Further Pure Mathematics | The quadratic function

Welcome to the study notes for "The Quadratic Function"! This chapter forms the foundation for many advanced topics in Further Pure Mathematics. Don't worry if some concepts look familiar from core maths; here, we dive deeper into the relationships between the roots and the coefficients, which is essential for problem-solving.

We will break down complex concepts like the Discriminant and the Sum and Product of Roots into easy-to-manage steps. Let’s get started!

The Basics: Standard Form and Structure

1. The General Form

The standard definition of a quadratic function (or equation) is:
\[ ax^2 + bx + c = 0 \] Where \(a\), \(b\), and \(c\) are constants, and crucially, \(a \neq 0\).

The graph of a quadratic function \(y = ax^2 + bx + c\) is a parabola.

• If \(a > 0\), the parabola opens upwards (U-shape) and has a minimum turning point.
• If \(a < 0\), the parabola opens downwards (n-shape) and has a maximum turning point.

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2. The Discriminant: Predicting the Roots

In Further Pure Maths, we often need to know how many real solutions (roots) an equation has without actually solving it. This is where the Discriminant comes in.

The discriminant is the part of the quadratic formula that lives under the square root sign:

\[ \text{Discriminant, } \Delta = b^2 - 4ac \]

Think of the discriminant as a fortune teller: it tells you the nature of the roots!

Key Cases for the Discriminant

The sign of \(\Delta\) determines the nature of the roots (the \(x\)-intercepts of the graph):

Case 1: Two Distinct Real Roots (\(\Delta > 0\))

If \(b^2 - 4ac > 0\):

• The equation has two different real numbers as solutions.
• The graph crosses the \(x\)-axis twice.
• Did you know? If \(\Delta\) is also a perfect square (e.g., 9, 25), the roots will be rational (they can be written as fractions).

Case 2: One Repeated Real Root (\(\Delta = 0\))

If \(b^2 - 4ac = 0\):

• The equation has exactly one real solution (often called a repeated root or tangent point).
• The graph touches the \(x\)-axis at exactly one point (the turning point lies on the \(x\)-axis).

Case 3: No Real Roots (\(\Delta < 0\))

If \(b^2 - 4ac < 0\):

• Since you cannot take the square root of a negative number in the real number system, the equation has no real solutions.
• The graph does not cross or touch the \(x\)-axis. It lies entirely above or below the axis.

Step-by-Step Example

Question: Determine the nature of the roots for \(2x^2 - 5x + 4 = 0\).

1. Identify \(a, b, c\): \(a=2\), \(b=-5\), \(c=4\).
2. Calculate the Discriminant: \[ \Delta = b^2 - 4ac = (-5)^2 - 4(2)(4) \]
3. Simplify: \[ \Delta = 25 - 32 = -7 \]
4. Conclude: Since \(\Delta = -7\) (which is less than 0), there are no real roots.

Key Takeaway: The Discriminant is the fastest way to check if a quadratic function intersects the \(x\)-axis.

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3. Relationships Between Roots and Coefficients

This section is central to Further Pure Mathematics. When a quadratic equation \(ax^2 + bx + c = 0\) has roots \(\alpha\) (alpha) and \(\beta\) (beta), there are powerful relationships connecting these roots directly to the coefficients \(a\), \(b\), and \(c\).

3.1 Sum of the Roots

The sum of the roots is given by:

\[ \alpha + \beta = -\frac{b}{a} \]

Remember: The negative sign is crucial here!

3.2 Product of the Roots

The product of the roots is given by:

\[ \alpha \beta = \frac{c}{a} \]

Memory Aid: S-P Formula

We can use these relations to instantly write down a quadratic equation if we know the sum (\(S\)) and product (\(P\)) of its roots.

If the equation is written with \(a=1\), it becomes:

\[ x^2 - (\text{Sum of Roots})x + (\text{Product of Roots}) = 0 \] \[ x^2 - Sx + P = 0 \]

Analogy: Always remember the minus sign for the sum! \(S\)tart with \(S\)ubtracting \(S\).

3.3 Forming New Equations

A common FPM task is using known roots to form a new quadratic equation whose roots are related to the originals.

Step-by-Step Process:

1. Start with the original equation, \(ax^2 + bx + c = 0\). Find \(S_o = \alpha + \beta = -b/a\) and \(P_o = \alpha \beta = c/a\).
2. Define the New Roots (e.g., \(\alpha' = 2\alpha\) and \(\beta' = 2\beta\)).
3. Calculate the New Sum (\(S_n\)): \(S_n = \alpha' + \beta'\). Express this in terms of \(S_o\).
4. Calculate the New Product (\(P_n\)): \(P_n = \alpha' \beta'\). Express this in terms of \(P_o\).
5. Form the new equation: \(x^2 - S_n x + P_n = 0\). (If \(a \neq 1\), you may need to multiply through to clear fractions).

Common Mistake to Avoid: When calculating the new product, remember to multiply the entire expressions. If new roots are \((\alpha-1)\) and \((\beta-1)\), the product is \((\alpha-1)(\beta-1)\), which expands to \(\alpha\beta - (\alpha+\beta) + 1\). Don't just multiply the individual modifications!

Key Takeaway: The relationships \(\alpha + \beta = -b/a\) and \(\alpha \beta = c/a\) are incredibly powerful tools for solving problems without finding the actual values of \(\alpha\) and \(\beta\).

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4. Completing the Square and the Vertex

While the standard form \(ax^2 + bx + c\) is useful for the discriminant and roots, the Completed Square Form is vital for identifying the turning point (the vertex) and understanding the range of the function.

4.1 The Completed Square Form

Any quadratic expression can be written in the form:

\[ a(x+p)^2 + q \]

Where \(p\) and \(q\) are constants derived from \(a\), \(b\), and \(c\).

4.2 Finding the Vertex (Turning Point)

The main advantage of the completed square form \(a(x+p)^2 + q\) is that the Vertex (the minimum or maximum point of the parabola) is immediately visible:

Vertex Coordinates: \((-p, q)\)

The line of symmetry is the vertical line passing through the vertex, given by \(x = -p\).

4.3 Step-by-Step Guide to Completing the Square (When \(a \neq 1\))

Let's complete the square for \(y = 3x^2 + 12x - 5\).

1. Factor out \(a\) from the first two terms only: \[ y = 3(x^2 + 4x) - 5 \]
2. Complete the square inside the bracket: Take half of the coefficient of \(x\) (which is 4/2 = 2). Square it (2² = 4). Add and immediately subtract this value inside the bracket: \[ y = 3(x^2 + 4x + 4 - 4) - 5 \]
3. Group the perfect square: \[ y = 3((x+2)^2 - 4) - 5 \]
4. Distribute the \(a\) value (3) back to the constant term inside the bracket: \[ y = 3(x+2)^2 - 3(4) - 5 \]
5. Simplify to find \(q\): \[ y = 3(x+2)^2 - 12 - 5 \] \[ y = 3(x+2)^2 - 17 \]

From this result, the vertex is \((-2, -17)\). Since \(a=3\) (positive), this is the minimum point.

Alternative Way to Find the Vertex (Using the Formula)

You can always find the \(x\)-coordinate of the vertex using the formula for the line of symmetry:

\[ x = -\frac{b}{2a} \]

Once you find the \(x\)-value, substitute it back into the original equation \(y = ax^2 + bx + c\) to find the \(y\)-coordinate.

Key Takeaway: The form \(a(x+p)^2 + q\) immediately identifies the turning point \((-p, q)\), which is crucial for determining the maximum or minimum value of the function (the range).

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5. Sketching Quadratic Graphs

To sketch a quadratic graph accurately, you need three key pieces of information:

i. Shape (Concavity)

Determined by \(a\). Is it U-shaped (\(a>0\)) or n-shaped (\(a<0\))?

ii. \(y\)-intercept

Set \(x=0\). The \(y\)-intercept is always the constant term, \(c\).

iii. \(x\)-intercepts (Roots)

Set \(y=0\). These are the roots found by factoring, the quadratic formula, or if the discriminant suggests there are no real roots, the curve will not cross the \(x\)-axis.

iv. Turning Point (Vertex)

Found using Completing the Square or the symmetry formula \(x = -b/2a\). This determines the minimum or maximum height of the curve.

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Quick Review: FPM Quadratic Essentials

You’ve covered the core theory of quadratic functions for Further Pure Mathematics! Practice using the Sum and Product relations, as they are the most frequent examination application of this chapter. Good luck!