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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.

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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.

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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?

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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

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number. So it's gonna be negative negative 2 comma 4, or in other words,

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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

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figure out what the slope is. And what is the slope of this guy? There's no coefficient,

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well the coefficient is 1 and there's no number out here at 1. So the slope is going to be

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1. So once we find the vertex, we just know we go to the right and go to the left.

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So the piecewise definition, it's either going to be 2 times x minus 4 if x is greater than

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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.

207
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There it is.

208
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So it's still a V, but it's uh, the slopes are changed.

209
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Okay, this is another type of transformation.

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What do we call this type of transformation?

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It definitely translated down by four.

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it translate left or right?

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No because there was no, there was nothing, it's just the x inside the function.

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It translated down by four.

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What did the two do to it?

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Definitely changed the slope.

217
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but what kind of transformations that go a dilation okay sometimes we call it a

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stretch because we're only dilating in one direction only in the y direction so

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So it actually, like if we took y, well, y equals 2 without the minus 4, it takes this

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function and multiplies all the y values by 2.

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So it moved up here.

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This one moved up to 4.

223
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It stretched it.

224
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Okay, and it stretched it, and then we graphed that and shifted it down by four.

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Okay.

226
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So.

227
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Let's not even write the piecewise definition.

228
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Let's just use transformations.

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This determines left to right, replacing x with x plus one.

230
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So what does it go?

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So the opposite of that number.

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So it goes left one.

233
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So we're going left one unit.

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What does the minus 2 do?

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Down 2.

236
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Down 2.

237
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And then what does this do?

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Stretch 3.

239
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With linear equations, stretching by a factor of 3

240
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is really just changing the slope to 3.

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So it stretches 3.

242
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

243
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the down.

244
00:28:31,510 --> 00:28:40,370
So this is going to go to negative one, left one, and down to negative two.

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So this is our vertex.

246
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And then for our two values, this determines our slope.

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It's going to be three and negative three.

248
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So, when we go to the right, we go up 3 over 1.

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Up 3 over 1.

250
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Then to the left, it's going to be up 3, left 1.

251
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So, we can graph these quickly with transformations if you understand each part and how it affects

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it.

253
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All right, let's try another one.

254
00:29:44,390 --> 00:29:45,140
Okay.

255
00:29:46,890 --> 00:29:50,810
So the x minus 3 is going to shift everything.

256
00:29:51,330 --> 00:29:52,890
Right 3.

257
00:29:56,110 --> 00:30:06,090
So 0, 0, our vertex goes to 3.

258
00:30:08,250 --> 00:30:09,870
And then the plus 1 does what?

259
00:30:10,950 --> 00:30:12,350
Up 1.

260
00:30:12,810 --> 00:30:17,310
instead of a 0, 0, it's going to be 3, 1.

261
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

262
00:30:33,690 --> 00:30:41,390
half and negative 1 half. So rise 1, run 2.

263
00:30:52,750 --> 00:30:57,790
So the one half actually says stretched it, compressed it by one half.

264
00:30:58,570 --> 00:31:01,890
Or say it's stretched by a factor of one half, which means it compressed it.

265
00:31:01,890 --> 00:31:05,370
And there's the quick graph.

266
00:31:06,290 --> 00:31:08,950
Where is the new vertex going to be?

267
00:31:13,870 --> 00:31:16,210
Goes to the right by 3.

268
00:31:18,390 --> 00:31:21,270
So 3, and there's no number on the outside,

269
00:31:21,270 --> 00:31:22,910
so it's not going up or down.

270
00:31:24,250 --> 00:31:25,130
So 3, 0.

271
00:31:29,030 --> 00:31:31,930
How is this negative going to affect it?

272
00:31:34,550 --> 00:31:35,300
Y.

273
00:31:37,130 --> 00:31:37,880
Right.

274
00:31:39,010 --> 00:31:43,350
A negative coefficient reflects the graph across the x-axis.

275
00:31:43,350 --> 00:31:44,730
and makes them the opposite.

276
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

277
00:31:57,730 --> 00:32:05,450
For x ≥ 3, the piece is y = −2(x − 3).

278
00:32:05,450 --> 00:32:27,770
For x < 3, the piece is y = −2(−x + 3).

279
00:32:27,770 --> 00:32:30,770
zero if I simplify that.

280
00:32:48,010 --> 00:32:51,230
Okay, so to the right of three

281
00:32:52,190 --> 00:32:54,870
is the part that has the negative slope now.

282
00:32:55,990 --> 00:32:58,810
To the left of 3 it has the positive slope.

283
00:33:00,310 --> 00:33:03,550
So it's going to do this instead.

284
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.

285
00:33:47,390 --> 00:33:50,050
Where is the new vertex going to be?

286
00:33:52,230 --> 00:33:53,830
opposite of the inside number, comma the outside number.

287
00:33:54,450 --> 00:33:55,550
Negative two, five.

288
00:34:02,070 --> 00:34:07,470
And then the slopes are going to be four and negative four.

289
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.

290
00:34:20,130 --> 00:34:20,880
So,

291
00:34:37,530 --> 00:34:38,370
That's kind of what I got.

292
00:34:39,330 --> 00:34:46,630
And we didn't need to write any of the piecewise definitions to figure that out.

293
00:34:50,170 --> 00:34:53,110
Once you know where the vertex is and how this affects the slope.

294
00:34:55,790 --> 00:34:56,730
It's easy.

295
00:34:58,150 --> 00:34:58,900
Sound good?

296
00:34:58,990 --> 00:34:59,740
Yeah.

297
00:34:59,750 --> 00:35:00,500
Okay.

298
00:35:00,110 --> 00:35:00,860
Okay.

299
00:35:06,130 --> 00:35:07,630
Now what's the equation?

300
00:35:16,590 --> 00:35:22,670
Okay, so what's the coefficient in front of the absolute value going to be?

301
00:35:23,630 --> 00:35:25,270
It's going to be 2. How can we tell?

302
00:35:27,730 --> 00:35:29,130
Rise 2, run 1.

303
00:35:30,210 --> 00:35:31,510
Okay, is it positive 2 or negative 2?

304
00:35:32,330 --> 00:35:38,070
Because the B's going up, it's definitely going to be positive 2.

305
00:35:39,090 --> 00:35:40,890
Times the absolute value.

306
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

307
00:35:54,490 --> 00:35:55,240
1.

308
00:35:54,770 --> 00:35:55,520
That's it.

309
00:35:58,250 --> 00:35:59,170
That's cool.

310
00:36:02,730 --> 00:36:03,530
Any questions on that?

311
00:36:03,670 --> 00:36:04,420
Does this make sense?

312
00:36:04,950 --> 00:36:09,570
Alright, let's try another one.

313
00:36:14,490 --> 00:36:17,570
So the slope is going to be...

314
00:36:21,190 --> 00:36:23,850
So y is 2, y is 3.

315
00:36:25,630 --> 00:36:26,410
Definitely negative.

316
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,

317
00:36:38,510 --> 00:36:42,390
x minus four plus two.

318
00:36:43,050 --> 00:36:43,910
Easy peasy.