Integrated Math 2–3 Honors — Piecewise Definitions and Transformations of Absolute-Value Functions
Corrected lesson transcript


## 0:00 — Piecewise Definitions

[0:04] Last class we discussed the piecewise definition of the absolute value of x.
[0:10] Remember this?
[0:12] Okay, what was the definition?
[0:14] The absolute value of x equals x if x is greater than or equal to zero,
[0:20] and equals −x if x is less than zero.

## 0:43 — Example: |x - 3|

[0:43] What would be this?
[0:51] The piecewise definition of this.
[0:55] It's very similar.
[0:57] It is either x − 3 or −(x − 3).
[1:06] Is this true?
[1:25] There's some different discussions going on.
[1:32] Is this true?
[1:44] Let's pretend this is true.
[1:46] It's not.
[1:50] If I plug in 0 here, that's negative 3.
[1:57] That becomes 3, right?
[1:58] Yeah.
[2:00] But 0 is here.
[2:02] If I plug in 0 here, you get negative 3,
[2:06] which is not correct.
[2:09] OK, so this is the way we do it.
[2:15] It is equal to this.
[2:21] It's not x greater than or equal to 0
[2:23] if the whole inside is positive or 0.
[2:28] So we write if x minus three is greater than or equal to zero.
[2:36] And if x minus three is less than zero.
[2:52] And then x minus three, and you can either leave the negative on the outside or go
[2:58] So, distribute it, negative x plus 3.
[3:04] And then, let's, for each of these, let's solve for x. So,
[3:08] plus 3 plus 3, if x is greater than or equal to 3.
[3:19] If x is less than 3.
[3:26] Okay, so this is the piecewise definition of this absolute value expression, absolute value
[3:33] of x minus 3. The key is when you set up the inequality, you just take the entire inside
[3:44] and set the entire inside greater

## 3:46 — More Piecewise Examples

[4:05] The absolute value of 2x + 4 is 2x + 4 or −(2x + 4).
[4:07] is going to be 2x plus 4 or negative 2x plus 4. So it's 2x plus 4 if 2x plus 4 is greater
[4:25] than or equal to 0, or if 2x plus 4 is less than 0.
[4:34] We get 2x greater than or equal to negative 4 divided by 2.
[4:47] And then x is greater than or equal to negative two if x is less than negative two.
[5:08] Make sense?
[5:10] Yeah.
[5:10] Try this one. It's gonna be a little bit different because when we solve the inequality you're gonna have to...
[5:16] Sorry, yeah, divide by an a is what the size, try this one on your own.
[5:36] When we divide by negative two here, we have to flip.
[6:22] You get that?
[6:24] Yeah.
[6:25] Okay.

## 6:28 — The Parent Function y = |x|

[6:28] Okay.
[6:29] Okay.
[6:36] Oops.
[6:42] What is this thing I've written down?
[6:48] An equation. The true false or open?
[6:52] Are the solutions numbers or ordered pairs?
[6:55] Four pairs. Four pairs. Because?
[6:57] Two variables. Two variables.
[7:00] This is a
[7:02] absolute value
[7:03] Because the equation has two variables, its solutions are ordered pairs.
[7:06] Is it also a function?
[7:08] It definitely is.
[7:09] You can plug in any number for x and get
[7:11] one y value.
[7:13] So we call this
[7:15] absolute
[7:18] value
[7:21] functions.
[7:25] The solution set is a collection of ordered pairs, and we will graph it.
[7:32] solution set.
[7:38] Okay. In order to do this, we have to break it up into his two cases. Okay. Well, let's
[7:59] And what is the piecewise definition
[8:04] Ok, so this has two different pieces.
[8:07] The first piece is y equals x, if x is greater than or equal to zero.
[8:14] And the other piece is y equals negative x.
[8:20] And these are both linear functions.
[8:24] What's the slope of this one?
[8:26] One.
[8:27] what's the y-intercept?
[8:29] Zero.
[8:30] Zero, there's nothing here.
[8:33] OK, so it's this line right here.
[8:41] With the slope of one, y-intercept to zero.
[8:45] except this is only true if x is greater than or equal to 0.
[8:50] So we have to erase everything to the left of 0 here.
[9:05] So we erase all this guy.
[9:15] And then this guy is also a linear equation with a slope of negative 1 and a y intercept
[9:22] of zero.
[9:48] Which is that line, except it's only true when x is less than zero.
[9:52] So we got that it's only true for the left here until we erase all these
[10:06] This left one does have an open circle right here
[10:11] Because it's not true however
[10:13] It's already filled in that point for that with this function, so we don't put an open circle there
[10:25] There we go.
[10:27] There it is.
[10:27] That's the graph of the absolute value.
[10:31] It looks like it's a V looking thing with the slope on the right of 1 and slope on the
[10:37] left of 1.
[10:40] This is the solution set.
[10:43] This is the graph of this very important function.
[10:48] Did you guys graph these last year?

## 10:51 — Horizontal Translations and the Vertex

[10:51] Okay, so absolute value is a type of piecewise,
[10:58] Okay, let's try another one.
[11:15] Let's scope that.
[11:29] Okay, let's write the piecewise definition.
[11:44] So it is x, no, is x plus 3 if what?
[11:55] If x plus 3 is greater than or equal to 0.
[11:59] and it is −(x + 3) when x + 3 is less than zero.
[12:22] Okay, then we subtract three here, subtract three, subtract three, subtract three.
[12:37] I'm missing the equation.
[12:40] Which one? X plus three is greater than zero.
[12:43] yeah, just X plus three.
[12:47] Yeah, it's the equality sign.
[12:59] Oops, this is supposed to be less than, right? Is that what you were talking about? Thanks.
[13:12] Now we have y = x + 3 when x ≥ −3,
[13:15] and y = −x − 3 when x < −3.
[13:24] And we got y is negative x minus 3 if x is less than 0.
[13:39] This is a linear equation with the y-intercept of 3 and a slope of 1.
[13:57] Okay. But only if x is, did I write zero? Geez. If x is greater than or equal to three,
[14:10] negative 3 if x is less than negative 3. So here's negative 3 right here. It's only true
[14:22] when x is greater than that number, so we erase everything here.
[14:41] And then, this is also a linear equation with a slope of negative one when it steps to negative
[14:49] three but we gotta erase everything or keep it only when X is less than negative three
[15:02] only here.
[15:04] So we erase all this.
[15:14] There we go.
[15:26] How does the graph of this guy compare to the graph of this one?
[15:39] This moves to the left by how much?
[15:41] Three.
[15:42] Three units.
[15:45] What did we call last year when we moved?
[15:51] There's a type of transformation called a translation.
[15:54] We translated this function y equals absolute value of x to the left by three.
[16:04] Okay.
[16:06] And how can we tell that from here?
[16:15] Did you guys do many transformations last year of actual functions?
[16:25] Okay.
[16:29] That's okay.
[16:31] So, the only thing different, our previous one was y equals absolute value of x, right?
[16:36] so when you replace x when you replace x with x plus some number let's call it a
[16:58] Let me just go with 3.
[17:03] It just takes this function and shifts it.
[17:10] Everything left, the opposite of that sign, so it's positive, it goes left by 3 units.
[17:21] We're going to do lots of transformations of functions.
[17:30] I'll come back to that.
[17:38] This little point right here is called the vertex.
[17:43] I don't know if we learned that last year.
[17:45] okay so here now the vertex is at negative 3 comma 0 instead of 0 comma 0
[17:56] all right let's try another one

## 17:59 — Vertex Form: y = a|x - h| + k

[18:00] Okay, let's do its piecewise definition.
[18:08] It's going to be x minus 2 minus 4 if the inside x minus 2 is greater than or equal
[18:23] to zero.
[18:25] The other piece is −(x − 2) − 4 when x − 2 is less than zero.
[18:46] We always want y by itself.
[18:49] Oh
[18:52] You could but then I get super complicated
[18:56] Okay
[18:58] Right, so this this guy right here is this piece and this piece or not piece but the
[19:06] expression and then the minus four
[19:08] Just stays on the outside
[19:12] And now let's add two plus two plus two plus two
[19:17] And then let's combine like terms here.
[19:24] We get x minus 6.
[19:29] If x is greater than or equal to 2.
[19:34] And negative x plus 2 minus 4. Is that minus 2?
[19:44] less than 2.
[19:52] So again we have two different pieces.
[19:54] One is y equals x minus 6.
[19:59] One piece is y = x − 6 when x ≥ 2.
[20:04] The other piece is y = −x − 2 when x < 2.
[20:07] minus 2
[20:11] x is less than 2
[20:17] again each piece here is a linear equation
[20:26] with a slope of 1
[20:27] and a y-intercept for negative 6
[20:33] uh oh, it's way down here
[20:53] except we need... it's only true when x is greater than zero, so...
[21:00] It's greater than negative 2, so we erase this part.
[21:05] Ah!
[21:23] This is a way of accepting negative 2.
[21:32] That's something like that.
[21:50] So compared to our function y equals the absolute value of x,
[21:55] What did the replacing x with x minus 2 do to it?
[22:01] Right, this was y equals absolute value of x.
[22:08] It moved it to the right or left? Right by 2.
[22:14] Like this.
[22:16] And then what did minus 4 on the outside of the function do?
[22:19] Move it down four.
[22:20] Move it down four.
[22:26] OK.
[22:28] So the new vertex is at 2 comma negative 4.
[22:45] is that always going to happen
[22:47] when we have x plus or minus number? It is. You can immediately look at this,
[22:53] you know where the vertex is. It's always the opposite of that number, comma that
[23:01] number. So it's gonna be negative negative 2 comma 4, or in other words,
[23:08] sorry, negative 4. 2 negative 4. So we're actually gonna graph these much faster
[23:15] without doing all the piecewise definition. We're just going to find the vertex and then
[23:24] figure out what the slope is. And what is the slope of this guy? There's no coefficient,
[23:33] well the coefficient is 1 and there's no number out here at 1. So the slope is going to be
[23:37] 1. So once we find the vertex, we just know we go to the right and go to the left.

## 23:44 — Vertical Stretch and Compression

[23:44] So the piecewise definition, it's either going to be 2 times x minus 4 if x is greater than
[23:57] to zero and 2 times negative x minus 4.
[25:06] So these are both linear equations.
[25:10] Y is up to negative four.
[25:13] But now the slope is two.
[25:15] So we rise two, run one.
[25:17] Rise two, run one.
[25:24] Oops, that's the green.
[25:39] There it is.
[25:41] So it's still a V, but it's uh, the slopes are changed.
[25:46] Okay, this is another type of transformation.
[25:51] What do we call this type of transformation?
[25:54] It definitely translated down by four.
[25:56] it translate left or right?
[25:59] No because there was no, there was nothing, it's just the x inside the function.
[26:08] It translated down by four.
[26:10] What did the two do to it?
[26:13] Definitely changed the slope.
[26:14] but what kind of transformations that go a dilation okay sometimes we call it a
[26:28] stretch because we're only dilating in one direction only in the y direction so
[26:34] So it actually, like if we took y, well, y equals 2 without the minus 4, it takes this
[26:48] function and multiplies all the y values by 2.
[26:54] So it moved up here.
[26:55] This one moved up to 4.
[26:58] It stretched it.
[27:01] Okay, and it stretched it, and then we graphed that and shifted it down by four.
[27:10] Okay.
[27:15] So.
[27:18] Let's not even write the piecewise definition.
[27:24] Let's just use transformations.
[27:32] This determines left to right, replacing x with x plus one.
[27:37] So what does it go?
[27:39] So the opposite of that number.
[27:41] So it goes left one.
[27:42] So we're going left one unit.
[27:49] What does the minus 2 do?
[27:51] Down 2.
[27:52] Down 2.
[27:57] And then what does this do?
[28:00] Stretch 3.
[28:02] With linear equations, stretching by a factor of 3
[28:05] is really just changing the slope to 3.
[28:08] So it stretches 3.
[28:19] OK, so the old vertex, 0, 0, of y equals absolute value of x is only affected by the left and
[28:29] the down.
[28:31] So this is going to go to negative one, left one, and down to negative two.
[28:43] So this is our vertex.
[28:51] And then for our two values, this determines our slope.
[28:56] It's going to be three and negative three.
[28:58] So, when we go to the right, we go up 3 over 1.
[29:04] Up 3 over 1.
[29:11] Then to the left, it's going to be up 3, left 1.
[29:25] So, we can graph these quickly with transformations if you understand each part and how it affects
[29:31] it.
[29:41] All right, let's try another one.
[29:44] Okay.
[29:46] So the x minus 3 is going to shift everything.
[29:51] Right 3.
[29:56] So 0, 0, our vertex goes to 3.
[30:08] And then the plus 1 does what?
[30:10] Up 1.
[30:12] instead of a 0, 0, it's going to be 3, 1.
[30:17] is our new vertex. And then what's our slope going to be? Our slopes are going to be 1
[30:33] half and negative 1 half. So rise 1, run 2.
[30:52] So the one half actually says stretched it, compressed it by one half.
[30:58] Or say it's stretched by a factor of one half, which means it compressed it.
[31:01] And there's the quick graph.

## 31:05 — Reflections Across the x-axis

[31:06] Where is the new vertex going to be?
[31:13] Goes to the right by 3.
[31:18] So 3, and there's no number on the outside,
[31:21] so it's not going up or down.
[31:24] So 3, 0.
[31:29] How is this negative going to affect it?
[31:34] Y.
[31:37] Right.
[31:39] A negative coefficient reflects the graph across the x-axis.
[31:43] and makes them the opposite.
[31:44] it's a reflection about the x-axis. Okay, so if we broke down the piecewise real
[31:57] For x ≥ 3, the piece is y = −2(x − 3).
[32:05] For x < 3, the piece is y = −2(−x + 3).
[32:27] zero if I simplify that.
[32:48] Okay, so to the right of three
[32:52] is the part that has the negative slope now.
[32:55] To the left of 3 it has the positive slope.
[33:00] So it's going to do this instead.
[33:11] So the negative coefficient of... simply makes it an upside down V instead of a regular V.
[33:47] Where is the new vertex going to be?
[33:52] opposite of the inside number, comma the outside number.
[33:54] Negative two, five.
[34:02] And then the slopes are going to be four and negative four.
[34:07] It can be an upside down V or a regular V. Upside down because of that, this negative.
[34:20] So,
[34:37] That's kind of what I got.
[34:39] And we didn't need to write any of the piecewise definitions to figure that out.
[34:50] Once you know where the vertex is and how this affects the slope.
[34:55] It's easy.
[34:58] Sound good?
[34:58] Yeah.
[34:59] Okay.
[35:00] Okay.

## 35:05 — Write an Equation from a Graph

[35:06] Now what's the equation?
[35:16] Okay, so what's the coefficient in front of the absolute value going to be?
[35:23] It's going to be 2. How can we tell?
[35:27] Rise 2, run 1.
[35:30] Okay, is it positive 2 or negative 2?
[35:32] Because the B's going up, it's definitely going to be positive 2.
[35:39] Times the absolute value.
[35:43] We went left 3, so we go x plus 3, or x minus the negative 3, and then we shift it up by
[35:54] 1.
[35:54] That's it.
[35:58] That's cool.
[36:02] Any questions on that?
[36:03] Does this make sense?
[36:04] Alright, let's try another one.
[36:14] So the slope is going to be...
[36:21] So y is 2, y is 3.
[36:25] Definitely negative.
[36:26] Put the V opens down, negative two thirds times the absolute value of one, two, three, four,
[36:38] x minus four plus two.
[36:43] Easy peasy.