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Diego is walking home from school at a constant rate. This graph represents function , which gives his distance from home, in kilometers, minutes since leaving the school.
Use the graph to find or estimate:
A toy rocket and a drone were launched at the same time.
Here are the graphs that represent the heights of the two objects as a function of time since they were launched.
Height is measured in meters above the ground, and time is measured in seconds since launch.
2 graphs. Horizontal axis, 0 to 7, time in seconds. Vertical axis, 0 to 50 by 10’s, height in meters. Graph R is parabolic and opens down. It starts at 0 comma 25. Its vertex is at 2 comma 45. It ends at 5 comma 0. Graph D is piecewise linear. It starts at 0 comma 0. Increases until 2 comma 20, horizontal until 5 comma 20, then decreases to 7 comma 0.
Analyze the graphs and describe—as precisely as you can—what was happening with each object. Your descriptions should be complete and precise enough that someone who is not looking at the graph could visualize how the objects were behaving.
In a bungee jump, the height of the jumper is a function of time since the jump begins.
Function defines the height, in meters, of a jumper above a river, seconds since leaving the platform.
Here is a graph of function , followed by five expressions or equations and five graphical features.
Match each description about the jump to a corresponding expression or equation and to a feature on the graph.
One expression or equation does not have a matching verbal description. Its corresponding graphical feature is also not shown on the graph. Interpret that expression or equation in terms of the jump, and describe the feature of the graph it represents. Record your answers in the last row of the table.
| description of jump | expression or equation |
feature of graph |
|---|---|---|
| a. the greatest height that the jumper is from the river | ||
| b. the height from which the jumper was jumping | ||
| c. the time at which the jumper reached the highest point after the first bounce | ||
| d. the lowest point that the jumper reached in the entire jump | ||
| e. |
Use the graph to:
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 function is decreasing if its outputs get smaller as the inputs get larger. This results in a downward sloping graph as it goes from left to right. A function can also be decreasing just for a restricted range of inputs.
This graph shows the function given by . It is decreasing for because the graph slopes downward to the right of the vertical axis.
A horizontal intercept of a graph is a point where the graph crosses the horizontal axis. If the axis is labeled with the variable , a horizontal intercept is also called an -intercept. The term can also refer to only the -coordinate of the point where the graph crosses the horizontal axis.
For example, the horizontal intercept of the graph of is , or just 6.
A function is increasing if its outputs get larger as the inputs get larger. This results in an upward sloping graph as it goes from left to right. A function can also be increasing just for a restricted range of inputs.
This graph shows the function given by . It is increasing for because the graph slopes upward to the left of the vertical axis.
A maximum of a function is a value of the function that is greater than or equal to all the other values. The maximum of the function’s graph is the highest point on the graph.
A minimum of a function is a value of the function that is less than or equal to all the other values. The minimum of the function’s graph is the lowest point on the graph.
A vertical intercept of a graph is a point where the graph crosses the vertical axis. If the axis is labeled with the variable , a vertical intercept is also called a -intercept. The term can also refer to only the -coordinate of the point where the graph crosses the vertical axis.
For example, the vertical intercept of the graph of is , or just -5.