SOLUTION: Write an equation for a rational function with vertical asymptotes at x=-5 and x=6 x intercepts at x=2 and x=-2 y intercept at 7 y=

Algebra ->  Rational-functions -> SOLUTION: Write an equation for a rational function with vertical asymptotes at x=-5 and x=6 x intercepts at x=2 and x=-2 y intercept at 7 y=      Log On


   

Question 1197127: Write an equation for a rational function with
vertical asymptotes at x=-5 and x=6
x intercepts at x=2 and x=-2
y intercept at 7
y=
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52818) About Me  (Show Source):
You can put this solution on YOUR website!
.
Write an equation for a rational function with
vertical asymptotes at x=-5 and x=6
x intercepts at x=2 and x=-2
y intercept at 7
y=
~~~~~~~~~~~~~~ 

Obviously, this rational function must have the binomial factors (x+5) and (x-6)
in the denominator and binomial factors (x-2) and (x+2) in the numerator.


So, we write  

    y = %28a%2A%28x-2%29%2A%28x%2B2%29%29%2F%28%28x%2B5%29%2A%28x-6%29%29 = %28a%2A%28x%5E2-4%29%29%2F%28%28%28x%2B5%29%2A%28x-6%29%29%29.      (1)


Real coefficient "a" in the formula is intended to provide the necessary value y= 7 at x= 0.


To find "a", we equate function (1) to 7 at x= 0

    7 = %28a%2A%280%5E2-4%29%29%2F%28%280%2B5%29%2A%280-6%29%29 = %28a%2A%28-4%29%29%2F%28-30%29 = %282a%29%2F15.


From this equation, we find 

    a = %287%2A15%29%2F2 = 105%2F2.


So, finally our function is

    y = %28105%2F2%29%2A%28%28x%5E2-4%29%2F%28%28x%2B5%29%2A%28x-6%29%29%29.

Solved.



Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Write an equation for a rational function y=p%28x%29%2Fq%28x%29+ with:

vertical asymptotes at x=-5 and x=6+ => we have in denominator
%28x%2B5%29 and %28x-6%29 as factors

q%28x%29=%28x%2B5%29%28x-6%29=>denominator

so far, your function is:

y=p%28x%29%2F%28%28x%2B5%29%28x-6%29%29+

x intercepts at x=2+and x=-2+ => we have in numerator
%28x-2%29 and %28x%2B2%29 as factors

p%28x%29=%28x-2%29%28x%2B2%29+
y=%28%28x-2%29%28x%2B2%29+%29%2F%28%28x%2B5%29%28x-6%29%29+

y intercept at 7

y=k%28%28%28x-2%29%28x%2B2%29+%29%2F%28%28x%2B5%29%28x-6%29%29%29

=> plug in+x=0 and y+=7 in above function and find k

7=k%28%280-2%29%280%2B2%29%29+%2F%28%280%2B5%29%280-6%29%29+
7=k%282%2F15%29+
7%2A15=2k
k=52.5

y=%2852.5%28x-2%29%28x%2B2%29+%29%2F%28%28x%2B5%29%28x-6%29%29

y=%2852.5%28x%5E2-4+%29%29%2F%28x%5E2+-+x+-+30%29



zoom in to see intercepts