1. Quadratic Functions

🧠 What This Chapter Is About

In Additional Mathematics, quadratics are no longer just equations to solve — they are functions.

This chapter focuses on:

• understanding how a quadratic function behaves,

• linking algebraic forms to graph features, and

• interpreting quadratics in real-world contexts.

📌 Unlike E-Math, we often do not solve for $x$ immediately. Instead, we study the shape, position, and conditions of the curve.

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🔹 1.1 Quadratic Functions of the Form $y=a(x-p)(x-q)$

🧠 Key Concepts

The form:

$$y=a(x-p)(x-q)$$

shows:

• where the graph cuts the $x$-axis, and

• how the graph opens.

📌 This form highlights the roots directly.

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🏷️ Definitions & Terminology

• Root / zero: the value of $x$ when $y=0$.

• $x$-intercept: the point where the graph crosses the $x$-axis.

From $y=a(x-p)(x-q)$:

• Roots are $x=p$ and $x=q$.

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📐 Understanding the Parameters

• $a$ controls the opening direction and steepness.

  • $a>0$: opens upwards

  • $a<0$: opens downwards

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✏️ Worked Example

Sketch the graph of:

$$y=2(x-1)(x-3)$$

• Roots: $x=1$, $x=3$

• Opens upwards since $a=2>0$

📌 Always mark the intercepts before sketching.

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❌ Common Mistakes ⚠️

• ❌ Treating $(x-p)(x-q)$ as an equation instead of a function.

• ❌ Forgetting to write $y=$ before sketching.

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🔹 1.2 Quadratic Functions of the Form $y=a(x-h{)}^2+k$

🧠 Key Concepts

This form shows:

• the turning point directly,

• how the graph is shifted.

📌 This is the most powerful form for graph interpretation.

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🏷️ Definitions & Terminology

• Turning point (vertex): the maximum or minimum point of the curve.

From $y=a(x-h{)}^2+k$:

• Turning point is $(h,k)$.

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📐 Understanding the Parameters

• $a$: opening and steepness

• $h$: horizontal shift

• $k$: vertical shift

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✏️ Worked Example

Find the turning point of:

$$y=-3(x+2{)}^2+5$$

Turning point:

$$(-2,5)$$

Graph opens downwards since $a<0$.

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❌ Common Mistakes ⚠️

• ❌ Writing turning point as $(2,5)$ instead of $(-2,5)$.

• ❌ Mixing up $h$ and $k$.

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🔹 1.3 Conditions for Quadratic Curve to Lie Completely Above or Below the $x$-Axis

🧠 Key Concepts

A curve lies:

• above the $x$-axis if $y>0$ for all $x$,

• below the $x$-axis if $y<0$ for all $x$.

This depends on:

• the sign of $a$, and

• whether the curve cuts the $x$-axis.

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📐 Conditions

For $y=a{x}^2+bx+c$:

• Curve lies completely above $x$-axis if:

  • $a>0$ and $b^2-4ac<0$

• Curve lies completely below $x$-axis if:

  • $a<0$ and $b^2-4ac<0$

📌 The discriminant tells us whether the curve has real roots.

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❌ Common Mistakes ⚠️

• ❌ Checking only the sign of $a$.

• ❌ Forgetting the discriminant condition.

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🔹 1.4 Quadratic Functions in Real-World Contexts

🧠 Key Concepts

Quadratic functions model situations involving:

• maximum or minimum values,

• projectile motion,

• optimisation problems.

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✏️ Worked Example

The height $h$ of a ball (in metres) is given by:

$$h=-5t^2+20t+1$$

• Maximum height occurs at the turning point.

• Graph opens downwards, so turning point is a maximum.

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✅ Quick Strategy 🎯 (A-Math Quadratics)

1. Decide which form of the quadratic is most useful.

2. Identify key features (roots or turning point).

3. Sketch with correct shape and intercepts.

4. Use the discriminant when reasoning about positions relative to the $x$-axis.

5. Interpret results in context — not just algebraically.