SOLUTION: What function has the following characteristics? A zero at x = 3 A hole when x = 5 A vertical asymptote at x = -1 A horizontal asymptote at y = 3 A y-intercept at y = -2

Question 1158026: What function has the following characteristics?
A zero at x = 3
A hole when x = 5
A vertical asymptote at x = -1
A horizontal asymptote at y = 3
A y-intercept at y = -2
Answer by KMST(5385) (Show Source): You can put this solution on YOUR website!
There could be other ways to get a function that does that, but we could easily find a ratio of polynomials (a rational function) that could do that.
Characteristic #1: A zero at x = 3
Having as a factor would do that.

Characteristic #2: A hole when x = 5
Having as a factor (having as a factor in numerator and denominator) would do that.

Characteristic #3: A vertical asymptote at x = -1
Having as a factor only in the denominator would do that.

So far we have the building blocks for a function that has the 3 characteristics above,

,
with a graph that looks like this:
.
It even has a horizontal asymptote, but at ,
and its y-intercept is .
The asymptote and intercept are not yet what we want.
If we add , the horizontal asymptote becomes , but then is not a zero.
If we include as another factor, the he horizontal asymptote becomes , but then the y- intercept becomes .

We need to meet the required characteristics #4 and #5 without loosing what we already have achieved.

Characteristic #4: A horizontal asymptote at y = 3
Characteristic #5: A y-intercept at y = -2

has a horizontal asymptote because it is the ratio of polynomials of the same degree, and when divide numerator by denominator the quotient is , and the remainder polynomial degree is less,
so that we could re-write as as

,
and we know that
Of course the reason the quotient is is that the leading coefficients of are both , and their ratio is .
The leading coefficient of is the product of the leading coefficients of the factors and that we had to include to get characteristics #1 and 2.
The leading coefficient of is the product of the leading coefficients of the factors and that we had to include to get characteristics #2 and #3.
The value of the y-intercept depended only on the independent terms of factors , , and .
We could include an extra factor in the denominator,
and an extra factor in the numerator
to try to achieve characteristics #4 and #5.
Horizontal asymptote is achieved by
because the ratio of leading coefficients would be .
To get "A y-intercept at y = -2"we need -->

The graph would then be

Closeup:
RELATED QUESTIONS
What function has the following characteristics? A zero at x = 4 and at x = -2 A... (answered by KMST)

I need to make up a rational function that has the following characteristics : crosses... (answered by josgarithmetic)

Write a function with the following characteristics: A vertical asymptote at x=3 A... (answered by josgarithmetic)

Write the equation of the rational function with vertical asymptotes at x = 2 and x = 1,... (answered by solver91311)

Make up a rational function that has the following characteristics: crosses the x-axis at (answered by stanbon)

Find the equation of a rational function which has an oblique asymptote at y=2x+1, a... (answered by josgarithmetic)

Find the equation of a rational function. Vertical Asymptote at x= -6 Horizontal... (answered by greenestamps)

Which of the following is true of f(x)? f(x)= (x^2+8x+15)/(x^2-2x-15). One of the... (answered by MathLover1)