Problem 59 - Algebra and Trigonometry

Question

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$ r(x)=\frac{x^{2}-x-6}{x^{2}+3 x} $$

Step-by-Step Solution

Answer

The intercepts are \(x = -2, 3\). Asymptotes: \(x = -3, 0\) (vertical), \(y = 1\) (horizontal). Domain: all \(x \neq -3, 0\). Range: all \(y \neq 1\).

Step 1: Find x-intercepts

To find the x-intercepts, set the numerator equal to zero and solve for \(x\):\[x^2 - x - 6 = 0.\]This factors to \((x - 3)(x + 2) = 0\), so the x-intercepts are \(x = 3\) and \(x = -2\).

Step 2: Find y-intercept

To find the y-intercept, evaluate \(r(x)\) at \(x = 0\):\[r(0) = \frac{0^2 - 0 - 6}{0^2 + 3 \, 0} = \frac{-6}{0}.\]Since the denominator is \(0\), there is no y-intercept.

Step 3: Identify vertical asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not zero at those points.Solve \(x^2 + 3x = 0\):\[x(x + 3) = 0\]Thus, \(x = 0\) and \(x = -3\) are potential places.Since the numerator is not zero at these points, these are indeed vertical asymptotes.

Step 4: Identify horizontal asymptotes

For horizontal asymptotes, compare the degrees of the polynomial in the numerator and denominator.Both are degree 2, so the asymptote is determined by the leading coefficient ratio:\[y = \frac{1}{1} = 1.\]So, the horizontal asymptote is \(y = 1\).

Step 5: Define the domain

The domain excludes values where the denominator is zero.From the factorization, these values are \(x = -3\) and \(x = 0\).Thus, the domain is all real numbers except \(x = -3\) and \(x = 0\).

Step 6: Determine the range

By observation of the horizontal asymptote and behavior near vertical asymptotes, the range is all real numbers except \(y = 1\).

Step 7: Sketch the graph and confirm

Sketch the graph by marking the intercepts and asymptotes found earlier, and draw rough curves. Use a graphing device to check the sketch. The intercepts imply the function crosses the x-axis at \(x = -2\) and \(x = 3\). Vertical asymptotes exist at \(x = -3\) and \(x = 0\), and the horizontal asymptote is \(y = 1\).

Key Concepts

x-intercepts

To find the x-intercepts of a rational function, we need to examine the numerator. Specifically, the x-intercepts occur where the numerator equals zero, because at these points, the whole expression becomes zero (provided the denominator isn't zero at the same point). For the function given, \(r(x) = \frac{x^2 - x - 6}{x^2 + 3x}\), this means setting the numerator \(x^2 - x - 6\) equal to zero:

• Solve \(x^2 - x - 6 = 0\) by factoring.
• This quadratic expression factors into \((x - 3)(x + 2) = 0\).
• This gives us the solutions \(x = 3\) and \(x = -2\).

Thus, the x-intercepts of the function are at these points: \(x = 3\) and \(x = -2\). Remember, these are the points where the graph intersects the x-axis.

asymptotes

Asymptotes of a rational function are lines that the graph of the function approaches but never actually touches. They come in two types: vertical and horizontal (or occasionally oblique).For vertical asymptotes:

• Solve \(x^2 + 3x = 0\) to find points where the denominator is zero and the numerator is not zero at these points: \(x(x + 3) = 0\).
• This gives us vertical asymptotes at \(x = 0\) and \(x = -3\).

For a horizontal asymptote:

• Compare the degrees of the polynomials in the numerator and the denominator. Both are degree 2, so the horizontal asymptote is determined by the leading coefficient ratio, which is \(\frac{1}{1} = 1\).
• Hence, the horizontal asymptote is \(y = 1\).

These asymptotes describe crucial boundaries for the graph of the function, indicating how it behaves at the extremes of the x-axis.

domain and range

The domain and range of a rational function describe where the function "lives" in terms of inputs and outputs. To find the domain, look for values of x where the function is undefined. For function \(r(x)\), these occur wherever the denominator is zero, specifically at \(x = 0\) and \(x = -3\). Thus, the domain can be expressed except for these points:

• Domain: all real numbers \(x\), except \(x = -3\) and \(x = 0\).

The range of the function is about the possible outputs (y-values) it can take. Observing that there is a horizontal asymptote at \(y = 1\), the function never actually equals this, suggesting that:

• Range: all real numbers \(y\), except \( y = 1\).

Understanding the domain and range helps to paint a complete picture of how the function behaves across its entire graph.

graphing rational functions

Graphing rational functions involves putting together all the pieces we've figured out: x-intercepts, vertical asymptotes, horizontal asymptotes, and domain/range information.Here’s how you sketch it:

• Start by plotting the x-intercepts: \(x = 3\) and \(x = -2\).
• Draw dashed lines to represent the vertical asymptotes at \(x = 0\) and \(x = -3\); the graph will approach but never cross these lines.
• Draw a dashed line for the horizontal asymptote \(y = 1\) as the limit the graph approaches as \(x\) goes to infinity or negative infinity.
• Fill in the sketch by drawing curves approaching the asymptotes from the intercepts, making sure the curves reflect the behavior of the function across its domain.

Using a graphing calculator or software can confirm your hand-drawn sketch, ensuring all features (like intercepts and asymptotes) are accurately represented on the graph.