MCF3M Functions & Applications · Unit 1 — Introduction to the Quadratic Function
Lesson 1.4
Exploring Transformations of Quadratic Functions
Use the Parabola Transformation Lab to discover how a, h and k move, flip and reshape the parent parabola y = x².
Estimated time: 30–45 minutes
Learning goals
- recognize y = x² as the quadratic parent function, with its vertex and axis
- investigate the effects of k, h and a
- distinguish translations, reflections, vertical stretches and compressions
- identify the vertex from vertex form
Step 1 of 4
Warm Up
Real-world idea
Meet the parent
Every parabola in this chapter comes from one “parent”: y = x².
Look at its key points. Where is the lowest point? Which line splits it into two mirror halves? Which way does it open?
y = x²
y(x) = x²
Step 2 of 4
Learn
The Parabola Transformation Lab
A transformation changes a graph's position, its orientation (up or down), or its shape.
In the lab, the grey curve is always y = x². The green curve is g(x) = a(x − h)² + k. You will change one letter at a time — but predict first.
Investigation A — what does k do?
a and h are locked. Change k.
1 · Predict first
What will happen when k changes from 0 to 4?
- y = x²
- g(x) = x²
2 · Test it
g(x) = x²
Make a prediction to unlock the controls.
- Vertex
- (0, 0)
- Axis of symmetry
- x = 0
- Opens
- up
- Shape
- same width as y = x²
What you found
Investigation B — what does h do?
a and k are locked. Change h — try positive AND negative values.
1 · Predict first
What will happen when h changes from 0 to 3 — so the equation becomes (x − 3)²?
- y = x²
- g(x) = x²
2 · Test it
g(x) = x²
Make a prediction to unlock the controls.
- Vertex
- (0, 0)
- Axis of symmetry
- x = 0
- Opens
- up
- Shape
- same width as y = x²
WATCH THE SIGN
Investigation C — what does a do?
h and k are locked. Test a = 2, 0.5, −1, −2 and −0.5.
1 · Predict first
What will happen when a changes from 1 to 2?
- y = x²
- g(x) = x²
2 · Test it
g(x) = x²
Make a prediction to unlock the controls.
- Vertex
- (0, 0)
- Axis of symmetry
- x = 0
- Opens
- up
- Shape
- same width as y = x²
Key term
Vertical stretch / compression
Multiplying every y-value by |a|. If |a| > 1 it is a vertical stretch (looks narrower). If 0 < |a| < 1 it is a vertical compression (looks wider).
Key term
Reflection in the x-axis
When a < 0, every y-value changes sign, so the parabola flips and opens down.
Free lab
Nothing is locked. Build a parabola that opens down, is narrower than y = x², and has its vertex at (−2, 3).
- y = x²
- g(x) = x²
g(x) = x²
- Vertex
- (0, 0)
- Axis of symmetry
- x = 0
- Opens
- up
- Shape
- same width as y = x²
Checkpoint — predict, then reveal
Before you see it: what will g(x) = −2(x − 3)² + 4 look like?
Vertex?
Opens…
Horizontal shift?
Vertical shift?
Compared with y = x², it is…
Putting it together
Now that you've seen each letter in action, here's the summary for g(x) = a(x − h)² + k.
hhorizontal translation (right if h > 0, left if h < 0)kvertical translation (up if k > 0, down if k < 0)sign of aa > 0 opens up; a < 0 reflects in the x-axis (opens down)|a||a| > 1 vertical stretch; 0 < |a| < 1 vertical compressionvertex = (h, k), axis: x = h
Worked example
Describe g(x) = −0.5(x − 1)² + 6.
Step 1
Read the parameters.
a = −0.5, h = 1, k = 6
Step 2
h = 1: right 1. k = 6: up 6.
vertex (1, 6), axis x = 1
Step 3
a < 0: reflection, opens down. |a| = 0.5: vertical compression, wider.
Answer: Right 1, up 6, reflected in the x-axis, vertically compressed by 0.5. Vertex (1, 6).
Step 3 of 4
Practice
Build it
Identify a, h, k and the vertex.
Apply it
Describe the transformations.
Challenge
Turn instructions into an equation.
Explain it
Find and fix the reasoning error.
Want more practice? (2 optional questions)
Step 4 of 4
Mastery Check
Lesson 1.4 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.