Lesson 6

Features of Graphs

Let’s use graphs of functions to learn about situations.

6.1: Walking Home

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. 

Graph of line. Horizontal axis, 0 to 24 by 4’s, minutes since leaving school. Vertical axis, 0 to 3 by point 5’s, kilometers from home. Line starts at about 0 comma 2 point 2 5, and ends at 20 comma 0.

Use the graph to find or estimate:

  1. \(d(0)\)
  2. \(d(12)\)
  3. the solution to \(d(m)=1\)
  4. the solution to \(d(m)=0\)

6.2: A Toy Rocket and a Drone

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.

2 graphs. Horizontal axis, 0 to 7, time in seconds. Vertical axis, 0 to 50 by 10’s, height in meters. Graph R is a parabolic and opens down. Graph D is piecewise linear.
  1. 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.

  2. Which parts or features of the graphs show important information about each object’s movement? List the features or mark them on the graphs.

6.3: The Jump

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.

Woman bungee jumping.
Graph of curve. Horizontal axis, 0 to 35 by 5’s, t, seconds. Vertical axis, 0 to 90 by 10’s, h, meters. Curve starts a 0 comma 80, decreases then increases, 4 times.

\(h(0)\)

\(h(t)=0\)

\(h(4)\)

\(h(t)=80\)

\(h(t)=45\)

  • first dip in the graph
  • vertical intercept
  • first peak in the graph
  • horizontal intercept
  • maximum
  1. 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
    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. 
     
  2. Use the graph to:

    1. estimate \(h(0)\) and \(h(4)\)
    2. estimate the solutions to \(h(t)=45\) and \(h(t) = 0\)


Based on the information available, how long do you think the bungee cord is? Make an estimate and explain your reasoning.

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.

Graph of curve. Horizontal axis, 0 to 4, time, seconds. Vertical axis, 0 to 25 by 5’s, height, meters. Curve goes through labeled point 0 comma 20, 1 comma 25, and 3 point 2 comma 0.
  • 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 feet, 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 points 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.

Glossary Entries

  • decreasing (function)

    A function is decreasing if its outputs get smaller as the inputs get larger, resulting in a downward sloping graph as you move from left to right.

    A function can also be decreasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is decreasing for \(x \ge 0\) because the graph slopes downward to the right of the vertical axis.

  • horizontal intercept

    The horizontal intercept of a graph is the point where the graph crosses the horizontal axis. If the axis is labeled with the variable \(x\), the horizontal intercept is also called the \(x\)-intercept. The horizontal intercept of the graph of \(2x + 4y = 12\) is \((6,0)\).

    The term is sometimes used to refer only to the \(x\)-coordinate of the point where the graph crosses the horizontal axis.

  • increasing (function)

    A function is increasing if its outputs get larger as the inputs get larger, resulting in an upward sloping graph as you move from left to right.

    A function can also be increasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is increasing for \(x \le 0\) because the graph slopes upward to the left of the vertical axis.

  • maximum

    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 graph of the function is the corresponding highest point on the graph.

  • minimum

    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 graph of the function is the corresponding lowest point on the graph.

  • vertical intercept

    The vertical intercept of a graph is the point where the graph crosses the vertical axis. If the axis is labeled with the variable \(y\), the vertical intercept is also called the \(y\)-intercept.

    Also, the term is sometimes used to mean just the \(y\)-coordinate of the point where the graph crosses the vertical axis. The vertical intercept of the graph of \(y = 3x - 5\) is \((0,\text-5)\), or just -5.