Question

How do I write an equation for this graph?

Answer 1

The equation for the function is `(2x(x-2))/((x+1)(x-4))`.

Explanation:

This equation will be a rational function, which means that there will be a polynomial divided by another polynomial.

The polynomial in the numerator will have the zeroes of the function, and the polynomial in the denominator will have the (vertical) asymptotes of the function.

The vertical asymptotes of the function are marked at `x=-1` and `x=4`, and the horizontal asymptote is marked at `y=2`.

This means that the denominator will need to have the factors `(x+1)` and `(x-4)` so that when `x` is `-1` or `4`, the function will be undefined:

`?/((x+1)(x-4))`

Next, we identify the zeroes of the function, which are `(0,0)` and `(2,0)`. This means that in the numerator, there will be the factors `x` and `(x-2)` so that when `x` is `0` or `2`, the function will be equal to `0`:

`(x(x-2))/((x+1)(x-4))`

Lastly, we need to figure how to get the horizontal asymptote. Since the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator, that means the horizontal asymptote is calculated by dividing the leading coefficients of each.

Therefore, we have to add a `2` out in front of the numerator so that it will become the leading coefficient, leaving the horizontal asymptote to be `y=2/1`, `y=2`:

`(2x(x-2))/((x+1)(x-4))`

This is the final rational function.