determine asymptotic behavior. This breakdown simplifies sketching the graph and understanding where it will cross the x-axis or diverge towards infinity.