# 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?

03:14

### Video Transcript

π΄π΅πΆπ· is reflected in the π₯-axis and then translated five units to the right. What is the image of point π΅?

So we have a combination of transformations in this question, firstly a reflection in the π₯-axis and then a translation five units to the right. Weβre asked to determine the coordinates of the image of point π΅ after this pair of transformations. π΅ is the point with coordinates negative seven, six.

Now if we think first about reflecting in the π₯-axis, the coordinate grid weβve been given in the question isnβt big enough for us to draw the image of this quadrilateral after this reflection on, which suggests that weβ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 π₯- and π¦-coordinates of a general point. So letβs think about a point which has coordinates π₯, π¦. When we reflect a point in the π₯-axis, this has a vertical effect, because the mirror line itself is horizontal.

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

Now letβs think about the effect of translating a point five units to the right. Well this is a horizontal transformation, which means the π¦-coordinate will be unaffected, but the π₯-coordinate will increase by five. Therefore, a point with coordinates π₯, π¦ will be mapped to the point with coordinates π₯ plus five, π¦.

Now remember weβre performing both of these transformations, so we need an overall effect. The overall effect of reflecting in the π₯-axis and then translating five units to the right is to map the point with coordinates π₯, π¦ to the point with coordinates π₯ plus five, negative π¦.

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

Therefore, the image of point π΅ after both transformations is the point with coordinates negative two, negative six.