Quadratic functions - Mathematics - Additional (0606) - Cambridge IGCSE

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Chapter 2: Quadratic Functions – Understanding the Curve of Life

Hello everyone! Welcome to the exciting world of Quadratic Functions. You’ve met these functions before in your regular IGCSE Maths, but in Additional Mathematics, we dive much deeper into what makes these curves tick.

This chapter is fundamental because quadratic functions, often resulting in a beautiful U-shaped curve called a parabola, appear everywhere—from modeling the trajectory of a thrown ball to determining the optimum price for a product. Mastering these techniques—especially completing the square and the discriminant—is essential for success in 0606.

1. The Anatomy of a Quadratic Function

A quadratic function is typically written in one of two forms.

Standard Form

The most common form is:
$$f(x) = ax^2 + bx + c$$

$x$ is the variable.
$a$, $b$, and $c$ are constants, and crucially, $a$ cannot be zero (otherwise, it wouldn't be quadratic!).

The Graph: The Parabola Shape

The graph of a quadratic function is always a parabola. The sign of $a$ determines the shape:

If $a > 0$ (positive): The parabola opens upwards, like a smiley face. It has a minimum point (the lowest point).
If $a < 0$ (negative): The parabola opens downwards, like a frowning face. It has a maximum point (the highest point).

2. Finding the Maximum or Minimum Value (The Vertex)

The maximum or minimum point of the parabola is called the vertex. Finding the vertex is critical for sketching graphs and determining the range.

Method 1: Completing the Square (CTS)

Completing the square transforms the standard form into the Vertex Form:
$$f(x) = a(x + p)^2 + q$$

Did you know? The method of completing the square gives you the exact coordinates of the vertex, $(-p, q)$.

Step-by-Step CTS Process

Let's find the vertex for $f(x) = 2x^2 - 12x + 5$:

Factor out $a$ from the $x^2$ and $x$ terms:
$$f(x) = 2(x^2 - 6x) + 5$$
Complete the square inside the bracket using $(\frac{b}{2})^2$. (Here, half of -6 is -3, and $(-3)^2 = 9$):
$$f(x) = 2(x^2 - 6x + 9 - 9) + 5$$
Rewrite the perfect square and move the extra constant out of the bracket (remembering to multiply by the factor $a$):
$$f(x) = 2((x - 3)^2 - 9) + 5$$ $$f(x) = 2(x - 3)^2 - 18 + 5$$
Simplify to get the vertex form $a(x+p)^2 + q$:
$$f(x) = 2(x - 3)^2 - 13$$

The vertex is at $(3, -13)$. Since $a=2$ is positive, this is a minimum value of $-13$.

Method 2: Using Differentiation (Advanced Technique - 2.1)

If you are confident with Calculus (Chapter 14), you can find the x-coordinate of the vertex by setting the derivative to zero:

Find the derivative, $\frac{dy}{dx}$ (or $f'(x)$).
Set $\frac{dy}{dx} = 0$ and solve for $x$.
Substitute this $x$-value back into the original function to find the maximum or minimum $y$-value.

Quick Takeaway: Completing the square is the most reliable way to find the exact vertex and is crucial for advanced applications and accurate sketching.

3. Sketching and Determining the Range (2.2)

Once you know the maximum/minimum point, you can accurately sketch the graph and define the Range of the function.

Range Determination

The range is the set of all possible output values (y-values) the function can take.

For $f(x) = 2(x - 3)^2 - 13$, since it has a minimum value of -13, the function is always greater than or equal to -13.
Range: $f(x) \ge -13$
If the function had a maximum value of 5 (e.g., $f(x) = - (x+1)^2 + 5$), the function is always less than or equal to 5.
Range: $f(x) \le 5$

Required Sketch Features

When sketching $y = f(x)$, you must clearly label the key features:

Vertex (Maximum or Minimum Point).
Y-intercept (found by setting $x=0$).
X-intercepts (if they exist, found by setting $y=0$ and solving the quadratic equation).

4. Solving Quadratic Equations (2.4)

Solving a quadratic equation, $ax^2 + bx + c = 0$, means finding the roots (the values of $x$ where the graph crosses the x-axis, i.e., the x-intercepts).

Methods for Finding Roots

Factorisation: Quickest, but only works if the roots are rational.
Completing the Square: Works for all real roots. Often used to prove or derive formulas.
Quadratic Formula: Always works for finding real roots.
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
(Remember this formula is provided in the exam!)

Common Mistake: When using the formula, ensure you use brackets correctly, especially when $b$ or $c$ are negative. A negative $b$ in the numerator becomes positive!

5. The Discriminant: Nature of the Roots (2.3)

The most powerful tool for quadratic analysis in Additional Mathematics is the Discriminant. It tells us about the nature of the roots without needing to solve the equation fully.

The discriminant is the expression found under the square root sign in the quadratic formula:
$$\Delta = b^2 - 4ac$$

Conditions for the Nature of Roots

The value of $\Delta$ tells us how many times the graph $y = ax^2 + bx + c$ intersects the x-axis:

Condition	Nature of Roots	Graph Connection
$\Delta > 0$	Two distinct real roots	The curve crosses the x-axis at two separate points.
$\Delta = 0$	Two equal real roots (or one repeated root)	The curve just touches the x-axis (it is a tangent to the axis).
$\Delta < 0$	No real roots	The curve lies entirely above or entirely below the x-axis (no intersection).

Memory Aid: Think of the discriminant as deciding the relationship status of the parabola with the x-axis!

6. Intersections between Lines and Curves (2.3 Related)

The discriminant is not just for the x-axis. It is used whenever you analyze the intersection between a line ($y = mx + c$) and a curve ($y = ax^2 + bx + c$).

The Process

Equate the two functions: Set the linear equation equal to the quadratic equation (Substitution or Elimination).
$$ax^2 + bx + c = mx + d$$
Rearrange into a single quadratic equation: Move all terms to one side to get the form:
$$Ax^2 + Bx + C = 0$$ (Note: A, B, and C are the new coefficients resulting from the rearrangement.)
Calculate the Discriminant $\Delta = B^2 - 4AC$.

Relationship Conditions

The value of the discriminant determines the relationship between the line and the curve:

$\Delta > 0$ (Intersect): The line intersects the curve at two distinct points.
$\Delta = 0$ (Tangent): The line touches the curve at exactly one point. (This is the condition for tangency).
$\Delta < 0$ (No Intersect): The line does not cross or touch the curve.

7. Solving Quadratic Inequalities (2.5)

Quadratic inequalities ask, "For what values of $x$ is the parabola above or below a certain line (usually $y=0$)?"

The Crucial 3-Step Strategy

Step 1: Find the Critical Values (The Roots)

Temporarily replace the inequality sign ($<$, $>$, $\le$, $\ge$) with an equals sign ($=$) and solve the resulting quadratic equation to find the roots (the critical values).

Example: Solve $x^2 - 4x - 5 > 0$
Set $x^2 - 4x - 5 = 0$. Factorise: $(x-5)(x+1) = 0$.
Critical values are $x = 5$ and $x = -1$.

Step 2: Sketch the Graph

Draw a simple sketch of the parabola, marking the critical values on the x-axis. This is the fastest way to determine the correct region.

Since the coefficient of $x^2$ is positive ($a=1$), the parabola is a minimum/happy face.

Step 3: Determine the Solution Set

Look at your sketch and determine which $x$-values satisfy the original inequality.

We want $x^2 - 4x - 5 > 0$ (i.e., where the curve is above the x-axis).

From the sketch, the curve is above the axis when $x$ is to the left of $-1$ or to the right of $5$.
Solution: $x < -1$ or $x > 5$

Important Notation: Always write your solution set clearly and correctly.

If the required region is between the roots (e.g., $x^2 < 0$), use a compound inequality: $-1 < x < 5$.
If the required region is outside the roots (e.g., $x^2 > 0$), use two separate inequalities joined by 'or': $x < -1$ or $x > 5$.
Be careful with the signs! If the inequality includes 'equal to' ($\le$ or $\ge$), then the solution must also include 'equal to' ($\le$ or $\ge$).

Quick Review: Quadratic Functions Checklist

Can I complete the square to find the vertex?
Can I use the vertex to determine the range and sketch the graph?
Do I know how the discriminant $\Delta = b^2 - 4ac$ relates to the number of roots?
Can I apply the discriminant to simultaneous equations to determine if a line is a tangent?
When solving quadratic inequalities, do I sketch the graph to find the solution set accurately?

Keep practising these skills, and you will find quadratic functions manageable and rewarding!