Questions: The figure below shows the graph of a rational function f. It has vertical asymptotes x=2 and x=-4, and horizontal asymptote y=0. The graph does not have an x-intercept, and it passes through the point (-3,1). 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. f(x)=a/(x-b) f(x)=a(x-b)/(x-c) f(x)=a/((x-b)(x-c)) f(x)=a(x-b)/((x-c)(x-d)) f(x)=a(x-b)(x-c)/((x-d)(x-e))

The figure below shows the graph of a rational function f. It has vertical asymptotes x=2 and x=-4, and horizontal asymptote y=0. The graph does not have an x-intercept, and it passes through the point (-3,1). 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. f(x)=a/(x-b) f(x)=a(x-b)/(x-c) f(x)=a/((x-b)(x-c)) f(x)=a(x-b)/((x-c)(x-d)) f(x)=a(x-b)(x-c)/((x-d)(x-e))
Transcript text: The figure below shows the graph of a rational function $f$. It has vertical asymptotes $x=2$ and $x=-4$, and horizontal asymptote $y=0$. The graph does not have an $x$-intercept, and it passes through the point $(-3,1)$. 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. $f(x)=\frac{a}{x-b}$ $\square$ $f(x)=\frac{a(x-b)}{x-c}$ $\square$ $f(x)=\frac{a}{(x-b)(x-c)}=\frac{\square}{\square \square}$ $f(x)=\frac{a(x-b)}{(x-c)(x-d)}=\frac{\square(\square)}{\square(\square)}$ \[ f(x)=\frac{a(x-b)(x-c)}{(x-d)(x-e)}=\frac{\square \prod D}{\prod D} \]
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Solution

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Solution Steps

Step 1: Determine the general form of the equation

The graph has two vertical asymptotes, indicating two factors in the denominator. Since the horizontal asymptote is $y=0$, the degree of the numerator must be less than the degree of the denominator. The graph does not have an $x$-intercept, meaning there are no factors in the numerator involving $x$. Thus, the appropriate form is $f(x) = \frac{a}{(x-b)(x-c)}$.

Step 2: Determine the values of b and c

The vertical asymptotes are at $x=2$ and $x=-4$. This tells us the denominator is zero at these values, giving us the factors $(x-2)$ and $(x+4)$. So, $b=2$ and $c=-4$.

Step 3: Determine the value of a

We know the graph passes through $(-3, 1)$. Substitute this point and the values of $b$ and $c$ into the equation: $1 = \frac{a}{(-3-2)(-3+4)}$ $1 = \frac{a}{(-5)(1)}$ $1 = \frac{a}{-5}$ $a = -5$

Final Answer

$f(x) = \frac{-5}{(x-2)(x+4)}$

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