20. (4 points) Write an equation for a rational function whose graph has all of the indicated features. vertical asymptote with equation x 3 horizontal asymptote with equation y =2 hole at x= 1 no x-intercepts
vertical asymptote at x=3
1/(x-3)
hole at x=1
(x-1)/((x-3)(x-1))
This has a horizontal asymptote at y=0. So, to get a
horizontal asymptote at y=2,
(x-1)/((x-3)(x-1)) + 2
The problem remaining is that this goes negative for x<3. So, square (x-3) and it stays positive
(x-1)/((x-3)^2(x-1)) + 2
Check the graph at
https://www.wolframalpha.com/input/?i=%28x-1%29%2F%28%28x-3%29%5E2%28x-1%29%29+%2B+2
Why did the rational function go to therapy?
Because it had commitment issues with its x-intercepts! It just couldn't seem to find any. Oh well!
Now, let me illustrate this for you. Since we want a vertical asymptote at x = 3, we can start with a denominator that has a factor of (x - 3). For the horizontal asymptote at y = 2, we want the numerator to have a lower degree than the denominator. So, let's have a simple numerator like 1.
Therefore, our rational function will be:
f(x) = 1 / (x - 3)
Now, because we want a hole at x = 1, we can just multiply both the numerator and denominator by (x - 1):
f(x) = (1 * (x - 1)) / ((x - 1) * (x - 3))
Simplifying, we get:
f(x) = (x - 1) / (x^2 - 4x + 3)
And there you have it! A rational function with a vertical asymptote at x = 3, a horizontal asymptote at y = 2, and a hole at x = 1.
To create an equation for a rational function with the given features, we can use the form:
f(x) = (ax + b) / (cx + d)
Based on the given information:
1. Vertical asymptote with equation x = 3:
This means that the denominator (cx + d) must have a factor of (x - 3), so c = 1 and d = -3.
f(x) = (ax + b) / (x - 3)
2. Horizontal asymptote with equation y = 2:
Since there is a horizontal asymptote at y = 2, the degree of the numerator and the denominator must be the same. Thus, the numerator must have a degree of 1 and the denominator must have a degree of 1.
f(x) = (ax + b) / (x - 3)
3. Hole at x = 1:
To have a hole at x = 1, both the numerator and the denominator must have a factor of (x - 1). This implies that a = b = 1.
f(x) = (x + 1) / (x - 3)
Therefore, the equation for the rational function with the given features is:
f(x) = (x + 1) / (x - 3)
To write an equation for a rational function with these specific features, we need to consider the characteristics of each feature.
1. Vertical asymptote:
A rational function has vertical asymptotes at values of x that make the denominator of the rational function equal to zero. Here, we have a vertical asymptote with the equation x = 3. This means that the function will approach infinity as x approaches 3.
2. Horizontal asymptote:
A rational function has a horizontal asymptote at y = a, where 'a' is a constant. In this case, the horizontal asymptote equation is y = 2, meaning that as x approaches positive or negative infinity, the function approaches the value of 2.
3. Hole:
A hole occurs when both the numerator and the denominator have the same factor and this common factor can be canceled out.
Based on the given information, we know that there is a hole at x = 1. To create a hole at x = 1, we need to ensure that x - 1 is a common factor in both the numerator and denominator of the rational function.
4. No x-intercepts:
If a rational function doesn't have any x-intercepts, the numerator should not be equal to zero for any value of x.
Taking all these features into account, we can construct the equation of the rational function:
f(x) = 2 * (x - 1) / ((x - 3) * (x - 1))
In this equation, we have (x - 1) in both the numerator and the denominator, creating a hole at x = 1. The vertical asymptote at x = 3 is created by the factor (x - 3). The horizontal asymptote is y = 2, indicating that as x approaches infinity, the function approaches 2.