Algebra and functions

Introduction: Mastering Algebra and Functions (Pure Mathematics 1)

Welcome to the foundation! Algebra and Functions are the bedrock upon which almost all higher mathematics is built. Don't worry if this chapter seems tricky at first; we are taking complex ideas and breaking them down into simple, manageable steps.

In this section of P1, you will learn to solve equations that go beyond simple linear forms, understand how graphs behave, and manipulate functions like an expert. This knowledge is crucial for future topics like Calculus and Coordinate Geometry. Let's dive in!

Section 1: Solving Quadratic Equations

A quadratic equation is any equation that can be written in the standard form:
\[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants, and \(a \ne 0\). The solutions to this equation are called the roots.

Method 1: Factorising (The Quick Fix)

Factorising works best when the roots are simple integers. You look for two numbers that multiply to make \(c\) and add to make \(b\).

Example: Solve \(x^2 + 5x + 6 = 0\)
1. Find two numbers that multiply to 6 and add to 5 (These are 2 and 3).
2. Write in factored form: \((x + 2)(x + 3) = 0\)
3. Since the product is zero, one of the brackets must be zero.
4. \(x + 2 = 0 \implies x = -2\).
5. \(x + 3 = 0 \implies x = -3\).

Method 2: Completing the Square (Finding the Turning Point)

Completing the Square is essential because it allows you to easily find the turning point (vertex) of the quadratic graph, and it's key for many P1 questions regarding maxima or minima.

The goal is to rewrite \(ax^2 + bx + c\) in the form \(a(x + p)^2 + q\).

Step-by-Step Guide (for \(x^2 + bx + c\)):
1. Half the coefficient of \(x\): \(p = \frac{b}{2}\).
2. Write the squared bracket: \((x + p)^2\).
3. Subtract the extra constant term created by squaring: \((x + p)^2 - p^2 + c\).

Key Takeaway for Graphing: If \(y = a(x + p)^2 + q\), the turning point is always at \((-p, q)\).

Method 3: The Quadratic Formula (The Lifeline)

If factorising is impossible, the formula will always give you the roots (if they exist).

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Did you know? The Quadratic Formula is actually derived by applying the method of Completing the Square to the general form \(ax^2 + bx + c = 0\)!

Section 2: The Discriminant and Quadratic Inequalities

The Discriminant (\(\Delta\))

The discriminant is the part of the quadratic formula under the square root: \(\Delta = b^2 - 4ac\). It tells you about the nature (type and number) of the roots without solving the whole equation. Think of it as asking: "How many times does the graph hit the x-axis?"

• Case 1: \(\Delta > 0\) (Positive): There are two distinct, real roots. (Crosses the x-axis twice).
• Case 2: \(\Delta = 0\) (Zero): There is exactly one real root (or two identical roots). (Touches the x-axis exactly once).
• Case 3: \(\Delta < 0\) (Negative): There are no real roots. (Never touches or crosses the x-axis).

Solving Quadratic Inequalities

Solving inequalities (like \(x^2 + x - 6 > 0\)) requires a crucial extra step: drawing a sketch! This is mandatory for P1 success.

Common Mistake Alert: When solving \( -x^2 > 4 \), do not try to take the square root of both sides, and be careful when multiplying by \(-1\) (remember to flip the inequality sign!).

Step-by-Step for Inequalities (e.g., \(x^2 + x - 6 > 0\)):
1. Find the critical values: Treat as an equality and solve \(x^2 + x - 6 = 0\). The roots are \(x=-3\) and \(x=2\).
2. Sketch the graph: Since the \(x^2\) term is positive (\(a=1\)), the parabola is U-shaped ("happy face") and crosses the x-axis at -3 and 2.
3. Determine the region: We want where the graph is \(> 0\) (i.e., above the x-axis).
4. Write the solution: The solution involves two separate regions.
Solution: \(x < -3\) or \(x > 2\).

Section 3: Simultaneous Equations

In P1, you frequently solve simultaneous equations involving one linear equation (straight line) and one non-linear equation (e.g., a parabola or circle). We use the substitution method.

Step-by-Step Substitution (e.g., \(y = x + 1\) and \(y = x^2 - 11\)):
1. Isolate a variable in the linear equation.
2. Substitute this expression into the non-linear equation, resulting in a single-variable quadratic.
Substitution: \(x + 1 = x^2 - 11\)
3. Solve the resulting quadratic equation. (\(x = 4\) or \(x = -3\)).
4. Find the corresponding \(y\) values. Use the simple linear equation for this step to minimise calculation errors.
If \(x = 4\), \(y = 4 + 1 = 5\).
If \(x = -3\), \(y = -3 + 1 = -2\).
5. Write down the pairs. The solutions must be given as coordinate pairs: \((4, 5)\) and \((-3, -2)\).

Key Takeaway: If you find two \(x\) values, you MUST find two corresponding \(y\) values. You should end up with the same number of pairs as the number of \(x\) solutions.

Section 4: Functions, Domain, and Range

Function Notation and Definition

A function is a rule that maps an input value (the domain) to exactly one output value (the range). We use the notation:
\[ f: x \to \text{rule involving } x \quad \text{or} \quad y = f(x) \]

Analogy: A function is like a vending machine. You put in one specific input (money + button), and you get exactly one corresponding output (a specific snack).

Domain and Range

• Domain: The set of all possible input values (\(x\)).
• Range: The set of all possible output values (\(f(x)\) or \(y\)).

For P1 functions, the domain is usually all real numbers (\(x \in \mathbb{R}\)) unless two common restrictions apply:
1. You cannot divide by zero: Exclude \(x\) values that make the denominator zero.
2. You cannot take the square root of a negative number: Ensure the term inside the root is \(\ge 0\).

To find the Range of a quadratic function, always find the turning point \((p, q)\).
If the graph is U-shaped (\(a>0\)), the range is \(f(x) \ge q\).
If the graph is N-shaped (\(a<0\)), the range is \(f(x) \le q\).

Section 5: Transformations of Graphs

Transformations allow us to draw a new graph \(y=g(x)\) by altering the graph of a known function \(y=f(x)\).

The Golden Rule of Transformations

This is the most important concept to master:

1. When a change is applied OUTSIDE the function (\(f(x) + a\)), it affects the y-coordinate and behaves intuitively (as expected).
2. When a change is applied INSIDE the function (\(f(x + a)\)), it affects the x-coordinate and behaves counter-intuitively (the opposite of what you expect).

1. Translations (Shifts)

Moving the graph without changing its shape.

• \(y = f(x) + a\): Vertical shift by \(a\) units (up if \(a>0\)).
• \(y = f(x + a)\): Horizontal shift by \(-a\) units (left if \(a>0\)).

Example: \(f(x+3)\) moves the graph 3 units to the LEFT (the opposite of +3).

2. Stretches

Changing the distance from the axis.

• \(y = a f(x)\): Vertical stretch by scale factor \(a\).
• \(y = f(ax)\): Horizontal stretch by scale factor \(\frac{1}{a}\).

3. Reflections

• \(y = -f(x)\): Reflection in the x-axis (negates the y-values).
• \(y = f(-x)\): Reflection in the y-axis (negates the x-values).

Tip for Combined Transformations: If a question involves both stretches/reflections and translations, always apply the stretches/reflections first, followed by the translations (shifting).

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