Features of Graphs
6.1: Walking Home (5 minutes)
This warm-up is an opportunity to practice interpreting statements in function notation. It also draws attention to statements that correspond to the intercepts of a graph of a function (for instance, \(d(0)\) and \(d(m) = 0\)), preparing students to reason about them in the lesson (particularly in the second activity).
Diego is walking home from school at a constant rate. This graph represents function \(d\), which gives his distance from home, in kilometers, \(m\) minutes since leaving the school.
Use the graph to find or estimate:
- the solution to \(d(m)=1\)
- the solution to \(d(m)=0\)
Ask students to interpret each statement in function notation before soliciting their response to each question. Make sure students understand, for instance, that \(d(0)\) represents Diego’s distance from school at the time of leaving (or at 0 minutes), and \(d(m)=0\) represents his distance from home being 0 km, \(m\) minutes after leaving school.
If time permits, discuss with students:
- “Is the relationship between Diego’s distance from school and time a linear function? How can we tell?” (Yes. The graph is a line, which means the function’s value changes at a constant rate.)
- “Can we tell from the graph how far away Diego’s house is from school? How?” (Yes. From the graph, we can see that at the time he leaves school, he is 2.25 km from home.)
- “Can we tell from the graph how long it took Diego to get home? How?” (Yes. From the graph, we can see the distance reaching 0 when the time is 20 minutes.)
- “Why does the graph slant downward (or have a negative slope)?” (As the input, \(m\), increases, the output, \(d(m)\), decreases.)
6.2: A Toy Rocket and a Drone (15 minutes)
In this activity, students examine graphs of functions, identify and describe their key features, and connect these features to the situations represented. These key features include the horizontal and vertical intercepts, maximums and minimums, and intervals where a function is increasing or decreasing (or where a graph has a positive or a negative slope).
Neither the features nor the terms are likely new to students. The idea of intercepts was introduced in middle school and further developed in earlier units. Graphical features such as maximums and minimums have been considered intuitively in various cases. They are simply more precisely defined in here. In a later activity, students will distinguish between a maximum of a graph and the maximum of a function.
Give students about 5 minutes of quiet work time. Follow with a whole-class discussion.
Design Principle(s): Maximize meta-awareness; Support sense-making
Supports accessibility for: Language; Organization
A toy rocket and a drone were launched at the same time.
Here are the graphs that represent the heights of 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.
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.
- Which parts or features of the graphs show important information about each object’s movement? List the features or mark them on the graphs.
When analyzing the graphs and describing what is happening with each object, some students may mistakenly think that the horizontal axis represents horizontal distance, neglecting to notice that it represents time. They may then describe how the objects were moving vertically as they traveled horizontally, rather than with respect to the number of seconds since they took off. Encourage these students to check the label of each axis and revisit their descriptions.
Select a few students to share their description of the graphs, and have each student describe the motion of one flying object.
Display a blank coordinate plane for all to see. As each student shares their response, sketch a graph to match what is being described.
For any gaps in their description, make assumptions and sketch accordingly. (For example, if a student states that the toy rocket reaches a height of 45 meters after 2 seconds but does not state its starting height, start the curve at \((0,0)\), \((0,40)\), or any other point besides \((0,25)\).) If requested, allow students to refine their descriptions and adjust the sketch accordingly.
Next, invite other students to share their response to the last question. On the graphs, highlight the features students noted. (See Student Response for an example.) Use the terms vertical intercepts, horizontal intercepts, maximum, and minimum to refer to those features and label them on the graphs.
Explain to students that:
- A point on the graph that is as high as or higher than all other points is called a maximum of the graph (or a relative maximum, because its height is viewed in relation to other points shown on the graph).
- A point on the graph that is as low as or lower than all other points is called a minimum of the graph (or a relative minimum).
A graph could have more than one relative maximum or minimum. For instance, the points \((2,(D(2))\) and \((5,D(5)\)) are both relative maximums, and \((0, D(0))\) and \((7,D(7))\) are both relative minimums.
If no students mentioned the intervals in which each function was increasing, staying constant, or decreasing, draw their attention to these features on the graphs and label them as such.
6.3: The Jump (15 minutes)
Earlier in the lesson, students identified key features of a graph of a function and related them to the features of a situation. In this activity, students continue to connect graphical and verbal representations of a function, applying the mathematical terms they learned. They also connect each feature (described in words and on the graph) to an expression or equation that could represent it mathematically, written in function notation.
During the activity synthesis, students learn to distinguish between a maximum or minimum of a graph (relative maximum or minimum) and the maximum or minimum of a function (absolute maximum or minimum). Students see that a maximum of a graph refers to a point on a graph that is as high as or higher than all other points, while the maximum of a function is a function value that is equal to or greater than all other values of that function.
Arrange students in groups of 2. Give students a few minutes of quiet think time, and then time to discuss their response with their partner.
Design Principle(s): Support sense-making; Maximize meta-awareness
Supports accessibility for: Visual-spatial processing
In a bungee jump, the height of the jumper is a function of time since the jump begins.
Function \(h\) defines the height, in meters, of a jumper above a river, \(t\) seconds since leaving the platform.
Here is a graph of function \(h\), followed by five expressions or equations and five graphical features.
- first dip in the graph
- vertical intercept
- first peak in the graph
- horizontal intercept
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 in terms of the graph of the function. Record your interpretation in the last row of the table.
description of jump expression
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:
- estimate \(h(0)\) and \(h(4)\)
- estimate the solutions to \(h(t)=45\) and \(h(t) = 0\)
Are you ready for more?
Based on the information available, how long do you think the bungee cord is? Make an estimate and explain your reasoning.
Select students to share how they made their matches and how they interpreted the expression or equation without a matching verbal description. After each student shares their thinking, ask if others also approached it the same way.
Make sure students can interpret \(h(t)=0\) to mean that the jumper is no longer in the air and is in fact on the surface of the water. Because the graph has no horizontal intercept, and because no verbal descriptions to this effect were given, we know \(h(t)=0\) had no match.
Next, ask students to share their response to the last set of questions. Discuss with students why \(h(t)=0\) has no solutions based on the information we have. It is possible that, if the graph is extended to include more time, we might see a representation of the jumper somehow reaching the water. But given the graph as is, it doesn’t have a solution.
This is a good time to introduce the distinction between a maximum (or minimum) of a graph and the maximum (or minimum) of a function. Ask students,
- “What is the greatest value of function \(h\)?” (80)
- “How do we know that 80 is the greatest, or that \(h(t)\) could not have greater values?” (The jumper could not be higher than the jumping platform.)
- “What is the least value of function \(h\)?” (about 10, based on the graph)
- “How do we know that 10 is the least value, or that \(h(t)\) could not have lesser values?” (We don’t. The jumper could go lower, say, if lowered into a receiving boat, or if released into the river.)
Explain that, in this case, the graph has one maximum, or one point that is higher than all the other points, and it coincides with the maximum of the function, which is the greatest value of the function.
The graph also has a minimum, \((4, h(4))\), which is lower than all other points shown on the graph, but this point does not represent the minimum of the function, or the lowest value the function could have. The function \(h\) could have 0 as its minimum (or a negative value if the jumper goes diving), but as far as the given graph is concerned, \((4,10)\) is a minimum.
Emphasize that a maximum (or minimum) of a graph is a point, and it gets labeled as such relative to other points visible on the graph. The maximum (or minimum) of a function, however, is a value that is the greatest (or least) for any input.
Keep students in groups of 2. Display the graph and the descriptions of two functions for all students to see:
- Function \(b\) gives the vertical distance (or the height) of a bee from the ground as a function of time, \(t\).
- Function \(d\) gives the distance of a child from where his mom is sitting as a function of time, \(t\).
Tell students that the outputs of both functions are measured in feet and the inputs are both measured in seconds.
Ask partners to choose a function. For that function, they should take turns identifying the following features on the graph and interpreting them in terms of the situation:
- vertical intercept
- horizontal intercept
- intervals where the function is increasing
- intervals where the function is decreasing
- intervals where the function is staying constant
- solution or solutions to \(b(t) = 3.5\) or \(d(t) = 3.5\)
Then, discuss questions such as:
- “How can you tell that a point on the graph is a maximum, a minimum, or neither?” (Look around it, left and right, to see whether the given point is higher than all the other points, lower than all the others, or neither.)
- “How many intercepts can the graph of a function have?” (An unlimited number on the horizontal axis, but only one on the vertical axis.)
6.4: Cool-down - The Squirrel (5 minutes)
Student Lesson Summary
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 \(h\), which gives the height, in meters, of a ball \(t\) seconds after it is tossed up in the air. From the graph, we can see that:
The point \((0,20)\) 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 \(t\) is 0, the value of \(h(t)\) is 20.
The statement \(h(0)=20\) captures this information.
The point \((1,25)\) 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 \(h\). 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 \((3.2,0)\) 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 \(t\) is 3.2, which we can write as \(h(3.2)=0\). Because \(h\) cannot have any lower value, 0 is also the minimum value of the function.
The height of the graph increases when \(t\) is between 0 and 1. Then, the graph changes direction and the height decreases when \(t\) is between 1 and 3.2. 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 \((0,b)\), where the first coordinate is 0 and \(b\) can be any number. The 0 is the input.
- A horizontal intercept is on the horizontal axis, so its coordinates have the form \((a, 0)\), where \(a\) can be any number and the second coordinate is 0. The 0 is an output.
- A graph that passes through \((0,0)\) intersects both axes, so that point is both a horizontal intercept and a vertical intercept. Both the input and output are 0.