Video Transcript
Find the image of point negative 10, negative nine after applying the translation 𝑥, 𝑦 maps onto 𝑥 minus eight, 𝑦 plus five followed by a 90-degree rotation counterclockwise about the origin.
So we’re going to be performing two transformations to the point at negative 10, negative nine. The first one is a translation. That’s when we slide a shape or, in this case, we slide the point. And we’re told that 𝑥, 𝑦 maps onto 𝑥 minus eight, 𝑦 plus five. In other words, we’re going to subtract eight from the 𝑥-coordinate and add five to the 𝑦-coordinate. We might sometimes see this written in column vector form as the vector negative eight, five. The point will move left eight spaces and up five.
Either way, we can map the point negative 10, negative nine onto the point negative 10 minus eight, negative nine plus five, which is the point with coordinates negative 18, negative four. And so we now have the image of the point negative 10, negative nine after the translation.
Our next job is to perform a rotation. So what we’re going to do next is, in fact, draw a diagram of the scenario. Now, due to space reasons, we have a slightly unusual scale, but we see that negative 18, negative four is around here. We want to rotate this 90 degrees counterclockwise. Firstly, we know that counterclockwise is this direction. And if that’s some any struggle to remember, make sure you have a look at an analog clock. Counterclockwise is the direction opposite to that which the actual hands of the clock move.
And then 90 degrees is a quarter of a turn, so the image of our point will end up somewhere in this quadrant. To work out exactly where the point ends up, we find the center, and the center is at the origin. That’s the point zero, zero, the point where our axes intersect.
And one way we have to perform this rotation is to use tracing paper. Alternatively, let’s join this point to the center of rotation at the origin. And we’re going to rotate this actual line by 90 degrees. That looks like this. And then the image of the point after this rotation sits at four, negative 18. And we’ve therefore finished. We’ve performed both the transformations, and the image of point negative 10, negative nine is four, negative 18.
