Question - Equation of Rational Function with Given Conditions
Solution:
To write an equation for a rational function that satisfies the given conditions, we can follow these steps:1. Vertical asymptotes at $$ x = 5 $$ and $$ x = -6 $$ are given by factors in the denominator that become zero at these x-values. Thus, the denominator will be $$ (x - 5)(x + 6) $$.2. X-intercepts at $$ x = -5 $$ and $$ x = 1 $$ indicate where the numerator of the function becomes zero. Thus, the numerator will be a product of factors that become zero at these x-values, which gives us $$ (x + 5)(x - 1) $$.3. The y-intercept is the point where the function crosses the y-axis, which happens when $$ x = 0 $$. To get a y-intercept at 6, we need the constant term of the numerator (when $$ x = 0 $$) to give us a y-value of 6. Since the denominator at $$ x = 0 $$ will be $$ (0 - 5)(0 + 6) = -30 $$, to obtain a y-intercept of 6, the numerator must evaluate to $$ -30 * 6 $$ when $$ x = 0 $$. Therefore, we can multiply the numerator by $$ -6 $$ to get the desired y-intercept.Putting it all together, the rational function is:$$ y = \frac{-6(x + 5)(x - 1)}{(x - 5)(x + 6)} $$This function satisfies the given conditions:- The vertical asymptotes are at $$ x = 5 $$ and $$ x = -6 $$.- The x-intercepts are at $$ x = -5 $$ and $$ x = 1 $$.- The y-intercept is at $$ y = 6 $$ because when $$ x = 0 $$, $$ y = \frac{-6(0 + 5)(0 - 1)}{(0 - 5)(0 + 6)} = \frac{30}{-30} = -1 \cdot -6 = 6 $$.
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