Problem 26 - 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 \(f\). $$ f(x)=\frac{x^{2}-2 x}{x+2} $$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = -2\), slant asymptote at \(y = x - 4\). Intercepts: \((0,0)\) and \((2,0)\).

Step 1: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function equals zero unless the numerator also equals zero at the same x-value (in which case it might form a hole instead). Set the denominator equal to zero:\[ x + 2 = 0 \]Solve for \(x\):\[ x = -2 \]Check the numerator at \(x = -2\):\[ x^2 - 2x = (-2)^2 - 2(-2) = 4 + 4 = 8 \]Since the numerator is not zero, there is a vertical asymptote at \(x = -2\).

Step 2: Find Slant Asymptotes

To find slant asymptotes, the degree of the numerator must be one more than the degree of the denominator. Here, the numerator's degree (2) is one more than the denominator's degree (1).Perform polynomial long division of \(x^2 - 2x\) by \(x + 2\):- Divide \(x^2\) by \(x\) to get \(x\).- Multiply \(x\) by \(x + 2\) to get \(x^2 + 2x\).- Subtract \((x^2 + 2x)\) from \((x^2 - 2x)\) to get \(-4x\).- Divide \(-4x\) by \(x\) to get \(-4\).- Multiply \(-4\) by \(x + 2\) to get \(-4x - 8\).- Subtract \((-4x - 8)\) from \(-4x\) to get \(8\).The quotient is \(x - 4\), thus the slant asymptote is \(y = x - 4\).

Step 3: Find the x-intercepts

The x-intercepts occur where the numerator is zero. Set the numerator equal to zero:\[ x^2 - 2x = 0 \]Factor the equation:\[ x(x - 2) = 0 \]Solve for \(x\):\[ x = 0 \quad \text{or} \quad x = 2 \]Thus, the x-intercepts are at \((0, 0)\) and \((2, 0)\).

Step 4: Find the y-intercept

The y-intercept occurs where the graph crosses the y-axis (i.e., where \(x = 0\)). Plug \(x = 0\) into the function:\[ f(0) = \frac{0 - 0}{0 + 2} = 0 \]Therefore, the y-intercept is \((0, 0)\).

Step 5: Sketch the Graph

Using the information found, sketch the graph:- Vertical asymptote at \(x = -2\).- Slant asymptote at \(y = x - 4\).- x-intercepts at \((0, 0)\) and \((2, 0)\).- y-intercept at \((0, 0)\).- As \(x\) approaches the vertical asymptote, the function will approach infinity or negative infinity, and as \(x\) goes to infinity or negative infinity, the function approaches the slant asymptote.

Key Concepts

Vertical Asymptotes

Vertical asymptotes in a graph represent values of \( x \) where the function grows infinitely large, as \( x \) approaches these values from the left or right. They often appear in rational functions at places where the denominator equals zero. In our exercise, the function \( f(x) = \frac{x^2 - 2x}{x + 2} \) has the denominator \( x + 2 \). To find the vertical asymptote, we set the denominator to zero:

• \( x + 2 = 0 \)
• Solving for \( x \), we find \( x = -2 \).

The numerator \( x^2 - 2x \) does not equal zero at this \( x \)-value, confirming there's no hole and a true vertical asymptote exists at \( x = -2 \). When we sketch the graph, the curve will move towards infinity or negative infinity as it nears \( x = -2 \). This distinctive feature helps us understand limits within functions involving rational expressions.

Slant Asymptotes

Slant or oblique asymptotes occur in rational functions where the degree of the numerator is exactly one more than the degree of the denominator. In our function \( f(x) = \frac{x^2 - 2x}{x + 2} \), the numerator's degree is 2, and the denominator's is 1, making the presence of a slant asymptote possible.To find this slant asymptote, we perform polynomial long division of the numerator by the denominator:

• Divide \( x^2 \) by \( x \) to get \( x \).
• Multiply \( x \) by \( x + 2 \) to get \( x^2 + 2x \).
• Subtract \( x^2 + 2x \) from \( x^2 - 2x \) to find \( -4x \).
• Divide \( -4x \) by \( x \) to obtain \( -4 \).
• Multiply \( -4 \) by \( x + 2 \) to produce \( -4x - 8 \).
• Subtract \( -4x - 8 \) from \( -4x \), revealing the remainder \( 8 \).

The resulting slant asymptote from the quotient is \( y = x - 4 \). This line indicates the behavior of \( f(x) \) as \( x \) approaches infinity, with the graph getting closer but not touching this line at extreme \( x \)-values.

X-Intercepts

X-intercepts are points on the graph where the function value is zero. This occurs where the numerator of a rational function becomes zero because the function value becomes \( 0 \) divided by some non-zero value, ultimately being zero.In the function \( f(x) = \frac{x^2 - 2x}{x + 2} \), set the numerator \( x^2 - 2x \) to zero:

• \( x^2 - 2x = 0 \)

Factoring gives us:

• \( x(x - 2) = 0 \)
• The solutions are \( x = 0 \) and \( x = 2 \).

These values make two x-intercepts for the graph at the points \((0, 0)\) and \((2, 0)\). When plotting the graph, you'll notice these locations where it crosses the x-axis, providing a clearer picture of the function's behavior.

Y-Intercepts

Y-intercepts are simpler to find since they reveal where the graph touches or crosses the y-axis. This intersection always occurs where \( x = 0 \), so we substitute \( x = 0 \) in our function \( f(x) = \frac{x^2 - 2x}{x + 2} \):

• \( f(0) = \frac{0^2 - 2 \, \times\, 0}{0 + 2} = 0 \)

The y-intercept is consequently at \( (0, 0) \). This same point occurred as an x-intercept, indicating that the graph of the rational function crosses at the origin. Whenever plotting or predicting values from a graph, locating the y-intercept is the simplest way to start.

Polynomial Division

Polynomial division is a process used to divide two polynomials to find an algebraic expression like a quotient and possibly a remainder. In functions with slant asymptotes like \( f(x) = \frac{x^2 - 2x}{x + 2} \), long division becomes essential to obtain the slant asymptote's equation.Here's a brief reminder of the division steps:

• Divide the first term of the numerator by the first term of the denominator.
• Multiply the entire divisor by this result, then subtract from the numerator.
• Repeat using the new polynomial formed from the subtraction until no lower degree term of the divisor can continue to divide.

This results in a quotient that represents the slant asymptote and a remainder that is less significant since it vanishes at larger \( x \)-values. Understanding polynomial division enhances comprehension of the behavior and major components of rational functions.