WEBVTT



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Last class we discussed the piecewise definition of the absolute value of x.

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Remember this?

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Okay, what was the definition?

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The absolute value of x equals x if x is greater than or equal to zero,

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and equals −x if x is less than zero.

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What would be this?

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The piecewise definition of this.

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It's very similar.

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It is either x − 3 or −(x − 3).

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Is this true?

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There's some different discussions going on.

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Is this true?

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Let's pretend this is true.

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It's not.

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If I plug in 0 here, that's negative 3.

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That becomes 3, right?

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Yeah.

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But 0 is here.

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If I plug in 0 here, you get negative 3,

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which is not correct.

00:02:09.060 --> 00:02:11.140
OK, so this is the way we do it.

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It is equal to this.

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It's not x greater than or equal to 0

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if the whole inside is positive or 0.

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So we write if x minus three is greater than or equal to zero.

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And if x minus three is less than zero.

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And then x minus three, and you can either leave the negative on the outside or go

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So, distribute it, negative x plus 3.

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And then, let's, for each of these, let's solve for x. So,

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plus 3 plus 3, if x is greater than or equal to 3.

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If x is less than 3.

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Okay, so this is the piecewise definition of this absolute value expression, absolute value

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of x minus 3. The key is when you set up the inequality, you just take the entire inside

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and set the entire inside greater

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The absolute value of 2x + 4 is 2x + 4 or −(2x + 4).

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is going to be 2x plus 4 or negative 2x plus 4. So it's 2x plus 4 if 2x plus 4 is greater

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than or equal to 0, or if 2x plus 4 is less than 0.

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We get 2x greater than or equal to negative 4 divided by 2.

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And then x is greater than or equal to negative two if x is less than negative two.

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Make sense?

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Yeah.

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Try this one. It's gonna be a little bit different because when we solve the inequality you're gonna have to...

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Sorry, yeah, divide by an a is what the size, try this one on your own.

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When we divide by negative two here, we have to flip.

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You get that?

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Yeah.

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Okay.

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Okay.

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Okay.

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Oops.

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What is this thing I've written down?

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An equation. The true false or open?

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Are the solutions numbers or ordered pairs?

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Four pairs. Four pairs. Because?

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Two variables. Two variables.

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This is a

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absolute value

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Because the equation has two variables, its solutions are ordered pairs.

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Is it also a function?

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It definitely is.

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You can plug in any number for x and get

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one y value.

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So we call this

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absolute

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value

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functions.

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The solution set is a collection of ordered pairs, and we will graph it.

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solution set.

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Okay. In order to do this, we have to break it up into his two cases. Okay. Well, let's

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And what is the piecewise definition

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Ok, so this has two different pieces.

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The first piece is y equals x, if x is greater than or equal to zero.

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And the other piece is y equals negative x.

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And these are both linear functions.

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What's the slope of this one?

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One.

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what's the y-intercept?

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Zero.

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Zero, there's nothing here.

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OK, so it's this line right here.

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With the slope of one, y-intercept to zero.

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except this is only true if x is greater than or equal to 0.

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So we have to erase everything to the left of 0 here.

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So we erase all this guy.

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And then this guy is also a linear equation with a slope of negative 1 and a y intercept

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of zero.

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Which is that line, except it's only true when x is less than zero.

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So we got that it's only true for the left here until we erase all these

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This left one does have an open circle right here

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Because it's not true however

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It's already filled in that point for that with this function, so we don't put an open circle there

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There we go.

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There it is.

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That's the graph of the absolute value.

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It looks like it's a V looking thing with the slope on the right of 1 and slope on the

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left of 1.

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This is the solution set.

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This is the graph of this very important function.

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Did you guys graph these last year?

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Okay, so absolute value is a type of piecewise,

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Okay, let's try another one.

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Let's scope that.

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Okay, let's write the piecewise definition.

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So it is x, no, is x plus 3 if what?

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If x plus 3 is greater than or equal to 0.

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and it is −(x + 3) when x + 3 is less than zero.

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Okay, then we subtract three here, subtract three, subtract three, subtract three.

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I'm missing the equation.

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Which one? X plus three is greater than zero.

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yeah, just X plus three.

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Yeah, it's the equality sign.

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Oops, this is supposed to be less than, right? Is that what you were talking about? Thanks.

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Now we have y = x + 3 when x ≥ −3,

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and y = −x − 3 when x < −3.

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And we got y is negative x minus 3 if x is less than 0.

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This is a linear equation with the y-intercept of 3 and a slope of 1.

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Okay. But only if x is, did I write zero? Geez. If x is greater than or equal to three,

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negative 3 if x is less than negative 3. So here's negative 3 right here. It's only true

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when x is greater than that number, so we erase everything here.

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And then, this is also a linear equation with a slope of negative one when it steps to negative

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three but we gotta erase everything or keep it only when X is less than negative three

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only here.

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So we erase all this.

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There we go.

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How does the graph of this guy compare to the graph of this one?

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This moves to the left by how much?

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Three.

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Three units.

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What did we call last year when we moved?

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There's a type of transformation called a translation.

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We translated this function y equals absolute value of x to the left by three.

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Okay.

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And how can we tell that from here?

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Did you guys do many transformations last year of actual functions?

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Okay.

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That's okay.

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So, the only thing different, our previous one was y equals absolute value of x, right?

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so when you replace x when you replace x with x plus some number let's call it a

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Let me just go with 3.

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It just takes this function and shifts it.

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Everything left, the opposite of that sign, so it's positive, it goes left by 3 units.

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We're going to do lots of transformations of functions.

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I'll come back to that.

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This little point right here is called the vertex.

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I don't know if we learned that last year.

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okay so here now the vertex is at negative 3 comma 0 instead of 0 comma 0

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all right let's try another one

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Okay, let's do its piecewise definition.

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It's going to be x minus 2 minus 4 if the inside x minus 2 is greater than or equal

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to zero.

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The other piece is −(x − 2) − 4 when x − 2 is less than zero.

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We always want y by itself.

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Oh

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You could but then I get super complicated

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Okay

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Right, so this this guy right here is this piece and this piece or not piece but the

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expression and then the minus four

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Just stays on the outside

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And now let's add two plus two plus two plus two

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And then let's combine like terms here.

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We get x minus 6.

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If x is greater than or equal to 2.

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And negative x plus 2 minus 4. Is that minus 2?

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less than 2.

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So again we have two different pieces.

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One is y equals x minus 6.

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One piece is y = x − 6 when x ≥ 2.

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The other piece is y = −x − 2 when x < 2.

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minus 2

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x is less than 2

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again each piece here is a linear equation

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with a slope of 1

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and a y-intercept for negative 6

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uh oh, it's way down here

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except we need... it's only true when x is greater than zero, so...

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It's greater than negative 2, so we erase this part.

00:21:05.700 --> 00:21:06.450
Ah!

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This is a way of accepting negative 2.

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That's something like that.

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So compared to our function y equals the absolute value of x,

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What did the replacing x with x minus 2 do to it?

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Right, this was y equals absolute value of x.

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It moved it to the right or left? Right by 2.

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Like this.

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And then what did minus 4 on the outside of the function do?

00:22:19.240 --> 00:22:20.620
Move it down four.

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Move it down four.

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OK.

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So the new vertex is at 2 comma negative 4.

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is that always going to happen

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when we have x plus or minus number? It is. You can immediately look at this,

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you know where the vertex is. It's always the opposite of that number, comma that

00:23:01.020 --> 00:23:07.360
number. So it's gonna be negative negative 2 comma 4, or in other words,

00:23:08.260 --> 00:23:15.260
sorry, negative 4. 2 negative 4. So we're actually gonna graph these much faster

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without doing all the piecewise definition. We're just going to find the vertex and then

00:23:24.900 --> 00:23:32.320
figure out what the slope is. And what is the slope of this guy? There's no coefficient,

00:23:33.680 --> 00:23:37.860
well the coefficient is 1 and there's no number out here at 1. So the slope is going to be

00:23:37.860 --> 00:23:44.400
1. So once we find the vertex, we just know we go to the right and go to the left.

00:23:44.540 --> 00:23:56.720
So the piecewise definition, it's either going to be 2 times x minus 4 if x is greater than

00:23:57.110 --> 00:24:03.710
to zero and 2 times negative x minus 4.

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So these are both linear equations.

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Y is up to negative four.

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But now the slope is two.

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So we rise two, run one.

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Rise two, run one.

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Oops, that's the green.

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There it is.

00:25:41.990 --> 00:25:46.990
So it's still a V, but it's uh, the slopes are changed.

00:25:46.990 --> 00:25:50.590
Okay, this is another type of transformation.

00:25:51.570 --> 00:25:53.510
What do we call this type of transformation?

00:25:54.510 --> 00:25:56.550
It definitely translated down by four.

00:25:56.750 --> 00:25:58.190
it translate left or right?

00:25:59.270 --> 00:26:05.990
No because there was no, there was nothing, it's just the x inside the function.

00:26:08.270 --> 00:26:09.670
It translated down by four.

00:26:10.770 --> 00:26:12.250
What did the two do to it?

00:26:13.630 --> 00:26:14.930
Definitely changed the slope.

00:26:14.930 --> 00:26:28.050
but what kind of transformations that go a dilation okay sometimes we call it a

00:26:28.050 --> 00:26:34.490
stretch because we're only dilating in one direction only in the y direction so

00:26:34.490 --> 00:26:48.090
So it actually, like if we took y, well, y equals 2 without the minus 4, it takes this

00:26:48.690 --> 00:26:54.270
function and multiplies all the y values by 2.

00:26:54.410 --> 00:26:55.270
So it moved up here.

00:26:55.410 --> 00:26:56.910
This one moved up to 4.

00:26:58.590 --> 00:27:00.190
It stretched it.

00:27:01.810 --> 00:27:07.030
Okay, and it stretched it, and then we graphed that and shifted it down by four.

00:27:10.930 --> 00:27:12.330
Okay.

00:27:15.310 --> 00:27:16.710
So.

00:27:18.070 --> 00:27:21.490
Let's not even write the piecewise definition.

00:27:24.190 --> 00:27:26.190
Let's just use transformations.

00:27:32.090 --> 00:27:36.270
This determines left to right, replacing x with x plus one.

00:27:37.130 --> 00:27:37.990
So what does it go?

00:27:39.850 --> 00:27:41.890
So the opposite of that number.

00:27:41.990 --> 00:27:42.930
So it goes left one.

00:27:42.930 --> 00:27:46.970
So we're going left one unit.

00:27:49.850 --> 00:27:51.530
What does the minus 2 do?

00:27:51.670 --> 00:27:52.420
Down 2.

00:27:52.830 --> 00:27:53.690
Down 2.

00:27:57.850 --> 00:27:59.190
And then what does this do?

00:28:00.670 --> 00:28:01.670
Stretch 3.

00:28:02.250 --> 00:28:05.090
With linear equations, stretching by a factor of 3

00:28:05.090 --> 00:28:07.210
is really just changing the slope to 3.

00:28:08.510 --> 00:28:09.850
So it stretches 3.

00:28:19.790 --> 00:28:29.590
OK, so the old vertex, 0, 0, of y equals absolute value of x is only affected by the left and

00:28:29.590 --> 00:28:30.340
the down.

00:28:31.510 --> 00:28:40.370
So this is going to go to negative one, left one, and down to negative two.

00:28:43.350 --> 00:28:44.810
So this is our vertex.

00:28:51.810 --> 00:28:54.690
And then for our two values, this determines our slope.

00:28:56.070 --> 00:28:57.610
It's going to be three and negative three.

00:28:58.850 --> 00:29:03.410
So, when we go to the right, we go up 3 over 1.

00:29:04.710 --> 00:29:06.470
Up 3 over 1.

00:29:11.050 --> 00:29:15.650
Then to the left, it's going to be up 3, left 1.

00:29:25.390 --> 00:29:31.870
So, we can graph these quickly with transformations if you understand each part and how it affects

00:29:31.870 --> 00:29:32.650
it.

00:29:41.210 --> 00:29:44.390
All right, let's try another one.

00:29:44.390 --> 00:29:45.140
Okay.

00:29:46.890 --> 00:29:50.810
So the x minus 3 is going to shift everything.

00:29:51.330 --> 00:29:52.890
Right 3.

00:29:56.110 --> 00:30:06.090
So 0, 0, our vertex goes to 3.

00:30:08.250 --> 00:30:09.870
And then the plus 1 does what?

00:30:10.950 --> 00:30:12.350
Up 1.

00:30:12.810 --> 00:30:17.310
instead of a 0, 0, it's going to be 3, 1.

00:30:17.310 --> 00:30:33.690
is our new vertex. And then what's our slope going to be? Our slopes are going to be 1

00:30:33.690 --> 00:30:41.390
half and negative 1 half. So rise 1, run 2.

00:30:52.750 --> 00:30:57.790
So the one half actually says stretched it, compressed it by one half.

00:30:58.570 --> 00:31:01.890
Or say it's stretched by a factor of one half, which means it compressed it.

00:31:01.890 --> 00:31:05.370
And there's the quick graph.

00:31:06.290 --> 00:31:08.950
Where is the new vertex going to be?

00:31:13.870 --> 00:31:16.210
Goes to the right by 3.

00:31:18.390 --> 00:31:21.270
So 3, and there's no number on the outside,

00:31:21.270 --> 00:31:22.910
so it's not going up or down.

00:31:24.250 --> 00:31:25.130
So 3, 0.

00:31:29.030 --> 00:31:31.930
How is this negative going to affect it?

00:31:34.550 --> 00:31:35.300
Y.

00:31:37.130 --> 00:31:37.880
Right.

00:31:39.010 --> 00:31:43.350
A negative coefficient reflects the graph across the x-axis.

00:31:43.350 --> 00:31:44.730
and makes them the opposite.

00:31:44.730 --> 00:31:57.730
it's a reflection about the x-axis. Okay, so if we broke down the piecewise real

00:31:57.730 --> 00:32:05.450
For x ≥ 3, the piece is y = −2(x − 3).

00:32:05.450 --> 00:32:27.770
For x < 3, the piece is y = −2(−x + 3).

00:32:27.770 --> 00:32:30.770
zero if I simplify that.

00:32:48.010 --> 00:32:51.230
Okay, so to the right of three

00:32:52.190 --> 00:32:54.870
is the part that has the negative slope now.

00:32:55.990 --> 00:32:58.810
To the left of 3 it has the positive slope.

00:33:00.310 --> 00:33:03.550
So it's going to do this instead.

00:33:11.010 --> 00:33:20.670
So the negative coefficient of... simply makes it an upside down V instead of a regular V.

00:33:47.390 --> 00:33:50.050
Where is the new vertex going to be?

00:33:52.230 --> 00:33:53.830
opposite of the inside number, comma the outside number.

00:33:54.450 --> 00:33:55.550
Negative two, five.

00:34:02.070 --> 00:34:07.470
And then the slopes are going to be four and negative four.

00:34:07.470 --> 00:34:15.530
It can be an upside down V or a regular V. Upside down because of that, this negative.

00:34:20.130 --> 00:34:20.880
So,

00:34:37.530 --> 00:34:38.370
That's kind of what I got.

00:34:39.330 --> 00:34:46.630
And we didn't need to write any of the piecewise definitions to figure that out.

00:34:50.170 --> 00:34:53.110
Once you know where the vertex is and how this affects the slope.

00:34:55.790 --> 00:34:56.730
It's easy.

00:34:58.150 --> 00:34:58.900
Sound good?

00:34:58.990 --> 00:34:59.740
Yeah.

00:34:59.750 --> 00:35:00.500
Okay.

00:35:00.110 --> 00:35:00.860
Okay.

00:35:06.130 --> 00:35:07.630
Now what's the equation?

00:35:16.590 --> 00:35:22.670
Okay, so what's the coefficient in front of the absolute value going to be?

00:35:23.630 --> 00:35:25.270
It's going to be 2. How can we tell?

00:35:27.730 --> 00:35:29.130
Rise 2, run 1.

00:35:30.210 --> 00:35:31.510
Okay, is it positive 2 or negative 2?

00:35:32.330 --> 00:35:38.070
Because the B's going up, it's definitely going to be positive 2.

00:35:39.090 --> 00:35:40.890
Times the absolute value.

00:35:43.970 --> 00:35:54.490
We went left 3, so we go x plus 3, or x minus the negative 3, and then we shift it up by

00:35:54.490 --> 00:35:55.240
1.

00:35:54.770 --> 00:35:55.520
That's it.

00:35:58.250 --> 00:35:59.170
That's cool.

00:36:02.730 --> 00:36:03.530
Any questions on that?

00:36:03.670 --> 00:36:04.420
Does this make sense?

00:36:04.950 --> 00:36:09.570
Alright, let's try another one.

00:36:14.490 --> 00:36:17.570
So the slope is going to be...

00:36:21.190 --> 00:36:23.850
So y is 2, y is 3.

00:36:25.630 --> 00:36:26.410
Definitely negative.

00:36:26.410 --> 00:36:36.690
Put the V opens down, negative two thirds times the absolute value of one, two, three, four,

00:36:38.510 --> 00:36:42.390
x minus four plus two.

00:36:43.050 --> 00:36:43.910
Easy peasy.