Question - Finding a Rational Function Given Vertical, X-intercepts, and Y-intercept
Solution:
To write an equation for a rational function that meets the given requirements, you need to consider the vertical asymptotes, x-intercepts, and y-intercept provided.Vertical asymptotes occur where the denominator of the rational function is zero. Thus, for vertical asymptotes at $$ x = -6 $$ and $$ x = 3 $$, the factors in the denominator will be $$ (x + 6) $$ and $$ (x - 3) $$, since setting either of these equal to zero gives the respective values of $$ x $$ for the asymptotes.X-intercepts occur where the numerator is zero. Given the x-intercepts at $$ x = -3 $$ and $$ x = -1 $$, the factors in the numerator will be $$ (x + 3) $$ and $$ (x + 1) $$, since setting these factors equal to zero will yield the x-intercepts.Combining these, we get a function of the form:\[ f(x) = \frac{a(x + 3)(x + 1)}{(x + 6)(x - 3)} \]where $$ a $$ is a constant we need to determine using the y-intercept.The y-intercept is found by setting $$ x = 0 $$ in the function. It is given that the y-intercept is 10; thus, we solve for $$ a $$ by substituting $$ x = 0 $$ into the equation to get $$ f(0) = 10 $$:\[ 10 = \frac{a(0 + 3)(0 + 1)}{(0 + 6)(0 - 3)} \]\[ 10 = \frac{3a}{-18} \]Now solve for $$ a $$:\[ a = \frac{10 \times (-18)}{3} = -60 \]Insert $$ a $$ back into the original equation to get the complete function:\[ f(x) = \frac{-60(x + 3)(x + 1)}{(x + 6)(x - 3)} \]
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