Lesson 11 ~25 min Unit 2 · Non-Linear +85 XP

The General Form $y = a(x - h)^2 + k$

Vertex form gives you the parabola's complete fingerprint in one expression: $a$ (direction + width), $(h, k)$ (vertex), and the axis $x = h$. Then we'll learn to expand it.

Today's hook: Three numbers $-$ $a$, $h$, $k$ $-$ pin down ANY parabola exactly. From $y = 2(x - 3)^2 - 5$, what's the vertex?

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Think First

warm-up

You've met four building blocks: dilation ($a$), reflection ($-a$), vertical shift ($+k$) and horizontal shift ($(x - h)$). Combine them all and you get the GENERAL or VERTEX form: $y = a(x - h)^2 + k$. Take $y = 2(x - 3)^2 - 5$. What is $a$? What is the vertex? Which way does it open? Write down what you can read off WITHOUT graphing.

Record your answer in your workbook.

The Big Idea

+5 XP

Every parabola can be written as $y = a(x - h)^2 + k$. Three numbers $-$ $a$, $h$, $k$ $-$ encode the entire shape and position.

The red curve $y = 2(x - 3)^2 - 5$ has $a = 2$ (opens up, narrow), vertex $(3, -5)$, axis $x = 3$. Compare with $y = x^2$ (the parent): it's been shifted right 3, down 5, and stretched by factor 2. All four transformations live in one neat formula.

$y = a(x - h)^2 + k$ — vertex $(h, k)$, axis $x = h$.

$a$ does two jobs

Sign = direction; size = width.

$h$ has a sign flip

$(x - 3)$ means $h = 3$, NOT $-3$.

$k$ keeps its sign

$+k$ moves up, $-k$ moves down.

What You'll Master

objectives

Know

Vertex form $y = a(x - h)^2 + k$ and the meaning of each parameter
Vertex is $(h, k)$, axis of symmetry is $x = h$
$a > 0$ opens up, $a < 0$ opens down; $|a|$ controls width

Understand

Why all four transformations combine cleanly into one formula
Why $(x - h)$ has a sign flip but $+k$ does not
How vertex form differs from expanded form $y = ax^2 + bx + c$

Can Do

Read $a$, vertex, axis and direction directly from vertex form
Expand vertex form to general form $y = ax^2 + bx + c$
Compare two parabolas by reading their vertex-form parameters

Words You Need

vocabulary

Vertex form$y = a(x - h)^2 + k$ — the form that reveals the vertex.

General/expanded form$y = ax^2 + bx + c$ — the form you get after multiplying out.

Parameter $a$Controls direction (sign) and width (magnitude).

Parameter $h$Horizontal position of the vertex. Note the sign flip: $(x - h)$.

Parameter $k$Vertical position of the vertex. No sign flip.

Axis of symmetryVertical line $x = h$ that the parabola is mirror-symmetric about.

Spot the Trap

heads-up

Wrong: Saying $y = 2(x + 3)^2 - 5$ has vertex $(3, -5)$.

Right: $(x + 3) = (x - (-3))$, so $h = -3$. Vertex $(-3, -5)$.

Wrong: Expanding $(x - 3)^2$ as $x^2 - 9$.

Right: $(x - 3)^2 = x^2 - 6x + 9$. There is a middle $-2 \times 3 \times x$ term.

Reading Vertex Form

+5 XP

Three reads, in this order:

$a$: the coefficient out the front. Sign $\to$ direction. Size $\to$ width.
$h$: inside the bracket, FLIP the sign. $(x - 3) \Rightarrow h = 3$. $(x + 5) \Rightarrow h = -5$.
$k$: the constant on the end. KEEP the sign.

Vertex is $(h, k)$. Axis of symmetry is $x = h$.

$y = a(x - h)^2 + k$ — read $a$, flip $h$, keep $k$.

$|a| > 1$ narrow

Steep walls, thin curve.

$|a| < 1$ wide

Flatter, lazier curve.

Min vs Max

$a > 0$: vertex is MIN. $a < 0$: vertex is MAX.

Converting Vertex Form to Expanded Form

+5 XP

To go from $y = a(x - h)^2 + k$ to $y = ax^2 + bx + c$, multiply out using $(x - h)^2 = x^2 - 2hx + h^2$.

Expand the bracket: $(x - h)^2 = x^2 - 2hx + h^2$.
Multiply by $a$: $a(x - h)^2 = ax^2 - 2ahx + ah^2$.
Add $k$: $y = ax^2 - 2ahx + (ah^2 + k)$.
Compare coefficients: $b = -2ah$, $c = ah^2 + k$.

Example: $y = 2(x - 3)^2 - 5 = 2(x^2 - 6x + 9) - 5 = 2x^2 - 12x + 18 - 5 = 2x^2 - 12x + 13$.

Vertex form $\longrightarrow$ expanded form via $(x - h)^2 = x^2 - 2hx + h^2$.

Square FIRST

Expand the bracket BEFORE multiplying by $a$.

Three terms not two

$(x - h)^2$ has THREE terms: $x^2$, $-2hx$, $h^2$.

Tidy at end

Combine the constant $ah^2 + k$ last.

Watch Me Solve It · 3 examples

Watch Me Solve It · Read features from $y = 2(x - 3)^2 - 5$

+15 XP per step

PROBLEM

State $a$, vertex, axis of symmetry, direction and width for $y = 2(x - 3)^2 - 5$.

1
Identify $a$
$a = 2$. Positive $\to$ opens UP. $|a| = 2 > 1 \to$ narrower than $y = x^2$.
2
Identify $h$ and $k$
$(x - 3) \Rightarrow h = 3$ (flip the sign). $-5 \Rightarrow k = -5$ (keep the sign).
3
State features
Vertex $(3, -5)$ (a MIN). Axis of symmetry: $x = 3$.
All four features come straight from the three parameters.

Answer$a = 2$ (up, narrow), vertex $(3, -5)$, axis $x = 3$.

Watch Me Solve It · Read features from $y = -\tfrac{1}{2}(x + 4)^2 + 3$

+15 XP per step

PROBLEM

State $a$, vertex, axis, direction and width for $y = -\tfrac{1}{2}(x + 4)^2 + 3$.

1
Identify $a$
$a = -\tfrac{1}{2}$. Negative $\to$ opens DOWN. $|a| = \tfrac{1}{2} < 1 \to$ WIDER than $y = x^2$.
2
Identify $h$ and $k$
$(x + 4) = (x - (-4)) \Rightarrow h = -4$. Constant $+3 \Rightarrow k = 3$.
3
State features
Vertex $(-4, 3)$ (a MAX since $a < 0$). Axis: $x = -4$.
Plus sign in the bracket flips to negative $h$.

Answer$a = -\tfrac{1}{2}$ (down, wide), vertex $(-4, 3)$ MAX, axis $x = -4$.

Watch Me Solve It · Expand $y = 3(x - 2)^2 + 1$

+15 XP per step

PROBLEM

Convert $y = 3(x - 2)^2 + 1$ to the expanded form $y = ax^2 + bx + c$.

1
Expand the bracket
$(x - 2)^2 = x^2 - 4x + 4$.
2
Multiply by $a = 3$
$3(x^2 - 4x + 4) = 3x^2 - 12x + 12$.
3
Add $k = 1$ and tidy
$y = 3x^2 - 12x + 12 + 1 = 3x^2 - 12x + 13$.
Check: vertex form had $a = 3$; expanded form starts with $3x^2$. Consistent.

Answer$y = 3x^2 - 12x + 13$.

Common Pitfalls

heads-up

Sign error on $h$

Reading $(x + 5)$ as $h = 5$.

Fix: rewrite as $(x - (-5))$ to expose $h = -5$. Sign FLIPS inside the bracket.

Forgetting middle term

Expanding $(x - 4)^2$ as $x^2 + 16$ or $x^2 - 16$.

Fix: $(x - 4)^2 = x^2 - 8x + 16$. Always THREE terms with a middle $-2hx$.

Multiplying $a$ too early

Trying $3(x - 2)^2 = (3x - 6)^2$ — wrong, because squaring isn't linear.

Fix: square the bracket FIRST, then multiply by $a$.

Copy Into Your Books

Vertex form

$y = a(x - h)^2 + k$
Vertex: $(h, k)$
Axis: $x = h$

Reading parameters

$a$: direction + width
$h$: FLIP sign in bracket
$k$: KEEP sign on end

Square identity

$(x - h)^2 = x^2 - 2hx + h^2$
THREE terms
Middle term $-2hx$

Expanding tips

Square FIRST
Multiply by $a$ SECOND
Combine constants LAST

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Vertex Form Drills

4 problems

Four quick reads and one expansion.

1 State vertex and axis for $y = 4(x - 1)^2 + 7$.
$h = 1$, $k = 7$.Vertex $(1, 7)$, axis $x = 1$
2 State vertex and direction for $y = -(x + 2)^2 - 6$.
$h = -2$, $k = -6$, $a = -1$.Vertex $(-2, -6)$, opens down (MAX)
3 Expand $(x - 5)^2$.
$x^2 - 2(5)x + 25$.$x^2 - 10x + 25$
4 Expand $y = 2(x - 1)^2 + 5$ to general form.
$2(x^2 - 2x + 1) + 5 = 2x^2 - 4x + 2 + 5$.$y = 2x^2 - 4x + 7$

Complete in your workbook.

Quick Check · 5 questions

The vertex of $y = 2(x - 3)^2 - 5$ is:

+10 XP

The axis of symmetry of $y = -\tfrac{1}{2}(x + 4)^2 + 3$ is:

+10 XP

For $y = -\tfrac{1}{2}(x + 4)^2 + 3$, the parabola is:

+10 XP

Expanded, $y = 3(x - 2)^2 + 1$ is:

+10 XP

For $y = 5(x + 1)^2 + 4$, the vertex is a:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyEasy3 MARKS

Q6. For $y = -2(x - 4)^2 + 9$, state $a$, the vertex, the axis of symmetry, the direction (up/down) and whether the vertex is a maximum or minimum.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. Expand $y = -(x - 5)^2 + 2$ into the form $y = ax^2 + bx + c$. Show all working.

Answer in your workbook.

ReasonHard3 MARKS

Q8. Parabola P: $y = 2(x - 1)^2 + 3$. Parabola Q: $y = -3(x + 2)^2 + 3$. (a) Which is wider? Justify. (b) Which has its vertex higher? Justify. (c) Which opens downward?

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $(x - 3) \Rightarrow h = 3$. Vertex $(3, -5)$.

2. A — $(x + 4) \Rightarrow h = -4$. Axis $x = -4$.

3. C — $a = -\tfrac{1}{2}$: down (sign) and wide (size $< 1$).

4. D — $3(x^2 - 4x + 4) + 1 = 3x^2 - 12x + 13$.

5. A — vertex $(-1, 4)$; $a = 5 > 0$ so MIN.

Show Your Working Model Answers

Q6 (3 marks): $a = -2$ [1]. $(x - 4) \Rightarrow h = 4$; $+9 \Rightarrow k = 9$. Vertex $(4, 9)$ [1]. Axis $x = 4$. $a < 0$ so opens DOWN; vertex is a MAXIMUM [1].

Q7 (3 marks): $(x - 5)^2 = x^2 - 10x + 25$ [1]. $-(x^2 - 10x + 25) = -x^2 + 10x - 25$ [1]. Add 2: $y = -x^2 + 10x - 23$ [1].

Q8 (3 marks): (a) P: $|a| = 2$; Q: $|a| = 3$. P is WIDER (smaller $|a|$) [1]. (b) P vertex $(1, 3)$, Q vertex $(-2, 3)$ — SAME height ($k = 3$ for both) [1]. (c) Q opens DOWN since $a = -3 < 0$; P opens up [1].

Stretch Challenge · +25 XP, +10 coins

Vertex Form from General Form (Completing the Square)

The parabola $y = x^2 - 6x + 11$ is in expanded form. Convert to vertex form $y = (x - h)^2 + k$ by completing the square. (a) Take half the coefficient of $x$ (that's $-3$); square it ($9$). (b) Write $x^2 - 6x + 9 - 9 + 11 = (x - 3)^2 + 2$. (c) State the vertex and axis. Now confirm by expanding $(x - 3)^2 + 2$.

Reveal solution

(a) Half of $-6$ is $-3$; square is $9$. (b) $y = x^2 - 6x + 9 - 9 + 11 = (x - 3)^2 + 2$. (c) Vertex $(3, 2)$, axis $x = 3$. Expanding $(x - 3)^2 + 2 = x^2 - 6x + 9 + 2 = x^2 - 6x + 11$. Matches.

Quick Review

Vertex form

$y = a(x - h)^2 + k$

Vertex

$(h, k)$

Axis

$x = h$

$h$ rule

FLIP sign inside bracket

$k$ rule

KEEP sign on end

Expand

$(x - h)^2 = x^2 - 2hx + h^2$

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