Question

the figure below shows the graph of a rational function f. it has vertical asymptotes x = 2 and x = 6, and horizontal asymptote y = 0. the graph has x-intercept -4, and it passes through the point (1, 2). the equation for f(x) has one of the five forms shown below. choose the appropriate form for f(x), and then write the equation. you can assume that f(x) is in simplest form. graph of the rational function is shown, along with five multiple-choice options for the form of f(x) and their corresponding visual representations with boxes and circles.

Explanation:

Step1: Determine the form of the rational function

Since the graph has vertical asymptotes \(x = 2\) and \(x=6\), the denominator of the rational function has factors \((x - 2)\) and \((x - 6)\). The horizontal asymptote \(y = 0\) implies that the degree of the numerator is less than the degree of the denominator. The \(x\)-intercept is \(x=-4\), so the numerator has a factor \((x + 4)\). The general form of a rational function \(y=\frac{a(x - b)}{(x - d)(x - e)}\) (where \(a\) is a non - zero constant, \(b\) is the \(x\) - intercept, and \(d,e\) are the vertical asymptotes).

Step2: Substitute the known values

We know \(b=-4\), \(d = 2\), \(e=6\). So the function is \(y=\frac{a(x+4)}{(x - 2)(x - 6)}\). The function passes through the point \((1,2)\). Substitute \(x = 1\) and \(y=2\) into the equation:

$$2=\frac{a(1 + 4)}{(1 - 2)(1 - 6)}$$

$$2=\frac{5a}{(-1)\times(-5)}$$

$$2=\frac{5a}{5}$$

$$a = 2$$

The function is \(y=\frac{2(x + 4)}{(x - 2)(x - 6)}\), which is in the form \(f(x)=\frac{a(x - b)}{(x - d)(x - e)}\)

Answer:

\(f(x)=\frac{a(x - b)}{(x - d)(x - e)}\)