Problem 33 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{-12}{x+6}$$

Step-by-Step Solution

Answer

After identifying the structure of the function \(f(x)=\frac{-12}{x+6}\), the vertical asymptote at \(x=-6\) and the horizontal asymptote at \(y=0\) are plotted. There is one y-intercept at y=-2 but no x-intercept. The graph is a smooth curve that does not cut across the x-axis and approaches the asymptotes as \(x\) moves towards positive or negative infinity.

Step 1: Identify the Structure of the Rational Function

The function \(f(x)=\frac{-12}{x+6}\) is a rational function of the form \(f(x)=\frac{a}{x-h}\) where a and h are constants. This function has a vertical asymptote at \(x=h\) and a horizontal asymptote at \(y=0\). For our function, a=-12 and h=-6.

Step 2: Plot the Asymptotes

In this function, the horizontal asymptote is \(y=0\). This can be drawn as a straight line along the x-axis. The vertical asymptote is \(x=-6\), this can be drawn as a straight line parallel to the y-axis at \(x=-6\). These lines divide the plane into regions and functions behave differently in every region.

Step 3: Find and Plot the x and y-intercepts

The x-intercept is found by setting \(f(x) = 0\) and solving for x. There is no x for which \(f(x) = 0\) because the numerator, -12, can't equal zero. So, there is no x-intercept. The y-intercept is found by setting \(x=0\), giving \(f(0)=\frac{-12}{0+6}=-2\). Plot this point on the graph.

Step 4: Sketch the Graph

Having plotted the asymptotes and intercept, draw a smooth curve that approaches the asymptotes as \(x\) moves towards positive or negative infinity. Since there is no x-intercept, the curve will not cross the x-axis.

Key Concepts

Vertical Asymptote

A vertical asymptote is a line where a function's value approaches infinity as the input approaches a specific point. Think of it as a boundary that your function will get extremely close to, but never actually touch. In the rational function \(f(x) = \frac{-12}{x+6}\), the vertical asymptote occurs at \(x = -6\). This happens because when \(x\) reaches -6, the denominator becomes zero, making the function undefined.

• To find a vertical asymptote in rational functions, set the denominator equal to zero and solve for \(x\).
• Vertical asymptotes are depicted as vertical dashed lines on a graph.

Understanding vertical asymptotes helps in predicting how the function behaves near certain input values, which is crucial for sketching accurate graphs.

Horizontal Asymptote

Horizontal asymptotes illustrate how a function behaves as the value of \(x\) becomes extremely large or extremely small. Unlike vertical asymptotes, which are about specific inputs, horizontal asymptotes deal with end behavior, or what happens "far out" on the graph.For **\(f(x) = \frac{-12}{x+6}\)**:- The horizontal asymptote is \(y = 0\). This is because, as \(x\) grows larger in both negative and positive directions, the fraction's denominator increases, making \(f(x)\) approach zero.

• Horizontal asymptotes tell you about the end behavior of the graph.
• They are usually drawn as horizontal dashed lines parallel to the x-axis.

Recognizing the horizontal asymptote helps in shaping the overall look of the function's graph and knowing its general trend.

Intercepts

Intercepts are the points where the curve intersects either the x-axis or the y-axis. In rational functions, finding intercepts provides pivotal points to aid in sketching the graph.**For our function \(f(x) = \frac{-12}{x+6}\):**- **X-Intercept**: This function does not have an x-intercept. Normally, x-intercepts occur when \(f(x) = 0\). However, since the numerator here is a constant \(-12\), \(f(x)\) cannot be zero.- **Y-Intercept**: Occurs when \(x = 0\). Substituting \(x = 0\) into the function gives \(f(0) = \frac{-12}{0+6} = -2\). Thus, the y-intercept is the point (0, -2).

• X-intercepts occur where a function crosses the x-axis, but may not exist for some rational functions.
• Y-intercepts are essential to plotting the function’s behavior around \(x = 0\).

By locating intercepts, you gain critical reference points that guide the accurate plotting of the function's graph, giving insight into its overall shape.