Create an original rational function that has at least one asymptote (vertical, horizontal, and/or slant) and possibly a removable discontinuity. List these features of your function: asymptote(s) (vertical, horizontal, slant), removable discontinuity(ies), x-intercept(s), y-intercept, and end behavior. Provide any other details that would enable another student to graph and determine the equation for your function. Do not state your function.
Vertical asymptote: x = a (where a is a constant)
Horizontal asymptote: y = b (where b is a constant)
Removable discontinuity: x = c (where c is a constant)
x-intercept(s): points where the function intersects the x-axis
y-intercept: point where the function intersects the y-axis
End behavior: behavior of the function as x approaches positive infinity and negative infinity
To graph and determine the equation for the function:
1. Determine the values of a, b, and c.
2. Use these values to determine the equations for the vertical asymptote, horizontal asymptote, and removable discontinuity.
3. Find the x-intercept(s) by setting y = 0 and solving for x.
4. Find the y-intercept by setting x = 0 and solving for y.
5. Analyze the end behavior by observing the function's behavior as x approaches positive infinity and negative infinity.
Remember to simplify the equation if possible and check for any restrictions on x or y (such as division by zero).