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- [Instructor] The graph of y

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is equal to absolute value of x

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is reflected across the x-axis

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and then scaled vertically[br]by a factor of seven.

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What is the equation of the new graph?

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So pause the video and see[br]if you can figure that out.

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Alright, let's work[br]through it together now.

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Now, you might not need[br]to draw it visually

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but I will just so that[br]we can all together

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visualize what is going on.

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So let's say that's my x-axis

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and that is my y-axis.

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y equals the absolute value of x.

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So for non-negative values of x,

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y is going to be equal to x.

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Absolute value of zero is zero.

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Absolute value of one is one.

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Absolute value of two is two.

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So it's gonna look like this.

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It's gonna have a slope of one

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and then for negative values,

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when you take the absolute value,

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you're gonna take the opposite.

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You're gonna get the positive.

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So it's gonna look like this.

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Let me see if I can draw[br]that a little bit cleaner.

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This is a hand drawn[br]sketch so bear with me

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but hopefully this is familiar.

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You've seen the graph

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of y is equal to absolute[br]value of x before.

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Now, let's think about the[br]different transformations.

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So first, they say is[br]reflected across the x-axis.

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So for example,

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if I have some x value right over here,

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before, I would take[br]the absolute value of x

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and I would end up there

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but now we wanna reflect across the x-axis

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so we wanna essentially get[br]the negative of that value

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associated with that corresponding x

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and so for example, this x,

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before, we would get[br]the absolute value of x

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but now we wanna flip across the x-axis

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and we wanna get the negative of it.

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So in general, what we are doing

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is we are getting the negative[br]of the absolute value of x.

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In general, if you're[br]flipping over the x-axis,

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you're getting the negative.

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You're scaling the expression[br]or the function by a negative.

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So this is going to be y[br]is equal to the negative

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of the absolute value of x.

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Once again, whatever absolute value of x

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was giving you before for given x,

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we now wanna get the negative of it.

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We now wanna get the negative of it.

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So that's what reflecting[br]across the x-axis does for us

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but then they say scaled[br]vertically by a factor of seven

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and the way I view that is if[br]you're scaling it vertically

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by a factor of seven,

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whatever y value you got for given x,

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you now wanna get seven times the y value,

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seven times the y value

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for a given x.

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And so if you think[br]about that algebraically,

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well, if I want seven times the y value,

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I'd have to multiply this thing by seven.

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So I would get y is[br]equal to negative seven

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times the absolute value of x

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and that's essentially[br]what they're asking,

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what is the equation of the new graph,

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and so that's what it would be.

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The negative flips us over the x-axis

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and then the seven scales[br]vertically by a factor of seven

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but just to understand[br]what this would look like,

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well, you multiply zero times seven,

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it doesn't change anything

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but whatever x this is,

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this was equal to negative x

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but now we're gonna get[br]to negative seven x.

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So let's see, two, three,[br]four, five, six, seven

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so it'd put it something around that.

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So our graph is now going to look,

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is now going to look like this.

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It's going to be stretched[br]along the vertical axis.

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If we were scaling vertically

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by something that had an[br]absolute value less than one

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then it would make the graph less tall.

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It would make it look,

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it would make it look wider.

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Let me make it at least look[br]a little bit more symmetric.

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So it's gonna look something,

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something like that but the key issue

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and the reason why I'm[br]drawing is so you can see

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that it looks like it's[br]being scaled vertically.

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It's being stretched in[br]the vertical direction

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by a factor of seven and the[br]way we do that algebraically

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is we multiply by seven

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and the negative here is what[br]flipped us over the x-axis.