Write an equation for a rational function

Question

Write an equation for a rational function with:

Vertical asymptotes at x = 5 and x = 6

x intercepts at x = -1 and x = 3

y intercept at 4

y =

Answer 1

Vertical asymptotes mean the denominator will equal zero at x = 5 and at x = 6 so it must be "(x - 5)(x - 6)". X-intercepts occur when y = 0 so, if written in the form y = A/B then the numerator A would be zero when x = -1 and x = 3 or it must be "(x + 1)(x - 3)".

So far, we have:

but we still need a y-intercept at 4. The y-intercept occurs when x = 0, Plugging in x = 0 into what we have yields y = -0.1 so we need to make an adjustment. If we add a multiplier to the numerator, we can maintain the other features (asymptotes and x-intercepts) but change the y-intercept. The multiplier would need to be -40. So:

Answer 2

Vertical asymptotes occur at values of x that make the denominator equal 0. Since we need vertical asymptotes at x = 5 and x = 6, this means we have to have factors of (x - 5) and (x - 6) in the denominator.

x intercepts are going to occur when our numerator is equal to 0. Thus, we need factors of (x + 1) and (x - 3) to guarantee we have zeros(x - intercepts) at x = -1 and x = 3.

Thus our rational function definitely must look like:

y = (x + 1)(x - 3)

(x - 5)(x - 6)

The only other stipulation we have is that we have a y intercept at y = 4. So when we let x = 0 in our function we need an output of 4. When we plug in 0 for x right now we get:

y = (0 + 1)(0 - 3) = -3 = -1

(0 - 5)(0 - 6) 30 10

Clearly this is not 4, so we need to multiply our function by some value to turn our -1/10 into a 4. So essentially we are solving the mini equation: -1/10A = 4. Multiplying both sides of the equation by -10 we see that A = -40.

So our function needs to be multiplied by -40 and in the end should look like:

y = -40(x + 1)(x - 3) we can distribute the top and bottom to get:

(x - 5)(x - 6)

y = -40x²+ 80x + 120

x² - 11x + 30

Answer 3

(x+1)(x-3) + 4

has x intercepts -1 and 3, and y intercept 4

for vertical asymptotes at x = 4 and 6, the denominator has to be zero at those x values

f(x) = (x+1)(x-3)/(x-5)(x-6) =y intercept when x=0

f(0) = 1(-3)/-5(-6) = -3/30=-1/10 -1/10 times -40 = -40/-10 = 4

f(x) = -40(x+1)(x-3)/(x-5)(x-6)

to be a rational function, limit x to rational values, not all real numbers