To find the x and y intercepts of a rational function, follow these steps:
Finding the Y-Intercept
The y-intercept occurs where the graph of the function crosses the y-axis. This happens when the value of x is 0. To find the y-intercept, substitute x = 0 into the function and solve for y.
For example, if you have a rational function f(x) = (2x + 3) / (x – 1), to find the y-intercept:
- Substitute x = 0: f(0) = (2(0) + 3) / (0 – 1) = 3 / -1 = -3
- Thus, the y-intercept is (0, -3).
Finding the X-Intercepts
The x-intercepts occur where the graph crosses the x-axis. This happens when the value of y is 0. To find the x-intercepts, set the numerator of the rational function equal to zero and solve for x. The x-intercepts represent the values of x for which the function equals zero.
Using the same example f(x) = (2x + 3) / (x – 1), to find the x-intercepts:
- Set the numerator equal to zero: 2x + 3 = 0
- Solve for x: 2x = -3, x = -3/2
- Thus, the x-intercept is (-3/2, 0).
Summary
In conclusion, to find the intercepts of a rational function: substitute x = 0 for the y-intercept and set the numerator equal to zero for the x-intercepts. This will give you the points where the graph crosses both axes, which are crucial for graphing the function.