Question

Find an equation of a rational function that satisfies the given conditions:
Vertical asymptotes: 𝑥 = −5, 𝑥 = 2
Horizontal asymptotes: 𝑦 = −2
𝑥-intercepts: 𝑥 = −6, 𝑥 = 4
𝑓(−2) = −4

Answer 1

𝑥-intercepts: 𝑥 = −6, 𝑥 = 4
----> we will need f(x) = a(x+6)(x-4) as a start
Vertical asymptotes: 𝑥 = −5, 𝑥 = 2
----> adjust to f(x) = a(x+6)(x-4)/(b(x+5)(x-2))
Horizontal asymptotes: 𝑦 = −2
----> a/b = -2 , or a = -2b

so far we have:
f(x) = -2b(x+6)(x-4)/(b(x+5)(x-2) = -2(x+6)(x-4)/((x+5)(x-2))
we also need
f(-2) = -4
f(-2) = -2(4)(-6) / (3)(-4) = 4 , well that was lucky

f(x) = -2(x+6)(x-4)/((x+5)(x-2))

verify by entering my equation into
www.desmos.com/calculator

Answer 2

extra credit: how would you satisfy the additional condition that the y-intercept is at (0,10) ?