Apply translations, rotations, reflections, and dilations on the plane.
Free, timed SHSAT practice sets on transformations— every question includes a full worked explanation, and there's no account required.
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Each transformation has a simple rule for a point \((x, y)\). Using \((2, 5)\) as the example:
Common traps: mixing up the x-axis and y-axis reflections (reflecting over the x-axis changes \(y\)), rotating clockwise instead of counterclockwise, and moving left or down in the wrong direction. A quick sketch on a grid catches most of these.
Moving 7 units left adds (-7) to \(x\), and moving 2 units up adds 2 to \(y\): \((4 + (-7), -5 + 2) = (-3, -3)\). (Moving the opposite way (subtracting the shifts) gives (11, -7). Adding the horizontal shift to \(y\) and the vertical shift to \(x\) gives (6, -12). Moving only 7 units left gives (-3, -5).)
Reflecting over the x-axis keeps \(x\) and changes the sign of \(y\): \((2, -3) \to (2, 3)\). (Changing the sign of \(x\) instead reflects over the y-axis, giving (-2, -3). Changing both signs is a \(180^\circ\) rotation, giving (-2, 3). Swapping the coordinates reflects over the line \(y = x\), giving (-3, 2).)
A \(90^\circ\) counterclockwise rotation sends \((x, y)\) to \((-y, x)\), so \((3, -7) \to (7, 3)\). (Rotating clockwise instead, \((y, -x)\), gives (-7, -3). Using the \(180^\circ\) rule, \((-x, -y)\), gives (-3, 7). Swapping the coordinates without changing a sign gives (-7, 3).)