Questions: Graph the rational function: f(x)=(4x-6)/(x+2) Start by drawing the vertical and horizontal asymptotes. Then plot two points on each piece of the graph. Finally, click on the graph to plot the points. This question includes a graph.

Graph the rational function:

f(x)=(4x-6)/(x+2)

Start by drawing the vertical and horizontal asymptotes. Then plot two points on each piece of the graph. Finally, click on the graph to plot the points.

This question includes a graph.

Transcript text: Graph the rational function: $f(x)=\frac{4x-6}{x+2}$ Start by drawing the vertical and horizontal asymptotes. Then plot two points on each piece of the graph. Finally, click on the graph to plot the points. Tools Help This question includes a graph. Show Me

Solution

Solution Steps

Step 1: Find the vertical asymptote.

The vertical asymptote occurs when the denominator is equal to zero. In this case, the denominator is x + 2. Setting this equal to zero and solving for x gives:

x + 2 = 0 x = -2

So the vertical asymptote is x = -2.

Step 2: Find the horizontal asymptote.

The horizontal asymptote is determined by the degrees of the numerator and denominator. Since both the numerator (4x - 6) and the denominator (x + 2) have a degree of 1, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = 4/1 = 4.

Step 3: Plot two points on each piece of the graph.

We need two points to the left and two points to the right of the vertical asymptote.

Left of x = -2:

Let's choose x = -3: f(-3) = (4(-3) - 6) / (-3 + 2) = (-18) / (-1) = 18. So, the point is (-3, 18). Let's choose x = -4: f(-4) = (4(-4) - 6) / (-4 + 2) = (-22) / (-2) = 11. So, the point is (-4, 11).

Right of x = -2:

Let's choose x = -1: f(-1) = (4(-1) - 6) / (-1 + 2) = (-10) / (1) = -10. So, the point is (-1, -10). Let's choose x = 0: f(0) = (4(0) - 6) / (0 + 2) = (-6) / (2) = -3. So, the point is (0, -3).