Write an equation for a rational function with

Question

Vertical asymptotes at x=3 and x=-1

x intercepts at x=-4 and x=-6

y intercept at 9

can someone help me with this question and how you get an answer from this? I am just not understanding.

Answer 1

Since there are vertical asymptotes at x = 3 and x = -1, x-3 and x+1 are factors of the denominator but not of the numerator.

Since the x-intercepts are x = -4 and x = -6, x+4 and x+6 are factors of the numerator but not of the denominator.

Form of the function: f(x) = a(x+4)(x+6) / [(x-3)(x+1)]

Since the y-intercept is 9, f(0) = 9.

So, a(4)(6) / [(-3)(1)] = -8a = 9

a = -9/8

Therefore, a function with the desired properties is f(x) = -9(x+4)(x+6) / [8(x-3)(x+1)]

Answer 2

To make an equation for a rational function you can use the basic setup of:

a( )( )...

y= _________

( )( )...

The vertical asymptotes tell you about the denominator, the x intercept tells you about the numerator and the y intercept tells you about a.

Let's start with the vertical asymptotes of 3 and -1. A vertical asymptote happens with the denominator equals 0 making the fraction irrational. There are 2 points in which the denominator in this equation is 0 so that means we are going to have 2 factors in the denominator. So let's deal with the first vertical asymptote of 3. Our factor is going to look like (x + some number) and we are trying to determine what "some number" is. We know that this factor must equal 0 when x is 3. This is what defines a vertical asymptote. Ok so to make (3 + some number) = 0, some number must be -3. So that means our first factor is (x-3). Great, let's put that in our equation:

a( )( )...

y= _________

(x-3)( )

Ok, now let's do the same thing for the other vertical asymptote which is -1. So (-1 + some number) = 0. Some number must be 1. Therefore our factor is (x+1). Let's add that in!

a( )( )...

y= _________

(x-3)(x+1)

Awesome, now we've finished the denominator. Let's move onto the numerator. The x intercepts are the values of x when y=0. We know that if the numerator of a fraction equals 0, then the entire fraction is zero. That means we can just deal with the numerator when we are dealing with x intercepts. Again, there are 2 intercepts, so that means we'll have 2 factors in the numerator. We can determine them the same way: y=(x+some number). The first x intercept is -4, so when x is -4, y is zero. Let's solve for "some number." 0=(-4+some number). Some number must be 4, so our first factor is (x+4). We can add this to our equation:

a(x+4)( )

y= _________

(x-3)(x+1)

Now let's deal with the last x-intercept of -6. We again have 0=(-6+some number) so "some number" is 6. Our final factor is (x+6)! Let's add it in!

a(x+4)(x+6)

y= _________

(x-3)(x+1)

The only thing left to solve for is a and the y intercept will help us solve a. The y intercept is the value when x = 0. So in this question, when x=0, y=9. Let's substitute 0 for x and 9 for y in our equation.

a(0+4)(0+6)

9= _________

(0-3)(0+1)

We can simplify this as:

9=4*6*a/(-3*1) or 9=24a/-3 or 9=-8a

now we solve for a and get:

a=-9/8

Yay! We have all the parts to our equation. Let's add the final part and see our answer!

-9/8(x+4)(x+6)

y= ____________

(x-3)(x+1)