C. Algebra · Linear and Non-Linear Relations

Transforming the Line y = ax: Translations, Reflections and Rotations

C4.3 A1About 70 minutes

Learning goals

I can translate a line y = ax up, down, left or right, and write the equation of the image.
I can reflect a line y = ax in the x-axis or the y-axis and write the equation of the image.
I can rotate a line y = ax by 90° or 180° about the origin and describe what happens to its slope.
I can describe how each transformation changes the graph and the equation.

Words to know

transformation
A change in the position or orientation of a figure, such as a translation, reflection or rotation.

image
The new figure (or line) after a transformation.

translation
A slide: every point moves the same distance in the same direction.

reflection
A flip over a line (the line of reflection). Each point and its image are the same distance from that line.

rotation
A turn about a fixed point (the centre of rotation) by a given angle.

perpendicular
Meeting at a right angle (90°).

Remember from Grade 8

In Grades 7 and 8 you translated, reflected and rotated shapes on the Cartesian plane and described them with coordinate rules. Now you will do the same to lines through the origin, y = ax, and see that every transformation changes the equation in a predictable way. In Grade 10 and beyond, the same ideas are used to transform curves like y = x².

Translations: slide the line

Up k units: add k to every y-coordinate. y = ax becomes y = ax + k.
Down k units: y = ax becomes y = ax − k.
Right h units: every point (x, y) moves to (x + h, y). The equation becomes y = a(x − h), which expands to y = ax − ah.
Left h units: y = ax becomes y = a(x + h).

A translation never changes the slope: the image is parallel to the original line. For a line, a horizontal slide always gives the same image as some vertical slide. For example, y = 2x moved right 1 is y = 2(x − 1) = 2x − 2, the same as moving it down 2.

y = 2x (solid) translated up 3 gives y = 2x + 3, and translated down 2 gives y = 2x − 2 (dashed). The arrows show single points sliding. All three lines are parallel with slope 2.

Reflections: flip the line

In the x-axis: (x, y) goes to (x, −y). y = ax becomes y = −ax.
In the y-axis: (x, y) goes to (−x, y). y = ax becomes y = a(−x) = −ax, the same image!

For a line through the origin, reflecting in either axis changes the sign of the slope. A rising line becomes a falling line just as steep.

Reflecting y = 2x (solid) in the x-axis: (1, 2) goes to (1, −2) and (2, 4) goes to (2, −4). The image is y = −2x (dashed): same steepness, opposite sign.

Rotations about the origin: turn the line

180°: (x, y) goes to (−x, −y). If y = ax, then −y = a(−x), which is still y = ax. A line through the origin lands on itself.
90° (either direction): the line turns to become perpendicular. The slope a becomes −1/a (flip the fraction and change the sign). For example, y = 2x becomes y = −(1/2)x.

Why? A 90° counterclockwise turn sends (x, y) to (−y, x). The point (1, a) on y = ax goes to (−a, 1). The slope from the origin to (−a, 1) is 1 ÷ (−a) = −1/a.

A 90° counterclockwise rotation about the origin takes (2, 4) on y = 2x to (−4, 2). The image line is y = −x/2 (dashed), perpendicular to the original. The slopes 2 and −1/2 multiply to −1.

Transformation of y = ax	Image equation	What happens to the slope	What happens to the y-intercept
Translate up k	y = ax + k	Same	0 becomes k
Translate right h	y = a(x − h) = ax − ah	Same	0 becomes −ah
Reflect in the x-axis or y-axis	y = −ax	Changes sign	Stays 0
Rotate 90° about the origin	y = −(1/a)x	Negative reciprocal	Stays 0
Rotate 180° about the origin	y = ax	Same (the line maps onto itself)	Stays 0

Summary of the effects on y = ax (a not 0).

Worked example

Example 1: A horizontal translation

Translate y = 3x 2 units to the right. Write the image equation and check it with a point.

Replace x with (x − 2): y = 3(x − 2) = 3x − 6.
Check: (0, 0) is on y = 3x. Moving it 2 right gives (2, 0). On the image: 3(2) − 6 = 0 ✓.
Notice: the image is the same as translating y = 3x down 6.

y = 3x − 6

Worked example

Example 2: A rotation

Rotate y = 3x by 90° counterclockwise about the origin.

Take a point on the line: (1, 3).
A 90° counterclockwise turn sends (x, y) to (−y, x): (1, 3) goes to (−3, 1).
The image passes through (0, 0) and (−3, 1): slope = 1 ÷ (−3) = −1/3.
Check with the rule: the negative reciprocal of 3 is −1/3 ✓.

y = −(1/3)x, perpendicular to y = 3x.

Worked example

Example 3: Two transformations in a row

Reflect y = 2x in the x-axis, then translate the image up 4. Write the final equation and describe the graph.

Reflect in the x-axis: y = −2x.
Translate up 4: y = −2x + 4.
The final line falls to the right with slope −2 and crosses the y-axis at 4.

y = −2x + 4

Just for fun

What did y = 5x say after being rotated 180° about the origin?

"Honestly, I feel exactly the same."

Hands-on activity

Slide, flip and turn a spaghetti line

You need: graph paper with axes drawn from −6 to 6, one piece of uncooked spaghetti (or a thin straight stick), a pencil.

Lay the spaghetti along y = x (through (0, 0) and (2, 2)).
Translate: slide it straight up 3 squares without turning it. Name two points it passes through and write its equation.
Reflect: put it back on y = x. Imagine the x-axis is a mirror: where would (2, 2) land? Lay the spaghetti through (0, 0) and that point. Write its equation.
Rotate: put it back on y = x. Hold the middle at the origin with one finger and turn it a quarter turn. Write its equation.
Repeat with y = 2x (through (0, 0) and (2, 4)).

What to look for: y = x becomes y = x + 3, then y = −x, then y = −x (a quarter turn of y = x gives y = −x, because −1/1 = −1). For y = 2x: y = 2x + 3, y = −2x, and y = −x/2.

Your turn

Translate y = 4x up 3 units.

Translate y = −2x down 5 units.

Reflect y = 5x in the x-axis.

Reflect y = −3x in the y-axis.

Rotate y = 2x by 90° about the origin.

Which transformation turns y = 3x into y = 3x − 4?

(2, 6) is on y = 3x. Where is its image after a reflection in the x-axis, and which line is it on?

More practice: warm-up

Does a translation change the slope of a line?

Translate y = x up 2.

Reflect y = 2x in the x-axis.

What is the image of y = ax after a 180° rotation about the origin?

Which point is on every line y = ax?

More practice: on your own

Translate y = 2x right 3. Expand your answer.

Translate y = −x left 4.

Rotate y = −(1/4)x by 90° about the origin. Check with a point.

y = 3x is reflected in the x-axis and then translated up 2. Write the final equation.

Describe transformations that take y = (1/2)x to y = −(1/2)x + 5.

Challenge

Show that translating y = 2x 1 unit right gives the same line as translating it 2 units down.

Does a 90° clockwise rotation of y = ax give a different line from a 90° counterclockwise rotation? Explain.

Reflect y = 2x in the line y = x. (Hint: a reflection in y = x swaps the coordinates of every point.)

Check your understanding

Quiz

y = 6x is translated down 4. What is the image?

y = 4x is reflected in the x-axis. What is the image?

y = 3x is rotated 90° about the origin. What is the slope of the image?

Which transformation of y = ax does not change its equation?

y = 2x is translated 3 units right. What is the image?

Which statement is true about every translation of y = ax?

For parents and teachers

Big idea: Expectation C4.3 asks students to translate, reflect and rotate lines defined by y = ax and describe how each transformation affects the graph and the equation. Key results: translations keep the slope; reflections in an axis change its sign; a 90° rotation gives the negative reciprocal (a perpendicular line); a 180° rotation leaves the line unchanged.

Common difficulties:

Horizontal translations: moving right h means y = a(x − h), with a minus sign. Checking one point clears this up.
Thinking a 90° rotation just changes the sign. It also flips the fraction: 2 becomes −1/2, not −2.

How to help: The spaghetti activity makes the transformations physical. Ask your child to predict the new equation before moving the spaghetti, then check two points.