Problem 502 - Precalculus

Question

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$k(x)=\frac{2 x^{2}-3 x-20}{x-5}$$

Step-by-Step Solution

Answer

Intercepts: \(x = -\frac{5}{2}, 4\); Vertical at \((0, 4)\). Vertical asymptote at \(x = 5\); Slant asymptote: \(y = 2x + 7\).

Step 1: Finding the Horizontal Intercepts

To find the horizontal intercepts, we set the numerator equal to zero and solve for \(x\): \(2x^2 - 3x - 20 = 0\). This quadratic can be factored into \((2x + 5)(x - 4) = 0\), giving us the intercepts \(x = -\frac{5}{2}\) and \(x = 4\).

Step 2: Finding the Vertical Intercept

To find the vertical intercept, we evaluate \(k(x)\) at \(x = 0\): \(k(0) = \frac{2(0)^2 - 3(0) - 20}{0 - 5} = 4\). Thus, the vertical intercept is \((0, 4)\).

Step 3: Finding Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Set the denominator equal to zero: \(x - 5 = 0\). Thus, there is a vertical asymptote at \(x = 5\).

Step 4: Finding Horizontal or Slant Asymptote

Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is a slant (oblique) asymptote. Perform polynomial long division: \((2x^2 - 3x - 20) \div (x - 5)\). The result is \(2x + 7\), so the slant asymptote is \(y = 2x + 7\).

Step 5: Sketching the Graph

Plot the horizontal intercepts \((-\frac{5}{2}, 0)\) and \((4, 0)\), and the vertical intercept \((0, 4)\). Draw the vertical asymptote at \(x = 5\) and the slant asymptote as a dashed line since the graph will approach it but never touch it. Analyze the behavior around the asymptotes and plot points to sketch the graph of \(k(x)\).

Key Concepts

Horizontal Intercepts

Horizontal intercepts, also known as x-intercepts, occur where the graph of a function crosses the x-axis. This means that the value of the function is zero at these points. For a rational function, like the one given, you find the horizontal intercepts by setting the numerator equal to zero and solving for the variable.
In the exercise, we have the function:

• \(k(x) = \frac{2x^2 - 3x - 20}{x - 5}\).

To find the horizontal intercepts, solve the equation:

• \(2x^2 - 3x - 20 = 0\).

This is a quadratic equation, and it can be factored into

• \((2x + 5)(x - 4) = 0\).

Each factor set to zero gives the intercepts:

• \(x = -\frac{5}{2}\) and
• x = 4.

These values indicate the points where the graph crosses the x-axis.

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. They occur in rational functions where the denominator is equal to zero, but the numerator is not zero. For the given function:

• \(k(x) = \frac{2x^2 - 3x - 20}{x - 5}\),

we find the vertical asymptotes by setting the denominator equal to zero:

• \(x - 5 = 0\).

Solving this equation, we find:

• \(x = 5\).

This result tells us that at \(x = 5\), the function has a vertical asymptote. The graph will approach this line as \(x\) becomes very close to 5, either from the left or the right, but it will not touch or cross the line at this point.

Slant Asymptote

A slant asymptote, also known as an oblique asymptote, is a line that the graph of a rational function approaches as the values of \(x\) move toward positive or negative infinity. Slant asymptotes occur in situations where the degree of the numerator is exactly one more than the degree of the denominator. In the function:

• \(k(x) = \frac{2x^2 - 3x - 20}{x - 5}\),

the degree of the numerator (2) is indeed greater than that of the denominator (1) by one.
To find the slant asymptote, perform polynomial long division of:

• \(2x^2 - 3x - 20\) by
• \(x - 5\).

The result of this division provides the equation of the slant asymptote:

• \(y = 2x + 7\).

As \(x\) approaches infinity, the function \(k(x)\) will get closer and closer to the line \(y = 2x + 7\), giving us a visual guide on how to graph the function.

Quadratic Equation

A quadratic equation is a polynomial equation of the form

• \(ax^2 + bx + c = 0\),

where \(a, b,\) and \(c\) are constants. Solving quadratic equations is fundamental in mathematics, and there are several methods to find their roots:

• Factoring,
• The quadratic formula,
• Completing the square.

In the context of the given problem, we solve a quadratic equation to find the horizontal intercepts of the function \(k(x)\):

• \(2x^2 - 3x - 20 = 0\).

The factoring method is used here, resulting in the factors:

• \((2x + 5)(x - 4) = 0\).

Each solution for \(x\), also known as a root, corresponds to a horizontal intercept of the graph. Quadratic equations themselves can also help predict the nature of roots, such as whether they are real or complex, by analyzing the discriminant.