MCR3U Functions · Unit 1 — Introduction to Functions
Lesson 1.7
Investigating Horizontal Stretches, Compressions, and Reflections
When a number multiplies x inside a function, the graph squeezes or stretches sideways — and it works the opposite way you might expect.
Estimated time: 30–45 minutes
Learning goals
- interpret y = f(kx)
- tell horizontal and vertical transformations apart
- calculate the horizontal scale factor 1/|k|
- identify a reflection in the y-axis when k is negative
- apply the point rule (x, y) → (x/k, y)
- recognize invariant points
- find a transformed equation from a graph or point mapping
Step 1 of 4
Warm Up
Vertical stretches, reflections and reading points.
Three quick questions from Lesson 1.6 and earlier.
Step 2 of 4
Learn
Inside vs outside, the scale factor 1/|k|, two worked examples and the explorer.
Real-world idea
Watching a video at 2× speed
At normal speed, the goal in a game highlight happens at 60 seconds. At 2× speed, it happens at 30 seconds.
Nothing about the goal changed — only when it happens. Every moment arrives in half the time. That is a horizontal compression.
What is happening?
If the normal video is f(t), the 2× video is g(t) = f(2t). At t = 30, g shows f(60) — the moment from 60 seconds.
Multiplying the input by 2 means you reach each output twice as fast. So every point moves to half its x-value: (60, y) → (30, y).
Key term
Horizontal stretch or compression
y = f(kx) scales the graph horizontally by a factor of 1/|k|.
y = f(3x) is a horizontal compression by a factor of 1/3.
|k| > 1 → horizontal compression by a factor of 1/|k| (narrower).
0 < |k| < 1 → horizontal stretch by a factor of 1/|k| (wider).
k < 0 → also a reflection in the y-axis.
Point rule: (x, y) → (x/k, y), for k ≠ 0.
Why it works backward
y = 3f(x): (6, 2) → (6, 6)Vertical stretch by 3 — multiply y by 3.y = f(3x): (6, 2) → (2, 2)Horizontal compression by 1/3 — divide x by 3.
Worked example
Example A — Given f(x) = √x, graph g(x) = √(4x).
Find k
g(x) = f(4x), so k = 4.
Describe
|k| = 4 > 1: horizontal compression by a factor of 1/4. No reflection (k > 0).
Map points
Divide each x-value by 4; keep y.
(0, 0) → (0, 0); (1, 1) → (1/4, 1); (4, 2) → (1, 2); (9, 3) → (9/4, 3)
Check one
g(1) = √(4 · 1) = √4 = 2. ✓ (1, 2) is on g.
Domain and range
4x ≥ 0 means x ≥ 0. Outputs are still never negative.
Domain x ≥ 0. Range y ≥ 0
Answer: g is √x squeezed toward the y-axis by a factor of 1/4. Domain x ≥ 0; range y ≥ 0.
Example A — parent √x (grey) and g(x) = √(4x)
g(x) = √(4x)
Domain: x ≥ 0 · Range: y ≥ 0
Worked example
Example B — Given f(x) = |x|, graph g(x) = |−0.5x|.
Find k
k = −0.5.
Size
|k| = 0.5 < 1: horizontal stretch by a factor of 1/0.5 = 2.
Sign
k < 0: reflection in the y-axis.
Map points
(x, y) → (x/(−0.5), y) = (−2x, y).
(1, 1) → (−2, 1); (−1, 1) → (2, 1); (2, 2) → (−4, 2)
Notice
Point (1, 1) went to (−2, 1), and point (−1, 1) went to (2, 1). The set of points is exactly the same as for |0.5x|, because |x| is symmetric about the y-axis.
Answer: A horizontal stretch by 2 and a reflection in the y-axis. The graph looks identical to y = |0.5x| — the reflection is invisible on a symmetric graph.
Invariant points
Horizontal Transformation Explorer
Predict, change k, then move the point. Turn on the comparison to see af(x) vs f(kx).
Predict first
Predict: what will y = f(2x) do to a graph?
- y = f(x)
- y = f(1x)
y = √x
Make a prediction to unlock k.
Horizontal scale factor 1/|k| = 1
no horizontal change
Original: (1, 1)
Transformed: (x/k, y) = (1, 1)
| x | y = f(x) | x/k | y |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 4 | 2 | 4 | 2 |
| 9 | 3 | 9 | 3 |
Inside the function, everything works backward: f(kx) divides x-values by k.
Outside multiplies y; inside divides x.
Step 3 of 4
Practice
Describe, map, model and explain. Hints wait until you ask.
Build it
Describe transformations and map points.
Apply it
Speed changes and slow-motion.
Challenge
Work backward from points and domains.
Explain it
Compare, justify and correct.
Step 4 of 4
Mastery Check
Six questions, no hints. Answer them all, then see how you did.
Lesson 1.7 mastery check
6 questions · no hints during the check. Answer them all, then check your score — you can retry as many times as you like.