# Video: Identifying the Image of a Point on a Trigonometric Graph Following a Transformation

The figure shows the graph of π(π₯). A transformation maps π(π₯) to π(2π₯). Determine the coordinates of π΄ following this transformation.

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### Video Transcript

The figure shows the graph of π π₯. A transformation maps π π₯ to π two π₯. Determine the coordinates of π΄ following this transformation.

As shown here in this question. You can see the point π΄ is at the coordinates 180, negative one. Weβre gonna have to see where this π΄ moves to when we actually transform our graph to π two π₯. The transformation of π two π₯ is actually involving stretches.

Iβm gonna have a couple of rules here for stretches. The first of these is actually π π π₯. And what this is is a stretch by the factor π in the π¦-axis. And what does this actually mean in practice? Well what it means in practice is that weβre going to multiply our π¦-coordinates by π. Okay, great!

So letβs move on to the next stretch. Well the next natural rule is that π ππ₯ β this time you can notice that the π is actually inside the parentheses. Well this is a stretch by the factor one over π in the π₯-axis. So what does this mean in practice? Well what this means in practice is weβre gonna actually multiply our π₯- coordinates by one over π, which actually is the same as dividing our π₯ coordinates by π.

Okay, great! So weβve now got two rules for stretches. So now letβs have a look at how we can transform our graph. Well actually the transformation thatβs gonna be taking place in this question is actually like the bottom one, because weβve got π two π₯ so therefore we know itβs going to be a stretch by factor one over π in the π₯-axis.

And if we take a look at why that might be the case, well weβve got π two π₯ and the two is inside the parentheses. So the two is like our π. So therefore, we actually know itβs gonna be a stretch by the factor one over two or a half in the π₯-axis. So what does this mean in practice? Well what it means in practice is that weβre gonna multiply the π₯-coordinates by a half.

Okay, great! So letβs go back to the graph and see if that can help us solve the problem. Okay, so what that Iβve actually done is Iβve actually sketched it on our graph in pink. Well as you can see, the graph itself actually looks as if itβs been like concertinaed or squashed. And thatβs actually because what weβve done is by multiplying our π₯-coordinates by half weβve actually halved each of our π₯-coordinates.

So therefore, if we take a look at what we wanted to find in this question, which is the coordinates of π΄, what we can see is that actually Iβve called our new π΄ π΄ dash. And the coordinates of our new π΄ are 90, negative one. And this is because weβve actually multiplied our 180 by a half because our 180 was our π₯-coordinate.

So multiply 180 by a half, we get 90. So therefore, we can say that following the transformation from π π₯ to π two π₯, the coordinates of π΄ are 90, negative one.