Geometric Transformations

Geometric transformations versus function transformations

A function transformation and a geometric transformation are easy to confuse, since both use words like shift, reflect, and stretch. A function transformation changes the graph of $y = f(x)$ by changing the input or output of the function, as covered in the Function Transformations and Piecewise Functions chapter. A geometric transformation, the subject of this chapter, moves or resizes points and figures directly in the coordinate plane, without reference to any function rule. The vocabulary overlaps; the object being transformed does not.

A geometric transformation moves a figure according to a fixed coordinate rule, without changing its shape (except for a dilation, which changes size). Reflections, rotations, and translations preserve size and shape; only a dilation changes the figure's dimensions, scaled by a constant factor from a center point.

The coordinate rules

Each transformation has its own coordinate rule, and mixing them up is the most common error on this topic. Learn each rule as a one-line instruction, and name the transformation before you touch the coordinates.

A dilation centered at the origin with scale factor $k$ multiplies both coordinates by $k$: $(x, y)$ becomes $(kx, ky)$. When a dilation is applied to an entire figure instead of a single point, its area scales by $k^2$, the same relationship covered in the similarity chapter of this Part. Lengths and perimeters scale by $k$; area scales by $k^2$.

Where this appears as a model

Think of an image resized for printing. A 4-by-6-inch photo enlarged by a scale factor of 3 becomes 12 by 18 inches, and its area grows from 24 to 216 square inches, scaling by the square of the linear factor instead of the factor itself.

Applying a rotation rule

Worked example. Point $(5, -2)$ is rotated $90^\circ$ counterclockwise about the origin. Find the image.

A $90^\circ$ counterclockwise rotation follows the rule $(x, y) \to (-y, x)$.

$(5, -2) \to (-(-2), 5) = (2, 5)$

Notice the double negative: the original $y$-value is already negative, so negating it gives a positive $x$-value. Sign slips like this are where most points are lost on rotation questions.

Combining two transformations

Worked example. Point $(3, -4)$ is reflected over the $y$-axis, then translated 2 units down. Find the final image.

Apply each transformation in order, using the result of the first as the input to the second.

Reflect over the $y$-axis: $(3, -4) \to (-3, -4)$

Translate 2 units down: $(-3, -4) \to (-3, -6)$

Worked example. A triangle has a vertex at $(0, 3)$. It is dilated by a scale factor of 2 centered at the origin, then reflected over the $x$-axis. Find the final image of that vertex.

Recognize. Two transformations are applied in sequence. Apply them one at a time, in the order given, using the result of the first as the input to the second.

Set up. Dilate $(0, 3)$ by scale factor 2: $(0, 6)$.

Solve. Reflect $(0, 6)$ over the $x$-axis: $(0, -6)$.

Check. The $y$-coordinate's magnitude doubled from 3 to 6, matching the dilation, and then flipped sign, matching the reflection.

Best route. Hand computation. Each rule is a one-line coordinate transformation.

Common error. Applying the two transformations in the reverse order, which does not always produce the same final image.

Strategy summary

• Name the transformation first, then write its rule, then substitute.
• For a sequence, write each intermediate point on its own line and never combine steps in your head.
• Reflections over an axis flip one sign; reflection over $y = x$ swaps; a $180^\circ$ rotation flips both signs.
• For counterclockwise rotations remember $90^\circ \to (-y, x)$, and build $270^\circ$ as "$90^\circ$ clockwise," $(y, -x)$.
• If a question gives an area and a scale factor, square the factor. If it asks about a perimeter or a length, do not.
• A translation never changes which way a figure faces; a rotation or reflection changes position and orientation but never size.