Standard Form

Master standard form and completing the square. Grade 11 Math 20-1 curriculum.

Lesson 1.4 of Quadratic Functions & Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Tracking a Point. Learn how to follow a single point through any transformation using mapping notation instead of redrawing the entire graph.
• Properties of y = a(x - p)² + q. Read every property of a parabola directly from vertex form: vertex, axis of symmetry, direction of opening, domain, range, and intercepts.
• Full Analysis Examples. Walk through complete parabola analyses including transformations, all key properties, and finding intercepts algebraically using the Square Root Principle.

Inside the lesson — a free preview

What Does Tracking a Point Mean?

When you transform a graph, every single point on it moves the same way. So instead of redrawing the whole graph, you can pick one point and track where it ends up after each transformation.

The Key Idea. For every transformation, just apply the rule to the point's (x, y) coordinates. This is faster than redrawing the entire parabola.

Mapping Notation

A mapping is a general rule for how any point (x, y) gets transformed. We write it as:

(x, y) → (new x, new y)

For example, a vertical stretch by a factor of 2 has the mapping (x, y) → (x, 2y). This says the x-coordinate stays the same and the y-coordinate gets doubled.

Reading Everything from the Equation

The form y = a(x - p)^2 + q is powerful because every property of the parabola can be read directly from the equation — no graph needed.

Worked example: Track the Point (2, 5)

Apply three different transformations to the point (2, 5).

Part a) — Vertical Stretch by a Factor of 2. The mapping is (x, y) → (x, 2y). Multiply the y-coordinate by 2.

(2, 5) → (2, 2 · 5) = (2, 10)

Mapped point: (2, 10)

Part b) — Reflection About the x-axis. The mapping is (x, y) → (x, -y). Flip the sign of the y-coordinate.

(2, 5) → (2, -5)

Mapped point: (2, -5)

Part c) — Horizontal Translation 4 Left, Vertical Translation 3 Up. The mapping is (x, y) → (x - 4, y + 3).

(2, 5) → (2 - 4, 5 + 3) = (-2, 8)

Mapped point: (-2, 8)

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