2.4 Features of Function Graphs

The y-intercept is the point where the graph of a function intersects the y-axis. It is represented as (0, y). For example, if a function has a y-intercept of (0, -3), it means the graph crosses the y-axis at -3.

A function is negative where its output (y-value) is less than zero. For example, if a function is negative for $$-5 < x < 0$$ and $$x > 6$$ , it means the function's values are below the x-axis in these intervals.

A function is positive where its output (y-value) is greater than zero. For example, if a function is positive for $$x < -3$$ and $$x > 3$$ , it means the function's values are above the x-axis in these intervals.

The x-intercept is the point where the graph of a function intersects the x-axis. It is represented as (x, 0). For example, if a function has an x-intercept of (6, 0), it means the graph crosses the x-axis at 6.

A function is negative in an interval if all the y-values of the function within that interval are below the x-axis. For example, if a function is negative for $$-7 < x < -3$$ and $$x > 5$$ , it means the function's values are below zero in these ranges.

A function is positive in an interval if all the y-values of the function within that interval are above the x-axis. For example, if a function is positive for $$x < -3$$ and $$x > 3$$ , it means the function's values are above zero in these ranges.

To find the y-intercept, set x = 0 in the function's equation and solve for y. The resulting point will be the y-intercept.

To find the x-intercept, set y = 0 in the function's equation and solve for x. The resulting point will be the x-intercept.

The x-intercept and y-intercept provide key points that help in sketching the graph of a function. They indicate where the graph crosses the axes, which is essential for understanding the function's behavior.

The general form of a linear function is $$y = mx + b$$ , where m is the slope and b is the y-intercept.