A rational function that passes through (0,0) and (4, 8/7) has the x-axis as a horiztonal asympotote. It also has 2 vertical asymptotes x=3, x=-3.
I understand that to set it up, it is
y= ax+b / (x+3)(x-3).
I don't understand how to solve for a and b.
Put in the points. You have two points, and two unknowns.
(0,0)
0= 0+b / (0+3)(0-3).
Solving this easy: b is zero.
Now that you have b, next point
y= ax / (x+3)(x-3). ( 8/7, 8/7)
8/7= a 4
solve for a.
thank you so much for your help
To solve for the values of a and b in the rational function, we can use the two given points: (0,0) and (4, 8/7).
For the point (0,0), we substitute x = 0 and y = 0 into the equation y = (ax + b) / ((x + 3)(x - 3)), resulting in:
0 = (0a + b) / ((0 + 3)(0 - 3))
Simplifying this equation, we have:
0 = b / (3)(-3)
Since we know that the numerator, b, is equal to zero, we have:
0 = 0
This statement is true, so the point (0,0) satisfies the equation.
Now that we have found that b = 0, we can rewrite the equation as:
y = ax / ((x + 3)(x - 3))
Using the second given point, (4, 8/7), we substitute x = 4 and y = 8/7 into the equation:
8/7 = a(4) / ((4 + 3)(4 - 3))
Simplifying further, we have:
8/7 = 4a / 7
To solve for a, we multiply both sides of the equation by 7 and divide by 4:
(a) = (8/7) * (7/4)
Simplifying this expression, we get:
a = 2/1 = 2
Therefore, the values of a and b in the rational function are a = 2 and b = 0. Substituting these values back into the original equation, we have the final form of the rational function:
y = (2x) / ((x + 3)(x - 3))
I hope this helps! Let me know if you have any further questions.