Problem 43 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$h(x)=\frac{-2 x}{(x-1)(x+4)}$$

Step-by-Step Solution

Answer

The rational function \(h(x)= -2x / (x-1)(x+4)\) has its vertical asymptotes at x = 1 and x = -4. The horizontal asymptote is at y = 0. Both the x and y intercept is at 0.

Step 1: Identify Points on the Graph

These are points that are good to show on the final plotted graph. For this function, let's get the y-intercept and the x-intercept. Y-intercept is calculated by just directly substituting \(x = 0\) into the function. So here, \(y = h(0)\) gives the y-intercept = 0. X-intercept is found by setting the y, or output of function rates to 0 and solving for \(x\). So, \(0 = -2x/(x-1)(x+4)\). Looking at the equation, if just the top (-2x) is equal to zero then the whole fraction becomes zero. So the x-intercept just comes out as \(x = 0\).

Step 2: Find the Vertical Asymptotes

Vertical asymptote(s) is/are the value(s) of \(x\) that cause the function to approach ± ∞. They are found by setting the denominator equal to zero and solving for \(x\), so for the equation (x – 1)(x + 4) = 0, x = 1 and x = -4 are the vertical asymptotes.

Step 3: Identify the Horizontal Asymptote

For rational functions, a horizontal asymptote depends on the degrees of the polynomials. If the degree of the denominator > degree of the numerator, as in this case, then y = 0 is the horizontal asymptote.

Step 4: Sketch the Graph

With these values plot points at (0,y) & (x,0) and x values at 1 and -4 (the vertical asymptotes). Also draw a line at y = 0 (Horizontal Asymptote). According to the behaviour around the asymptotes the graph is drawn.

Key Concepts

Vertical Asymptotes

In graphing rational functions, vertical asymptotes are key features that show where the function value tends towards infinity, either positively or negatively. In our function \( h(x) = \frac{-2x}{(x-1)(x+4)} \), vertical asymptotes occur where the denominator equals zero.

To find these, we solve the equation \( (x-1)(x+4) = 0 \). This means that:

• \( x = 1 \)
• \( x = -4 \)

These are the points where the function is undefined because division by zero is impossible. As \( x \) approaches these values from either side, \( h(x) \) will shoot off to \( +\infty \) or \( -\infty \). Vertical asymptotes are represented by vertical dashed lines on a graph.

Horizontal Asymptotes

Horizontal asymptotes help us understand the behavior of rational functions as \( x \) approaches infinity or negative infinity. For the function \( h(x) = \frac{-2x}{(x-1)(x+4)} \), we compare the degrees of the numerator and the denominator:

• Numerator: degree 1 (\(-2x\))
• Denominator: degree 2 (\((x-1)(x+4)\))

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \). This means that as \( x \) moves far to the positive or negative direction along the x-axis, the function's value approaches zero. This is represented as a horizontal dashed line on a graph along \( y=0 \).

X-Intercepts

X-intercepts of a rational function reflect where the graph crosses the x-axis. At these points, the output value of the function is zero. To find the x-intercepts for \( h(x) = \frac{-2x}{(x-1)(x+4)} \), we set the entire function equal to zero, \[0 = \frac{-2x}{(x-1)(x+4)} \]The function is zero when the numerator is zero. Solving \(-2x=0\) yields \(x=0\). This means the graph will intersect the x-axis at the origin \((0, 0)\). Keep in mind that the x-intercepts are invaluable in understanding where the curve of the function will pass through on the x-axis.

Y-Intercepts

The y-intercepts are equally important because they show where the function intersects the y-axis. To find the y-intercept of any rational function, substitute \( x = 0 \) into the function. In our function \( h(x)=\frac{-2x}{(x-1)(x+4)} \), \[h(0)=\frac{-2(0)}{(0-1)(0+4)}=0\]This results in a y-intercept at the point \( (0, 0) \). As a note, if your y-intercept does not equal zero, make sure you check your calculations again to ensure accuracy.