Video: Determining the Coordinates of a Point after Translation

Lauren McNaughten

𝐴𝐵𝐶𝐷 is reflected in the 𝑥-axis and then translated 5 units to the right. What is the image of point 𝐵?

03:13

Video Transcript

𝐴𝐵𝐶𝐷 is reflected in the 𝑥-axis and then translated five units to the right. What is the image of point 𝐵?

So we have a combination of transformations in this question, firstly a reflection in the 𝑥-axis and then a translation five units to the right. We’re asked to determine the coordinates of the image of point 𝐵 after this pair of transformations. 𝐵 is the point with coordinates negative seven, six.

Now if we think first about reflecting in the 𝑥-axis, the coordinate grid we’ve been given in the question isn’t big enough for us to draw the image of this quadrilateral after this reflection on, which suggests that we’re not intended to answer this question by drawing on the coordinate grid.

Instead, we need to think about what effect each of these transformations have on the 𝑥- and 𝑦-coordinates of a general point. So let’s think about a point which has coordinates 𝑥, 𝑦. When we reflect a point in the 𝑥-axis, this has a vertical effect, because the mirror line itself is horizontal.

The 𝑥-coordinate is unchanged, but the 𝑦-coordinate will be. As the image will be the same distance below the 𝑥-axis, as the original point was above the 𝑥-axis, the 𝑦-coordinate will change from 𝑦 to negative 𝑦. So, overall, reflection in the 𝑥-axis maps a point with coordinates 𝑥, 𝑦 to a point with coordinates 𝑥, negative 𝑦.

Now let’s think about the effect of translating a point five units to the right. Well this is a horizontal transformation, which means the 𝑦-coordinate will be unaffected, but the 𝑥-coordinate will increase by five. Therefore, a point with coordinates 𝑥, 𝑦 will be mapped to the point with coordinates 𝑥 plus five, 𝑦.

Now remember we’re performing both of these transformations, so we need an overall effect. The overall effect of reflecting in the 𝑥-axis and then translating five units to the right is to map the point with coordinates 𝑥, 𝑦 to the point with coordinates 𝑥 plus five, negative 𝑦.

Each coordinate has only been affected by one of the two transformations: the 𝑦-coordinate was affected by the reflection and the 𝑥-coordinate was affected by the translation. So applying this mapping to the point 𝐵, which had coordinates negative seven, six, we add five to the 𝑥-coordinate, so it becomes negative seven plus five, and we multiply the 𝑦-coordinate by negative one, so it becomes negative six.

Therefore, the image of point 𝐵 after both transformations is the point with coordinates negative two, negative six.