Problem 52 - 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. \(r(x)=\frac{x^{2}+3 x}{x^{2}-x-6}\)

Step-by-Step Solution

Answer

X-intercepts: \(x = 0, -3\); Vertical asymptotes: \(x = 3, -2\); Horizontal asymptote: \(y = 1\).

Step 1: Find the X-Intercepts

To find the x-intercepts of the function where the graph crosses the x-axis, set \(y = 0\), which means setting the numerator equal to zero: \(x^{2} + 3x = 0\). Factor the equation as \(x(x + 3) = 0\). This gives the x-intercepts at \(x = 0\) and \(x = -3\).

Step 2: Find the Y-Intercept

Find the y-intercept by setting \(x = 0\) in the function: \(r(0) = \frac{0^{2} + 3 imes 0}{0^{2} - 0 - 6} = \frac{0}{-6} = 0\). The y-intercept is at \((0, 0)\).

Step 3: Identify the Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Solve \(x^{2} - x - 6 = 0\) to find these points. Factor to \((x - 3)(x + 2) = 0\) which gives \(x = 3\) and \(x = -2\) as vertical asymptotes.

Step 4: Identify the Horizontal Asymptotes

The horizontal asymptote is determined by the degrees of the numerator and denominator. Since both the numerator and denominator are degree 2, compare coefficients of the highest degrees. The horizontal asymptote is \(y = \frac{1}{1} = 1\).

Step 5: Sketch the Graph

Using the intercepts and asymptotes, sketch the graph. The graph passes through \(x = 0\) and \(x = -3\) and approaches the vertical asymptotes \(x = 3\) and \(x = -2\), but never crosses them. It levels off at the horizontal asymptote \(y = 1\) as \(x\) goes to ±∞.

Key Concepts

X-Intercepts

When dealing with rational functions, finding x-intercepts is crucial. An x-intercept is where the graph of the function crosses the x-axis. This happens when the output value, or y, is zero. To find the x-intercepts, set the numerator of the rational function equal to zero since a fraction is zero when its numerator is zero (and the denominator is not). For the function \(r(x)=\frac{x^{2}+3x}{x^{2}-x-6}\), the numerator is \(x^2 + 3x\). Set this equal to zero:

• \(x^2 + 3x = 0\)
• Factor the equation: \(x(x + 3) = 0\)
• This results in \(x = 0\) and \(x = -3\)

Thus, the x-intercepts are \(x = 0\) and \(x = -3\). Each x-intercept represents a point on the graph where the rational function equals zero.

Y-Intercepts

The y-intercept is the point where the graph of the function crosses the y-axis. For rational functions, this occurs when \(x = 0\).To find the y-intercept, substitute \(x = 0\) into the function \(r(x)=\frac{x^{2}+3x}{x^{2}-x-6}\):

• \(r(0) = \frac{0^{2} + 3 \times 0}{0^{2} - 0 - 6} = \frac{0}{-6} = 0\)

This calculation shows that the y-intercept of the function is at the point \((0, 0)\). This point means that the graph will touch the origin.

Vertical Asymptotes

Vertical asymptotes are important features of the graph of a rational function. They occur at values of \(x\) where the function becomes undefined, typically when the denominator equals zero, and the numerator does not simultaneously become zero.For \(r(x)=\frac{x^{2}+3x}{x^{2}-x-6}\), determine where the denominator \(x^2 - x - 6\) is zero:

• Factor the denominator to \((x - 3)(x + 2) = 0\)
• Solving gives \(x = 3\) and \(x = -2\)

These values, \(x = 3\) and \(x = -2\), are locations of vertical asymptotes. As \(x\) approaches these values, the function values grow infinitely large, causing the graph to approach these vertical lines without crossing them.

Horizontal Asymptotes

Horizontal asymptotes reveal the behavior of a rational function as \(x\) approaches infinity. This is determined by comparing the degrees of the numerator and denominator polynomials.For the function \(r(x)=\frac{x^{2}+3x}{x^{2}-x-6}\):

• Both the numerator and denominator are of degree 2
• When degrees are equal, the horizontal asymptote is found by dividing the leading coefficients
• Since both leading coefficients are 1, the horizontal asymptote is \(y = \frac{1}{1} = 1\)

This horizontal line \(y = 1\) indicates that as \(x\) grows very large, either positively or negatively, the function approaches this constant value. This behavior helps sketch the tail end of the graph and understand how it behaves at extreme values.