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

The figure shows the graph of 𝑓(π‘₯). A transformation maps 𝑓(π‘₯) to 𝑓(2π‘₯). Determine the coordinates of 𝐴 following this transformation.

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Video Transcript

The figure shows the graph of 𝑓 π‘₯. A transformation maps 𝑓 π‘₯ to 𝑓 two π‘₯. Determine the coordinates of 𝐴 following this transformation.

As shown here in this question. You can see the point 𝐴 is at the coordinates 180, negative one. We’re gonna have to see where this 𝐴 moves to when we actually transform our graph to 𝑓 two π‘₯. The transformation of 𝑓 two π‘₯ is actually involving stretches.

I’m gonna have a couple of rules here for stretches. The first of these is actually π‘Ž 𝑓 π‘₯. And what this is is a stretch by the factor π‘Ž in the 𝑦-axis. And what does this actually mean in practice? Well what it means in practice is that we’re going to multiply our 𝑦-coordinates by π‘Ž. Okay, great!

So let’s move on to the next stretch. Well the next natural rule is that 𝑓 π‘Žπ‘₯ β€” this time you can notice that the π‘Ž is actually inside the parentheses. Well this is a stretch by the factor one over π‘Ž in the π‘₯-axis. So what does this mean in practice? Well what this means in practice is we’re gonna actually multiply our π‘₯- coordinates by one over π‘Ž, which actually is the same as dividing our π‘₯ coordinates by π‘Ž.

Okay, great! So we’ve now got two rules for stretches. So now let’s have a look at how we can transform our graph. Well actually the transformation that’s gonna be taking place in this question is actually like the bottom one, because we’ve got 𝑓 two π‘₯ so therefore we know it’s going to be a stretch by factor one over π‘Ž in the π‘₯-axis.

And if we take a look at why that might be the case, well we’ve got 𝑓 two π‘₯ and the two is inside the parentheses. So the two is like our π‘Ž. So therefore, we actually know it’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 we’re gonna multiply the π‘₯-coordinates by a half.

Okay, great! So let’s go back to the graph and see if that can help us solve the problem. Okay, so what that I’ve actually done is I’ve actually sketched it on our graph in pink. Well as you can see, the graph itself actually looks as if it’s been like concertinaed or squashed. And that’s actually because what we’ve done is by multiplying our π‘₯-coordinates by half we’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 𝐴, what we can see is that actually I’ve called our new 𝐴 𝐴 dash. And the coordinates of our new 𝐴 are 90, negative one. And this is because we’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 𝑓 π‘₯ to 𝑓 two π‘₯, the coordinates of 𝐴 are 90, negative one.

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