MPM2D · Lesson 1: Vertex Form ↔ Standard Form

BEFORE WE START

Quick recap — you've already got this

REVIEW · NOT TODAY'S NEW MATERIAL

In y = a(x−h)² + k, the vertex is (h, k) and the axis of symmetry is x = h.

a controls the shape — narrower if |a| > 1, wider if |a| < 1, flipped if a is negative.

h and k slide the graph left/right and up/down without changing its shape.

WHERE WE'RE GOING TODAY

Learning goals

Every parabola has two useful equations for it — vertex form (great for graphing) and standard form (great for other jobs later, like finding zeros). Today you learn to move between them.

Expand vertex form into standard form by multiplying out the brackets.

Identify a, b, and c correctly once an equation is expanded.

Complete the square to turn standard form back into vertex form.

Use completing the square to find the vertex directly from standard form.

Explain why both forms describe the exact same parabola.

NEW SKILL 1 · EXPANDING

From vertex form to standard form

y = (x−2)² + 3

WATCH FOR THESE

Three ways to lose marks expanding

Forgetting to FOIL fully

(x−h)² is NOT x² − h². You must multiply (x−h)(x−h) out completely — the middle term is easy to drop.

(x−h)² = x² − 2hx + h²

Not distributing a to every term

a multiplies ALL three terms inside the brackets, not just the x² term. This is where most sign errors happen.

a(x²−2hx+h²) = ax²−2ahx+ah²

Losing track of k

k is added at the very end, after distributing a — it combines with ah² to form the final constant c.

c = ah² + k

NEW SKILL 2 · COMPLETING THE SQUARE

From standard form back to vertex form

WHY ADD AND SUBTRACT (b/2a)²?

x² + bx is missing the piece that would make it a perfect square. Adding (b/2)² completes it — but that changes the value, so we subtract the same amount right back out. Net change: zero.

Half of b—

(half)²—

Vertex—

b (try your own, a = 1)6

PRACTICE · TAP A QUESTION TO REVEAL THE ANSWER

Convert both directions

QUICK GAME

Form Match

Pick the equivalent equation in the other form.

SCORE: 0 / 0

VERTEX FORMy = (x−1)² + 2

HOMEWORK

Before next class

1. Expand: y = (x−3)² + 4.
2. Expand: y = (x+5)² − 2.
3. Expand: y = 2(x−1)² + 3.
4. Expand: y = −(x+4)² + 6.
5. Complete the square: y = x² + 8x + 10.
6. Complete the square: y = x² − 6x + 1.
7. Complete the square: y = x² + 2x − 5.
8. Challenge — complete the square: y = 3x² + 12x + 7 (factor the 3 out first).
9. A classmate says (x−5)² expands to x² − 25. Explain their mistake and show the correct expansion.
10. Explain, in your own words, why completing the square always involves adding and then subtracting the same number.