Final answer:
Intercepts of a rational function include the y-intercept, which is the point where the function crosses the y-axis, and the x-intercepts, where it crosses the x-axis. The y-intercept is found by setting x to zero, and x-intercepts are obtained by setting the function to zero. These intercepts are crucial in understanding the graph's starting point on the y-axis and the function's roots respectively.
Step-by-step explanation:
When identifying the intercepts of a rational function, there are two types to consider: the y-intercept and the x-intercept. The y-intercept is the point where the graph of the function crosses the y-axis. This occurs when the input value (x) is zero. To find the y-intercept of a given function, such as y = mx + b, you would set x to zero and solve for y, which would simply be the value of b. The y-intercept provides valuable information about where the graph begins on the y-axis.
On the other hand, x-intercepts are points where the graph crosses the x-axis, which occurs when the output value (y) is zero. Finding the x-intercepts involves setting the function equal to zero and solving for x. These intercepts are important in understanding the roots or solutions of the function.
For example, consider the line graph with an equation of the form y = 3x + 9. The y-intercept is 9, which indicates that the graph crosses the y-axis at (0, 9). The slope is 3, which means the line rises 3 units for every 1 unit it moves to the right along the x-axis. The shape and position of the line are dictated by these intercepts and the slope. If a line has a larger intercept, it would shift upwards on the graph, while a smaller intercept would result in a downward shift.