Piecewise Functions

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

All we are saying is give absolute value, a circle and the y-axis a chance.

I'm guessing that this joke occurs to me every time I have to cover the material in class, which leaves me wondering if I've done this joke before or not. Longtime fans might remember a previous occurrence. All google products are incredibly slow on the machine I'm currently using today. Isn't that odd? You know what else is odd? [Pause and wait for the audience to shout out the answer.] Yeah, I might use that gag again, too.

I mean, c'mon. I've been doing this for over 13 years now. And Wikipedia still won't acknowledge it.

During a continuous month of blogging back in May of 2014, I already have a write-up of piecewise functions (along with a joke about the Indy 500).

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Come back often for more funny math and geeky comics.

June 2016 Common Core Algebra 1 Regents, Parts 3 and 4

What follows is a portion of the Common Core Integrate Algebra exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.

Part I is posted here.
Part II is posted here.

June 2016 Algebra Regents, Part III

33. The height, H, in feet, of an object dropped from the top of a building after t seconds is given by H(t) = -16t² + 144.
How many feet did the object fall between one and two seconds after it was dropped?
Determine, algebraically, how many seconds it will take for the object to reach the ground.

h(1) = -16(1)² + 144 = 128
h(2) = -16(2)² + 144 = 80
h(2) - h(1) = 128 - 80 = 48 feet between the 1st and 2nd second.

Solve for h(t) = 0.

-16t² + 144 = 0
-16(t² - 9) = 0
-16(t + 3)(t - 3) = 0
t = -3 or t = 3
It takes 3 seconds to reach the ground.

34. The sum of two numbers, x and y, is more than 8. When you double x and add it to y, the sum is less than 14.
Graph the inequalities that represent this scenario on the set of axes below.
Kai says that the point (6,2) is a solution to this system. Determine if he is correct and explain your reasoning.

The first inequality you need to graph is x + y > 8. The second one is 2x + y < 16.
The graph looks like this:

(graph will be uploaded later)

Looking at the graph, Kai is incorrect because (6, 2) is on a broken line which is not part of the solution set.

Note that if you drew the graph with solid lines, you lost a point for that. However, Kai would have been correct according to that mistaken graph. You have to be consistent.

35. An airplane leaves New York City and heads toward Los Angeles. As it climbs, the plane gradually increases its speed until it reaches cruising altitude, at which time it maintains a constant speed for several hours as long as it stays at cruising altitude. After flying for 32 minutes, the plane reaches cruising altitude and has flown 192 miles. After flying for a total of 92 minutes, the plane has flown a total of 762 miles.
Determine the speed of the plane, at cruising altitude, in miles per minute.
Write an equation to represent the number of miles the plane has flown, y, during x minutes at cruising altitude, only.
Assuming that the plane maintains its speed at cruising altitude, determine the total number of miles the plane has flown 2 hours into the flight.

There was much discussion over this question, but in the end, there was no arguing with the rubric for scoring this question. If you feel it's an unfair question, appeal to the state. In the meantime...

To find the speed at cruising altitude, use the two points given (32 minutes, 192 miles) and (92 minutes, 762 miles).
Speed in Miles per minute = changes in distance (miles) / change in time (minutes)
(762 - 192) / (92 - 32) = 9.5
If you showed your work, you have 1 point already.

The second part was where many students got caught up.
The word only was meant to apply to both the x and the y values, not just the x. In other words, you did not need to account for the distance traveled before reaching cruising altitude.
Because of this, the correct equation was y = 9.5x.
If you included "+ 192", you didn't get credit.

For the last part, you not only need to remember the initial 192 miles, but also the first 32 minutes of the flight. The question states that it is 2 hours into the flight, not 2 hours at cruising altitude. Also remember that you are dealing with miles per minute, so you need to convert.

2 hours = 120 minutes
120 - 32 = 88 minutes at cruising altitude
y = 9.5(88) + 192 = 836 + 192 = 1028 miles

36. On the set of axes below, graph:

How many values of x satisfy the equation f(x) = g(x)? Explain your answer, using evidence from your graphs.

(Graph will be posted later)

It is important that you had a break in the line at x = -1. The linear portion ends, the quadratic portion needed to have an open circle.
Most of the mistakes I saw fit into these categories:

• Draw three equations across the entire plane
• Connecting the two parts of the piecewise function
• Forgetting the open circle
• Shading the graph like it was an inequality
• Graphing a broken line for the quadratic because of the greater than symbol
• Saying that there were no solutions because the three lines didn't intersect at a single point (they didn't have to)

There was one solution because f(x) and g(x) only intersect one time.
You did not have to give the coordinates of the solution, and the solution was NOT a proper explanation. Seriously. You had to reference the fact that the lines only cross/intersect one time so there is one solution.

Also, if you had a graphing error, your final answer had to match the graph you drew. If, for example, you graph g(x) = -1/2x + 1, that would be one graphing error, but that line would intersect f(x) two times. Your answer had to match your graph.

June 2016 Algebra Regents, Part IV

37. Franco and Caryl went to a bakery to buy desserts. Franco bought 3 packages of cupcakes and 2 packages of brownies for $19. Caryl bought 2 packages of cupcakes and 4 packages of brownies for $24. Let x equal the price of one package of cupcakes and y equal the price of one package of brownies.
Write a system of equations that describes the given situation.
On the set of axes below, graph the system of equations.
Determine the exact cost of one package of cupcakes and the exact cost of one package of brownies in dollars and cents. Justify your solution.

The two equations were

3x + 2y = 19
2x + 4y = 24
Note: if you used c and b instead of x and y, you lost 1 point because the instructions said to use x and y.

(The graph will be loaded later.)

You could solve the system of equations using elimination. You could also solve them using the functions on the graphing calculator, but you needed to explain how you got your answer. A correct pair of answers without an explanation or procedure was only 1 point instead of 2.

Multiply the first equation by 2 and subtract

6x + 4y = 38
2x + 4y = 24
4x = 14
x = 3.50

2(3.50) + 4y = 24
7.00 + 4y = 24
4y = 17
y = 4.25

Cupcakes were 3.50 and brownies were 4.25.

Day 30 of 30: You Want a Piece of This Function?

This is day 30 of the 30-day blogging challenge. It's the checkered flag! Maybe I'll take tomorrow off.

I briefly thought about doing something concerned with the Indy 500, but it's almost 10pm ET as I write this and that race is long over. I'm sure the excitement isn't for those present, but it would be forced now. So, continuing from last night's discussion on functions, let's talk, by which I mean complain, about piecewise functions.

Okay, one Indy-related question:

Q: How many ways can 33 cars be arranged at the start of the race?

A: One. That's what Time Trials are for!

That could've been a comic for today had I had time to make it. Busy weekend. Busier with the cleaning and the grading. But back to functions.

Just explaining to a student what a piecewise function is is not a simple task. Explaining how to read one takes time and patience, along with repetition of the phrase, "when you see the comma, think 'when'". Oddly, I once said, "say 'when'", and it sounded like I was pouring beverages out. I had to switch that up.

Worst of all are the examples that they give. They make no sense whatsoever. They are purely abstract creations that you wonder if they might ever come up -- even just in another math class.

This isn't to say that piecewise functions aren't useful in the real world, or even in mathematics. I could even justify them in Algebra as opposed to waiting for Trigonometry/Algebra 2. But do they have to make them so confusing off the bat. (Hell, the name itself is confusing -- and might I add that it isn't even recognized by my spell checker!)

Examples of reasonable math functions that they could have brought up? First, the absolute value function, which looks like this:

"When" x is less than 0, you want to flip the sign, (i.e., take the negative of x because a negative of a negative number is positive). Otherwise, leave x alone. Note that the last condition has to cover all other possibilities in the domain. You don't want to leave a value out(*), and you definitely don't want to repeat a value, because then it won't be a function. (*) Yeah, there are times you'll leave something out, but not here, not now and not with absolute value!

Another one that they can use which makes for an interesting graph, but doesn't have any variables, is the Sign function, not to be confused with the Sine function:

Negative numbers return -1, positive numbers return positive 1 and zero returns 0. This was useful when I was programming computers, something that I'm glad I've done and something to which I'll refer often. Why not steer the kids in that direction if it's something challenging that might interest them? By the way, the Sign function would be the basis for some kind of trinary system of anything when binary gets boring.

So we have two good examples to start. So what do the books give us? Something like this:

Okay, maybe nothing that nuts, and maybe nothing with e or i in the exponent, but it might as well have been. Non-continuous functions that mean nothing even in the abstract.

On the other hand, finding relevant, relateable uses for piecewise functions was a little crazy. I could've tried my Financial Algebra textbook which is constantly trying to get the students to create some of these (and eventually did). And then there are the old standbys, which don't mean as much any more. I used to use the example of different phone plans when talking about systems of equations. This is easily adaptable into a piecewise function. There's a minimum charge for a certain number of minutes and then you have to pay, say, $.10 per minute after you used your allotment. There are two problems with this example: first, the minimum means that there will be a constant, a line with a slope of zero and no variable. Second, what kid in my class a) pays for their phone, or b) doesn't have unlimited minutes. Minute plans are already a thing of the past.

Pay phones are right behind them, but you can tell them that they're at the mall or the airports and they'll believe that they're there somewhere and just haven't noticed.

My other example suffers similar problems: if you live in an apartment and aren't allowed to have a washing machine, then you have to go a laundromat where you can do it yourself, or you can pay to have them do it for you. It's usually done by weight. I gave an example (and I haven't dropped off laundry in a long time) of a cost of $.60/pound with a $6.00 minimum, and let them figure out that you're paying for at least ten pounds whether or not you bring ten pounds. Same problem with the flat minimum and just as unrelateable.

On the other hand, if I put enough of these problems on the board, they'll see enough commas and say "When".

Like I am now because this is too much.

30 Days. When!