Step by step
The question
Question: translate the point P(2, 1) left 3, then reflect it in the x-axis.
P = (2, 1)
= translate left 3,
= then reflect in the x-axis
1 Draw the shape
Put the point on the grid.
2 First: translate
Translation left 3: left 3 takes 3 from x.
The rule is: (x, y) goes to (x − 3, y).
(x, y) → (x − 3, y)
P(2, 1) goes to P'(−1, 1).
P(2, 1) → P'(−1, 1)
3 Then: reflect in the x-axis
Reflection in the x-axis: x stays the same, y changes sign.
The rule is: (x, y) goes to (x, −y).
(x, y) → (x, −y)
P'(−1, 1) goes to P''(−1, −1).
P'(−1, 1) → P''(−1, −1)
Careful!
The second move starts from the new points.
Not from the first point.
Careful: start from P', not from P
So the final image is P''(−1, −1).
P'' = (−1, −1)
4 Check
Let's check in one go.
Put the first rule's answer (x − 3, y) into the second rule.
We get (x − 3, −y).
(x, y) → (x − 3, y) → (x − 3, −y)
P goes straight to (−1, −1).
That is P''.
Right!
P(2, 1) → P''(−1, −1)
Let's look back.
First: draw the shape.
Then: translate left 3.
Then: reflect in the x-axis.
Last: check.
Answer
P'' = (−1, −1)