MCR3U Functions · Unit 1 — Introduction to Functions
Lesson 1.6
Exploring Transformations of Parent Functions
Stretch, flip and slide four parent graphs. One equation — g(x) = af(x − d) + c — tells you exactly how every graph moves.
Estimated time: 30–45 minutes
Learning goals
- recognize the graphs of y = x², y = √x, y = 1/x and y = |x|
- explain vertical stretches, compressions and reflections in the x-axis
- explain horizontal and vertical translations
- interpret g(x) = af(x − d) + c
- predict a transformed graph's key points, domain and range
Step 1 of 4
Warm Up
Three quick questions about shapes, slides and flips.
You already know these parent shapes and how points move. Let's warm up before you transform whole graphs.
Step 2 of 4
Learn
Four parent graphs, one transformation rule, two worked examples and the lab.
Real-world idea
One photo, many edits
When you edit a photo on your phone, you can make it taller, flip it, or slide it on the screen. The picture is still the same picture — it's just been moved or resized.
Graphs work the same way. Start with a parent graph, then edit it with three numbers: a, d and c.
The four parent functions in this lesson
Quadratic
f(x) = x²
Domain: all real numbers · Range: y ≥ 0
A U shape. Lowest point (vertex) at (0, 0).
Square root
f(x) = √x
Domain: x ≥ 0 · Range: y ≥ 0
Starts at (0, 0) and only goes right — no negative inputs.
Reciprocal
f(x) = 1/x
Domain: x ≠ 0 · Range: y ≠ 0
Two branches. Never touches x = 0 or y = 0 (its asymptotes).
Absolute value
f(x) = |x|
Domain: all real numbers · Range: y ≥ 0
A sharp V. Vertex at (0, 0).
Key term
Transformation form
g(x) = a·f(x − d) + c
g(x) = 2(x − 1)² + 3 uses f(x) = x² with a = 2, d = 1, c = 3.
a — multiplies every y-value. |a| > 1 stretches vertically, 0 < |a| < 1 compresses vertically, and a < 0 reflects in the x-axis.
d — slides the graph horizontally. d > 0 moves right, d < 0 moves left. Watch the sign: x − 3 means right 3, x + 3 means left 3.
c — slides the graph vertically. c > 0 moves up, c < 0 moves down.
Point rule
Same rules, different shapes
Worked example
Example A — Transform f(x) = |x| into g(x) = −2|x − 3| + 1.
Read the values
Match g(x) to a·f(x − d) + c.
a = −2, d = 3, c = 1
Stretch
|a| = 2: every y-value doubles — a vertical stretch by a factor of 2.
Reflect
a is negative: the V flips upside down (reflection in the x-axis).
Translate
d = 3: shift 3 right. c = 1: shift 1 up.
Key points
Use (x, y) → (x + 3, −2y + 1).
(0, 0) → (3, 1); (1, 1) → (4, −1); (−1, 1) → (2, −1)
Domain and range
The V opens down from its vertex (3, 1).
Domain: all real numbers. Range: y ≤ 1
Answer: Vertex (3, 1); the V opens down and is twice as steep. Domain all real numbers; range y ≤ 1.
Example A — parent (grey) and g(x) = −2|x − 3| + 1
g(x) = −2|x − 3| + 1
Domain: all real numbers · Range: y ≤ 1
Worked example
Example B — Transform f(x) = √x into g(x) = 3√(x + 2) − 4.
Read the values
x + 2 = x − (−2).
a = 3, d = −2, c = −4
Stretch
a = 3: vertical stretch by a factor of 3. No reflection (a > 0).
Translate
d = −2: shift 2 left. c = −4: shift 4 down.
Starting point
(0, 0) → (0 − 2, 3·0 − 4).
(0, 0) → (−2, −4)
More points
(1, 1) → (−1, −1); (4, 2) → (2, 2).
Domain and range
The curve starts at (−2, −4) and rises to the right.
Domain: x ≥ −2. Range: y ≥ −4
Answer: Starting point (−2, −4). Domain x ≥ −2; range y ≥ −4.
Example B — parent (grey) and g(x) = 3√(x + 2) − 4
g(x) = 3√(x + 2) − 4
Domain: x ≥ −2 · Range: y ≥ −4
Parent Function Transformation Lab
Pick a parent, predict, then test. Finish with the target challenges.
- f(x) = x²
- your graph
g(x) = x²
- Vertex
- (0, 0)
- Domain
- all real numbers
- Range
- y ≥ 0
Change one value at a time. Watch the labelled key points move.
a scales and flips. d slides sideways. c slides up and down.
Same three numbers, any parent function.
Step 3 of 4
Practice
Identify, map, model and explain. Hints wait until you ask.
Build it
Read a, d and c and map points.
Apply it
Use transformations to read real situations.
Challenge
Work backward and find domain, range and asymptotes.
Explain it
Correct errors and describe transformations in words.
Step 4 of 4
Mastery Check
Six questions, no hints. Answer them all, then see how you did.
Lesson 1.6 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.