Create a rational function such that the graph of has vertical asymptotes at x=5 and x= -7, a hole at x=2 , and a horizontal asymptote at y = 14. By creating a rational function, you are to write rule for this function. There are many correct solutions here.
You know that the denominator has to be zero at x=5 and x=-7
The hole means that the numerator and denominator are both zero at x=2
so, we can start with
f(x) = (x-2)/[(x-2)(x-5)(x+7)]
Now, we need a horizontal asymptote at y=14. That means that the numerator and denominator must have the same degree, x^n, with the highest degree having a coefficient in the numerator 14 times that in the denominator. So, the simplest one I can think of is
f(x) = 14x^2(x-2)/[(x-2)(x-5)(x+7)]
Sure, here's a rational function that satisfies the given conditions:
f(x) = (x + 7)(x - 2) / ((x - 5)(x - 7))
This function has vertical asymptotes at x = 5 and x = -7 because the denominators of the rational function become zero at those values.
It has a hole at x = 2 because both the numerator and denominator have a factor of (x - 2), which cancels out.
Lastly, it has a horizontal asymptote at y = 14 because as x approaches positive or negative infinity, the function approaches the value 14.
Keep in mind that this is just one of many possible correct solutions!
To create a rational function that satisfies the given conditions, we can start by considering the asymptotes and the hole.
The given vertical asymptotes are x = 5 and x = -7. This means that the denominator of the rational function should have factors of (x - 5) and (x + 7).
The given hole is at x = 2. This implies that both the numerator and the denominator should have a factor of (x - 2). However, since there is a hole at x = 2, this common factor will cancel out.
A horizontal asymptote at y = 14 indicates that the degree of the numerator and denominator should be the same. So, let's assume both have a degree of 1.
Based on these observations, we can create a rational function with the desired properties:
f(x) = (x - 2) / [(x - 5)(x + 7)]
This rational function has a vertical asymptote at x = 5, x = -7, a hole at x = 2, and a horizontal asymptote at y = 14.
Please note that there may be other valid rational functions that satisfy these conditions, but this is one possible solution.
To create a rational function with the given specifications, we can start by identifying the factors that correspond to the vertical asymptotes, hole, and horizontal asymptote.
Vertical Asymptotes:
Since the graph has vertical asymptotes at x = 5 and x = -7, we can include the factors (x - 5) and (x + 7) in the denominator of our rational function.
Hole:
The graph has a hole at x = 2, which means there is a factor of (x - 2) in both the numerator and denominator of our rational function. However, this factor will cancel out, resulting in the hole.
Horizontal Asymptote:
The graph has a horizontal asymptote at y = 14. To achieve this, we need the degree (highest power) of the numerator to be less than or equal to the degree of the denominator. In this case, we can choose a constant value, such as 14, for the numerator.
Putting it all together, the rational function can be represented as:
f(x) = (x - 2)(14) / ((x - 5)(x + 7))
In this function, the factor of (x - 2) appears in both the numerator and denominator, creating the hole. The factors of (x - 5) and (x + 7) in the denominator correspond to the vertical asymptotes. Lastly, the constant value 14 in the numerator ensures that the function approaches y = 14 as x approaches positive or negative infinity, representing the horizontal asymptote.