Question

Write the equation of a possible rational

Answer 1

Answer:

The graph has a removable discontinuity at x=-2.5 and asymptoe at x=2, and passes through (6,-3)

Step-by-step explanation:

A rational equation is a equation where

$\frac{p(x)}{q(x)}$

where both are polynomials and q(x) can't equal zero.

1. Discovering asymptotes. We need a asymptote at x=2 so we need a binomial factor of

$(x - 2)$

in our denomiator.

So right now we have

$\frac{p(x)}{(x - 2)}$

2. Removable discontinues. This occurs when we have have the same binomial factor in both the numerator and denomiator.

We can model -2.5 as

$(2x + 5)$

So we have as of right now.

$\frac{(2x + 5)}{(x - 2)(2x + 5)}$

Now let see if this passes throught point (6,-3).

$\frac{(2x + 5)}{(x - 2)(2x + 5)} = y$

$\frac{(17)}{68} = \frac{1}{4}$

So this doesn't pass through -3 so we need another term in the numerator that will make 6,-3 apart of this graph.

If we have a variable r, in the numerator that will make this applicable, we would get

$\frac{(2x + 5)r}{(2x + 5)(x - 2)} = - 3$

Plug in 6 for the x values.

$\frac{17r}{4(17)} = - 3$

$\frac{r}{4} = - 3$

$r = - 12$

So our rational equation will be

$\frac{ - 12(2x + 5)}{(2x + 5)(x - 2)}$

or

$\frac{ - 24x - 60}{2 {x}^{2} + x - 10}$

We can prove this by graphing