Problem 9 - Precalculus: Functions and Graphs

Question

Sketch the graph of \(f\) $$f(x)=\frac{-3 x}{x+2}$$

Step-by-Step Solution

Answer

The graph is a hyperbola with a vertical asymptote at \(x = -2\), a horizontal asymptote at \(y = -3\), and passes through the origin (0,0).

Step 1: Identify Vertical Asymptote

Identify where the denominator equals zero by setting \(x + 2 = 0\). Solving this equation, we find \(x = -2\). This is a vertical asymptote of the function.

Step 2: Identify Horizontal Asymptote

To find the horizontal asymptote, compare the degrees of the polynomial in the numerator and the denominator. Both the numerator and the denominator are of degree 1, so the horizontal asymptote is given by the ratio of their leading coefficients, which is \(-3/1 = -3\). Thus, the horizontal asymptote is \(y = -3\).

Step 3: Determine the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the function. This yields \(f(0) = \frac{0}{2} = 0\). Thus, the \(y\)-intercept is at the point (0, 0).

Step 4: Find the x-intercept

To find the \(x\)-intercept, set the numerator equal to zero, \(-3x = 0\), which gives \(x = 0\). This means the \(x\)-intercept is also at (0, 0).

Step 5: Plot Asymptotes and Intercepts

On the graph, draw a vertical line at \(x = -2\) for the vertical asymptote, and a horizontal line at \(y = -3\) for the horizontal asymptote. Plot the intercept at (0, 0), which is shared by both the \(x\)- and \(y\)-intercepts.

Step 6: Analyze and Sketch the Graph

As \(x\) approaches negative infinity, \(f(x)\) approaches the horizontal asymptote from below at \(y = -3\). As \(x\) approaches negative 2 from the left, \(f(x)\) goes to positive infinity. As \(x\) approaches negative 2 from the right, \(f(x)\) goes to negative infinity. Finally, as \(x\) approaches positive infinity, \(f(x)\) approaches the horizontal asymptote from above. Sketch the curve respecting these observations.

Key Concepts

Vertical Asymptotes in Rational Functions

Vertical asymptotes are lines that the graph of a rational function approaches but never actually touches or crosses. These occur when the denominator of the rational function equals zero, causing the function to be undefined at that point. For the function \( f(x) = \frac{-3x}{x+2} \), we can find the vertical asymptote by setting the denominator equal to zero:

• Set \(x + 2 = 0\) which leads to \(x = -2\).

This means that \(x = -2\) is a vertical asymptote of the function. On a graph, this is represented by a dashed line parallel to the y-axis at \(x = -2\). As the values of \(x\) approach \(-2\), the values of \(f(x)\) will tend to increasingly large positive or negative values.

Horizontal Asymptotes of Rational Functions

Horizontal asymptotes give us information about the end behavior of a rational function’s graph. They tell us what value \(f(x)\) approaches as \(x\) either approaches positive infinity or negative infinity. For the function \( f(x) = \frac{-3x}{x+2} \), both the numerator and the denominator have the same degree, which is 1.

• This means we can find the horizontal asymptote by dividing the leading coefficients. Here, this would be \(-3/1 = -3\).

Thus, the horizontal asymptote is \(y = -3\). On a graph, this represents a horizontal line at \(y = -3\). As \(x\) approaches either direction of infinity, the value of \(f(x)\) nears \(-3\).

Intercepts: Finding Where the Graph Crosses the Axes

Intercepts tell us where the function crosses the x-axis and y-axis. They are helpful points to determine overall graph behavior.

• To find the y-intercept, set \(x=0\) in the function: \(f(0) = \frac{0}{2} = 0\). This shows that the y-intercept is at the point (0,0).
• For the x-intercept, set the numerator equal to zero: \(-3x = 0\), which results in \(x = 0\). This confirms the x-intercept is also at (0,0).

Thus, for \(f(x) = \frac{-3x}{x+2} \), both the x-intercept and y-intercept are at the origin (0,0). This point is shared by both intercepts, marking where the graph crosses both axes.

Graph Sketching of Rational Functions

Sketching the graph of a rational function involves combining the features we've identified, such as the intercepts and asymptotes.

• First, draw a vertical dashed line at \(x = -2\) for the vertical asymptote, indicating this boundary.
• Then, draw a horizontal dashed line at \(y = -3\) for the horizontal asymptote to indicate the end behavior of the graph.
• Plot the intercept at (0,0), where the graph crosses both the x-axis and y-axis.

Understanding the behavior near these asymptotes completes the picture. As \(x\) approaches \(-2\) from the left, the function approaches positive infinity, and from the right, it approaches negative infinity. As \(x\) tends towards positive or negative infinity, \(f(x)\) approaches the horizontal asymptote at \(y = -3\). Using these observations allows us to sketch a curve that accurately represents the rational function's behavior.