Functions
Relations and Functions
Graphing and Interpreting Functions
Piecewise Functions
Absolute Value Functions
Inverse Functions
Rates of Change

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FunctionsGraphing and Interpreting Functions

Reading time: ~90 min

The Olympics is full of incredible athletic feats. It’s also full of interesting data. Graphs help us visualize that data. During our time together today, we will watch Olympic competitions and analyse their graphs for interesting information. Let’s head over to the gymnastics arena!

Ri

Ri Se-gwang of the People’s Republic of Korea is about to vault. He won the gold medal for the vault in the 2016 Summer Olympic Games. Let’s watch.

vault mock-up

There are several things going on here. Move the video back and forth to see how the graph lines up with the motion.

First, we need to understand is what the axes represent. The x-axis in this graph is the horizontal distance Ri travels throughout his vault. It is measured in centimeters. The y-axis is the vertical distance in centimeters Ri travels. This gives us information about Ri’s position much like chess pieces on a board.

The graph does not include any information about time. For example, we cannot tell when Ri landed on the pit. Some of the graphs of later events will include time along the horizontal axis.

On this graph, we see the vault is at (, )), which means Ri ran about 3.5 meters in his approach. The starting point on the runway is at the origin. Ri lands at (3910, 30), which means the pit is about centimeters tall.

Let’s build some intuition for what graphs of different events look like. Match the graph to the event. Be sure to pay close attention to what the axes represent.

Triple Jump50 M Freestyle100 M HurdlesVaultDivingSkiing
GraphGraphGraphGraphGraphGraph
PhotoPhotoPhotoPhotoPhotoPhoto
VideoVideoVideoVideoVideoVideo
Françoise Mbango Etone of Cameroon holds the Olympic record in women’s triple jump with a length of 15.39 meters.César Cielo of Brazil holds the Olympic record for the men’s 50 meter with a time of 21.47 seconds.Sally Pearson of Australia holds the Olympic record in the women’s 100 meter hurdles with a time of 12.35 seconds.Ri Se-gwang of the People’s Republic of Korea won the gold medal for the vault in the 2016 Summer Olympic Games.Ren Qian of China won the gold medal for diving in the 2016 Summer Olympic Games.
Start: 985mFinish: 805mVertical drop: 180mGates: 66Finish time: 48.33

As you can see there are several different perspectives for graphing motion. One perspective is distance as a function of time. Select the events represented with this perspective.

We see the other two events are as a function of . We will look at two of these functions in more detail. Let’s head over to the pool.

We are just in time for the men’s 50 meter freestyle finals. Keep a close eye on César Cielo Filho of Brazil. The graph of his swim will appear as the video plays.

César Cielo of Brazil holds the Olympic record for the men’s 50 meter with a time of 21.47 seconds.

César Cielo

What an emotional race for Cielo! He broke the world and olympic records with this swim. The graph shows Cielo’s record-breaking swim. The shape is one we have seen before in graphing linear functions. Recall this graph represents all the ordered pairs matching inputs to outputs. We can use information from the graph to write the function describing Cielo’s swim. Let’s call the function f(t).

Remember the general form for linear function like this is y=mx+c, where m is the slope and c is the y-intercept. We replace y with the name of the function, and we replace x with the input variable for this function.

y=mx+c ⇒ft=mt+c

This means we need to find and from the graph.

Notice the horizontal-axis, in this graph shows in seconds. The vertical-axis is the distance from the starting block to the opposite end of the pool measured in . We see that the y-axis intercept is meters, which represent the of the race.

y=mx+c ⇒ft=mt+0

We are only missing slope. In this function, slope represents Cielo’s . How could we use the graph to see how fast he swims?

As we move from left to right along the line, the vertical distances measure meters. The horizontal distances measure second. Cielo’s speed for this race, which is a key feature of linear functions.

Recall that the slope is the vertical change divided by the horizontal change (rise over run). The slope of f(t) is meters per second. y=mx+c

f(t)=t+0

Let’s say we want to know how long it took Cielo to swim the first 10 meters. Ten meters into the race, This means we are looking at f(t)=10. The phrase “how long” indicates we are solving for t.

ft=2.35t 10=2.35t 102.35=t 4.25=t

Cielo swims the first 10 meters in just over seconds.

Take a look at the top four finishers during this race:

| Swimmer | | Function Name | | Color | | :------ | | :------------ | | :---- | | Cesar Cielo Filho | | f(t) | | green | | Amaury Leveaux | | l(t) | | purple | | Alain Bernard | | b(t) | | blue | | Ashley Callus | | c(t) | | red |

All the lines cross the y-axis at meters because this is the distance of the race. At first glance, we notice the graphs seem almost on top of each other. This must indicate that the swimmers’ speeds are similar.

Say we want to figure out how many seconds Cielo is ahead of Leveaux after 10 meters. We already know Cielo swam this distance in 4.25 seconds. We don’t know the function rule for l(t), Leveaux’s swim, but we do have the graph. Find the time, t, where l(t) = .

Leveaux swims the first 10 meters in seconds. That means Cielo was only seconds ahead of Leaveaux!

Leveaux and Bernard trained together, and it shows. They stay neck-in-neck the whole race. They finish only seconds apart.

Use the graph to find where everyone is at the 20 second mark. f(20)= l(20)= b(20)= c(20)=

Callus is 6 meters behind at the 20 second mark.

Let’s head over to the diving pool for the women’s 10 meter platform competition.

Meanwhile, on the other side of the Aquatics center, a diving competition is in progress. Ren Qian is among the youngest Olympic medalists. She is diving now - let’s watch

Ren Qian of China won the gold medal for diving in the 2016 Summer Olympic Games.

Let’s call the function representing Ren’s dive d(x). The input values, x, are horizontal distances from the platform. The output values, d(x), are Ren’s throughout the dive. Immediately, we notice the shape of this graph is different from the swimming graphs above. This graph has turning points compared to the linear functions turning points.

Graphs with this shape are called cubic functions. We can get important information from the graph even without knowing the function equation. Match the given statements to the graph.

| Place contextual statement cards on graph | | Target key feature appears when card is placed | |Function notation appears when card is placed | | --- | | --- | | --- | | Ren’s takes her place on the platform. | | Vertical intercept | | | | Ren reaches the highest point of her dive. | | maximum | | | | Ren completes 3.5 somersaults. | | decreasing | | 0.335<x<2.556 | | Ren’s entry is nearly flawless. | | Horizontal intercept | | d(2.056)=0 | | Ren turns around under water. | | minimum | | d(2.556)=-1.623 | | Ren surfaces after a nearly flawless dive. | | Horizontal intercept | | d(2.989)=0|

Recall the y-intercept is where x=. In function notation, this looks like d()=10. The pattern of d(0)=y-intercept is true for any function. // That's an interesting thought. +philipp@mathigon.org can text oder be dependent on how the student uses the interactive? In this case, what order they choose to place the cards on the graph?

Similarly, the x-intercepts are were =0. This graph has x-intercepts. They represent the surface of the water in the pool.

When we talk about the maximum, we are really talking about the highest value. Ren’s tallest height is meters. She reaches her maximum height when she is meters from the board. In function notation, this looks like .

The minimum is Ren’s lowest height. In this graph her lowest point is underwater. Because the x-axis represents the surface of the water, the minimum d(x) is . She turns around at meters underwater and meters from the diving platform.

Intuitively, we understand that the graph is increasing when Ren’s body is moving . The notation for increasing is different from tuning points and intercepts. Since the graph increases for more than one point, we represent the section of the graph using an interval. The interval communicates the values corresponding to Ren’s increasing height. Note that there are many different ways to write intervals, we use inequalities in this chapter.

Ren moves up during the intervals:

0<x<0.335 0.335<x<2.556 2.556<x<2.989

The unchecked interval is where the graph is . Ren is moving down from d(0.335)= meters to d(2.556)= meters.

Notice the is where the Ren’s path changes from increasing to decreasing heights. The minimum is where Ren’s path changes from decreasing to increasing.

Let’s think about the input and output values for d(x). Recall domain is the set of all possible input values for d(x). One method for finding the domain is starting with the set of Real numbers and narrowing the set down to a reasonable range for the given situation.

Ren’s horizontal distance starts at the diving platform and ends where she resurfaces in the pool. We know the diving platform is at x= meters. She resurfaces at x= meters. Therefore, we can write the domain as . {.fixme} SHOW AFTER PREVIOIUS BLANKS FILLED Note that the endpoints, 0 and 2.989, are included in the domain using <= and >=.

Recall range is the set of all heights Ren travels. Notice that Ren goes below the surface of the water. In fact, we can use the function’s to determine the lower bound on the range. The minimum d(x) is meters.

Similarly, the maximum d(x) gives us the upper bound on the range. Therefore, the range is <= d(x) <= .

Let’s head to the beach for the gold medal men’s volleyball match between Brazil and Italy. The teams engage in a beautiful volley (556-601). As you watch the video, notice the shape of the graph. Is it what you expect?

sketch

The graph shows the volley as a function of . That means for at each moment, the ball has a position marked by . For example, the ball is feet high at 2 seconds. We can find information about when certain things happen.

For instance, the ball spends about seconds above the net. Brazil scores a point after about seconds. Italy places a beautiful set at about seconds. The ball reaches its maximum height, about feet, off of the set.

Notice that the graph does not show when the ball changes direction. This could happen if the graph were a function of . Such a graph would tell us where certain things happen. We can’t tell by looking at this graph which side of the net the ball is on.

Creating Graphs

The women’s pole vault is just about to start. You will be drawing the graph for this event.

We like to start graphing using a table. Fill in the table below. Note the landing pad, called the pit, is 0.81 meters tall.

| Time (s) | | Height (m) | | ------ | | -------- | | 0 | | 0 | | 1 | | 0 | | 4 | | | | 5.5 | | | | 6 | | | | 6.5 | | | | 7 | | | | 7.5 | | | | 8 | | |

Plot these values on the coordinate plane.

sketch

This graph is interesting because between and about seconds, the graph is constant. Stefanidi’s maximum height is meters. The last point on the graph is at (, ) because she lands on the pit, not the ground.

Systems of Functions/ Simultaneous Functions

Let’s head over to the track for the women’s 800 meter final. Looks like we arrive in time to catch the last 200 meters of the race. 322-400

| Athlete | | Adelle Tracey | | Laila Boufaarirane | | Raevyn Rogers | | --------- | | ----------- | | ---------------- | | ------------| | Country | | GBR | | FRA | | USA | | @ 90.63 s | | 600 | | 598 | | 590 | | @ 800 m | | 121 | | 126 | | 120.2 | | color | | black | | green | | orange | | function | | g(t)=6.58545x+3.16101 | | f(t)=5.71105x+80.4071 | | u(t)=7.10179x-53.6354 |

800 M mock-up

When we have two or more function on the same coordinate plane, we call them a system of equations. Systems like this add key features that help us further understand what is happening in the race between Tracey, Boufaarirane, and Rogers. For example, something interesting is happening when the graphs intersect. Select all the true statements about the intersection points in this system.

| Tracey and Rogers are at the same location. | | g(t) = u(t) | | Rogers is ahead of Boufaarirane. | | u(t) < f(t) | | Boufaarirane and Rogers are at the same location. | | f(t) = u(t) | | Tracey is behind Rogers. | | g(t) > u(t) |

The graphs intersect when one runner passes another. When the runners have about 200 meters left in their race, is in third place.

Rogers passes Boufaarirane at about seconds when they are both about meters into the race. In function notation, this looks like f()=u()=.

About seconds later, overtakes . They have about meters to the finish line.

The slopes of each function tell us each runner’s . Rogers is running at about meters per second.

m=y2−y1x2−x1 m=y2−590x2−90.63 m=800−590120.2−90.63 m=21029.57 m=7.1

Roger’s speed is meters per second faster than Boufaarirane and meters per second faster than Tracey.

In this system of functions, we can see who is ahead at any given time during the race. For example, we write f(t) > u(t) when .

| Students label with the given contextual statement cards. | | This information appears after the functions notation card is correctly placed. | | Extra information. Not cards. | | --- | | --- | | --- | | Boufaarirane is ahead of Rogers. | | f(t) > u(t) | | 90.63=<t<96.382 | | Rogers is ahead of Boufaarirane. | | f(t) < u(t) | | 96.382<t<=120.2 | | Tracey is ahead of Rogers. | | u(t) < g(t) | | 90.63<=t<109.998 | | Tracey is ahead of Boufaarirane. | | g(t) > f(t) | | | | Rogers is ahead of Tracey. | | g(t) < u(t) | | 109.998<t<=120.2 |

When we talk about one function being greater than another, we are using the values to identify a range of . For example, we see Tracey is ahead of Boufaarirane for this entire stretch of the race. This is expressed as . We can think of this as “the range of time when Tracey has run a farther distance than Boufaarirane”. That range is <= t <= . We can do a similar analysis for each pair of functions.

Looking at the three functions on the coordinate plane, we can see that the relationship between u(t) and g(t) changes from g(t)>u(t) to u(t)>g(t) when . This means that the upper bound on g(t)>u(t) is where g(t) u(t), which is t= seconds. This t-value, 110 seconds, is also the lower bound on u(t)>g(t).

Finally, here is one more function that represents a sport. Can you think of what it is, and write a short story that explains the different features of the chart?

TODO: draw chart << free-form text input >>