If the graph of the function y-(ax^2+b)/(x^2+cx+4) has vertical asymptotes x=1,x=4, a horizontal asymptote of y=2 and x-intercepts at x=+-2, find a, b and c.
Answers: a=2, b=-8 and c=-5
I found a and b but I'm having trouble finding c.
I made y=0 and x=2
0=(4a+b)/(4+2c+4)
Then I simplified:
0=4a+b
-4a=b
I know that when the a term in the numerator and a term in the denominator both have the same highest exponent on the same variable then you take the coefficients of those two terms and divide them by each other. (You take the coefficient of that term in the numerator and divide it by the coefficient of that term in the denominator)
So the term in the numerator would be ax^2 with the coefficient of 'a' and the term in the denominator would be x^2 with the coefficient of '1'.
In the question, I'm given the information that the horizontal asymptote of this function is y=2, therefore I can conclude that a=2.
Referring back on the equation I previously formulated, -4a=b.
Sub a=2 into the equation:
b=-8
Now I'm having trouble finding c...
Any help would be greatly appreciated :)
If it has vertical asymptotes at x=1,4 then that means that
x^2+cx+4 = (x-1)(x-4)
So, c = -5
To find the value of c, we can use the given information about the graph's vertical asymptotes and x-intercepts.
Since there are vertical asymptotes at x = 1 and x = 4, it means that the denominator of the function, x^2 + cx + 4, must have factors of (x - 1) and (x - 4). This is because if a function has vertical asymptotes at specific x-values, the denominator must have factors of (x - a), where a is the x-value of the asymptote.
Hence, we can set up the following equation:
x^2 + cx + 4 = (x - 1)(x - 4)
Expanding the right side:
x^2 + cx + 4 = x^2 - 5x + 4
Now we can equate the coefficients of matching powers of x on both sides:
cx = -5x (coefficients of x on the left side and right side)
c = -5
Therefore, c = -5.
So the final values are:
a = 2
b = -8
c = -5