Start with point 𝐴: negative five, negative one. Translate it by five units to the left and two units up. Follow this with a translation of five units to the right and four units up. What are the new coordinates?
So we begin with the coordinates of a point, and we’re then told to translate it in two different ways. Remember a translation is just a shift of the point. First, we translate this point five units left and two units up. Then, we translate this new point five units right and four units up.
Let’s look at the effect of each of these translations. When translating a point five units left and two units up, the effect on the general point with coordinates 𝑥, 𝑦 is that five is subtracted from the 𝑥-coordinate and two is added to the 𝑦-coordinate. We’re subtracting five here because the translation is to the left, and therefore the 𝑥-coordinate is decreasing.
So for the point 𝐴, we subtract five from the 𝑥-coordinate, which gives negative five minus five, and then we add two to the 𝑦-coordinate, negative one plus two, which gives the image of point 𝐴 as negative 10, one after the first translation.
Now let’s consider the effect of the second translation. Translating a point five units right and four units up will cause the point with general coordinates 𝑥, 𝑦 to move to the point with coordinates 𝑥 plus five, 𝑦 plus four.
Remember we’ve already performed one translation of the point 𝐴, so the point that we’re translating here is the image of 𝐴, the point with coordinates negative 10, one. So the point with coordinates negative 10, one is mapped to negative 10 plus five, one plus four, which gives the coordinates of this new point as negative five, five.
Now these two translations could actually be combined into one overall translation rather than one followed by another. Translating a point five units to the left and then five units to the right has the overall effect of not moving it horizontally at all, so moving it zero units across.
Translating a point two units up and then a further four units up has the overall effect of translating it six units up. So we could achieve these two translations in one step by mapping the point with coordinates 𝑥, 𝑦 to the point with coordinates 𝑥, 𝑦 plus six.
We see that if we take the point 𝐴 with coordinates negative five, negative one, then this gets mapped to negative five, negative one plus six, which is the point with coordinates negative five, five, and we’ve already established that this is the answer to the problem.