The graph of the function can give us useful information about the quantities in a situation. Some points and features of a graph are particularly informative, so we pay closer attention to them.

Let’s look at the graph of function , which gives the height, in meters, of a ball seconds after it is tossed up in the air. From the graph, we can see that:

• The point is the vertical intercept of the graph, or the point where the graph intersects the vertical axis.

  This point tells us that the initial height of the ball is 20 meters, because when  is 0, the value of is 20​​​​​.

  The statement captures this information.

• The point is the highest point on the graph, so it is a maximum of the graph.

  The value 25 is also the maximum value of the function . It tells us that the highest point the ball reaches is 25 meters, and that this happens 1 second after the ball is tossed.

• The point is a horizontal intercept of the graph, a point where the graph intersects the horizontal axis. This point is also the lowest point on the graph, so it represents a minimum of the graph.

  This point tells us that the ball hits the ground 3.2 seconds after being tossed up, so the height of the ball is 0 when is 3.2, which we can write as  . Because cannot have any lower value, 0 is also the minimum value of the function.

• The height of the graph increases when is between 0 and 1. (A graph is increasing where the function’s values get greater as the graph is read left to right.) Then, the graph changes direction, and the height decreases when is between 1 and 3.2. (A graph is decreasing where the function’s values get lesser as the graph is read left to right.) Neither the increasing part nor the decreasing part is a straight line.

  This suggests that the ball increases in height in the first second after being tossed and then starts falling between 1 second and 3.2 seconds. It also tells us that the height does not increase or decrease at a constant rate.

Because the intercepts of a graph are points on an axis, at least one of their coordinates is 0. The 0 corresponds to the input or the output of a function, or both.

• A vertical intercept is on the vertical axis, so its coordinates have the form , where the first coordinate is 0 and can be any number. The 0 is the input.
• A horizontal intercept is on the horizontal axis, so its coordinates have the form , where can be any number and the second coordinate is 0. The 0 is an output.
• A graph that passes through intersects both axes so that point is both a horizontal intercept and a vertical intercept. Both the input and output are 0.