What’s a function that has a vertical asymptote at x= -2, a horizontal axis at y= 3, and a hole at (4,2)?
vertical asymptote at x= -2
y = 1/(x+2)
horizontal asymptote at y = 3 ... degree top and bottom must be the same
y = 3x/(x+2)
A hole at (4,2) means that we're trying to divide by x-4, so
y = 3x(x-4) / (x+2)(x-4)
To find a function that satisfies the given conditions, we can use the properties of rational functions. A rational function is defined as the ratio of two polynomials, and it can have vertical asymptotes, horizontal axes, and holes at certain points.
Let's break down the given conditions one by one:
1. Vertical asymptote at x = -2: A vertical asymptote occurs when the denominator of the rational function becomes zero at a particular value of x. So, for our function, we need a factor in the denominator that cancels out with (x + 2) so that it becomes zero at x = -2. We can write the denominator as (x + 2).
2. Horizontal axis at y = 3: A horizontal axis occurs when the degree of the numerator is less than or equal to the degree of the denominator. This means that the highest power of x in the numerator should be lower than or equal to the highest power of x in the denominator. To achieve this, we can consider the numerator as a constant (let's say 3) or a linear function of x (let's say mx).
3. Hole at (4, 2): A hole in the graph occurs when both the numerator and the denominator have a common factor that cancels out. To create a hole at (4, 2), we need a factor of (x - 4) in both the numerator and denominator. We can write it as (x - 4) in both the numerator and denominator.
Putting these conditions together, we can write the function as:
f(x) = (m(x - 4))/(x + 2)
The value of m will determine the slope of the linear function in the numerator. You can choose any value for m as long as it satisfies the conditions.
So, substituting the given conditions, the function could be, for example:
f(x) = (2(x - 4))/(x + 2)
or
f(x) = (-5(x - 4))/(x + 2)
Remember that there can be multiple correct answers based on the chosen values for m and other coefficients.