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 intercept
thank you
Sure! Here's an equation for a rational function that satisfies all the given features:
f(x) = (2(x-1))/(x-3)
This equation has a vertical asymptote at x = 3, meaning the graph gets arbitrarily close to x = 3 from both sides but never touches it. It also has a horizontal asymptote at y = 2, indicating that as x approaches positive or negative infinity, the function approaches y = 2.
Furthermore, this equation has a hole at x = 1. This means there is a removable discontinuity at x = 1, where the graph is not defined, but if you plug in x = 1 into the equation, you will get a valid output.
Lastly, this equation has no x-intercepts, implying that the graph does not intersect the x-axis.
To create a rational function with these features, following steps can be followed:
1. Start with the general form of a rational function:
f(x) = (Numerator)/(Denominator)
2. Include a vertical asymptote at x = 3:
To do this, include a factor of (x - 3) in the denominator.
3. Include a horizontal asymptote at y = 2:
To do this, make the degree of the numerator equal to or less than the degree of the denominator.
4. Incorporate a hole at x = 1:
To create a hole, include a factor of (x - 1) in both the numerator and denominator. This way, they will cancel out, leaving a hole at x = 1.
5. Combine all the elements:
Putting everything together, the equation for the rational function would be:
f(x) = [(x - 1)/(x - 1)] * [(x - 3)/(x - 1)]
Simplifying this, we get:
f(x) = (x - 3) / (x - 1)
Therefore, the equation for the rational function with the given features is:
f(x) = (x - 3) / (x - 1)
To write an equation for a rational function given the indicated features, follow these steps:
1. Start with the general equation for a rational function:
f(x) = (Numerator) / (Denominator)
2. Identify the features:
- Vertical asymptote at x = 3: This means the denominator must have a factor of (x - 3).
- Horizontal asymptote at y = 2: This means the degree of the numerator must be equal to or less than the degree of the denominator.
- Hole at x = 1: This means there will be a common factor in both the numerator and denominator of (x - 1).
- No x-intercept: This means the numerator should not have any real roots.
3. Determine the degrees of the numerator and denominator:
Since there is a horizontal asymptote at y = 2, we want the degree of the numerator to be less than or equal to the degree of the denominator. Therefore, we can choose the degree of the numerator to be lower.
4. Construct the equation:
Let's assume the numerator is of degree 0 (a constant). And since there is a hole at x = 1, we will introduce (x - 1) as a common factor in both the numerator and denominator.
f(x) = (k * (x - 1)) / (x - 3)
Where k is any nonzero constant.
Note: The value of k will affect the shape of the graph, but it won't affect the asymptotes or the hole.
So, an example equation for the given features could be:
f(x) = (2 * (x - 1)) / (x - 3)
vertical asymptote needs 1/(x-3)
horizontal asymptote needs equal degree, top/bottom = 2
hole needs (x-1) top and bottom
So, so far we have
2(x-1)/((x-1)(x-3))
but that has hor asymp y=0, so we need equal degree top and bottom
2(x-1)(x+1)/(x-1)(x-3)
but that also has x-intercept. So, let's square it:
y = 1+((x^2-1)/(x^2-4x+3))^2