Properties & Transformations of Quadratic Functions

Learn vertex form and transformations of parabolas. Grade 11 Math 20-1 Alberta - diploma prep.

Lesson 1.3 of Quadratic Functions & Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

Properties of Quadratic Functions. Identify whether an equation is quadratic, read key properties of a parabola from its graph, and practise with the three forms of a quadratic function: general, vertex, and factored.
Transformations — How a, p, and q Change the Graph. Discover how each parameter in the vertex form changes the parabola: the stretch factor controls width and direction, p shifts the graph left or right, and q shifts it up or down.
Putting It All Together — y = a(x - p)² + q. Combine all three transformations using the SRT order — Stretch, Reflect, Translate — to graph any quadratic in vertex form and read off every key property in one step.
Practice with Transformations. Apply a transformation cheatsheet to track points through multiple transformations, sketch parabolas from vertex form, and analyse trickier examples including reflections and compressions.

Recall: Quadratic Functions. A quadratic function has a degree of 2 — the highest exponent of x must be exactly 2. Its graph is always a parabola.

To be quadratic, the highest exponent of x must be exactly 2 — not 1, not 3, and the variable cannot appear inside a square root or in a denominator.

Every parabola has these five properties. You should be able to identify all of them from a graph or equation.

The Five Properties.

Look at the graph above. Let's find every property of y = (x - 2)^2 - 3.

a) Vertex. Reading from the graph, the vertex is at (2, -3).

b) Direction of Opening and Axis of Symmetry. The parabola opens up because a = 1 > 0.

The axis of symmetry is the vertical line through the vertex: x = 2.

Minimum Value. Since the parabola opens up, it has a minimum. The minimum value is the y-coordinate of the vertex: -3.

c) Domain and Range. Domain: The parabola extends left and right forever.

D: \x x R\

Range: The lowest y-value is -3 (the vertex), and it goes up forever.

R: \y y ≥ -3, y R\

d) Finding the x- and y-intercepts algebraically.

Step 1 — Start with the given equation. y = (x - 2)^2 - 3

Step 2 — Replace x with 0. y = (0 - 2)^2 - 3

y = 4 - 3

y = 1

y-intercept: (0, 1)

Step 3 — Start with the given equation. y = (x - 2)^2 - 3

Step 4 — Replace y with 0. 0 = (x - 2)^2 - 3

Add 3 to both sides:

3 = (x - 2)^2

Take the square root of both sides (remember ±):

±√(3) = x - 2

Add 2 to both sides:

x = 2 ± √(3)

x-intercepts: x = 2 + √(3) and x = 2 - √(3)

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