Chapter 1: Algebra and Functions

This chapter establishes the algebraic proficiency required for Mathematical Methods. We will move beyond basic calculations to focus on the relationships between variables, the nature of patterns, and the geometry of functions.

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1.1 Arithmetic and Geometric Sequences

In Methods, we view sequences as discrete functions where the input is the position $n$.

Arithmetic Sequences

A sequence where the difference between consecutive terms is constant ($d$).

• $n^{\text{th}}$ Term: $t_n = a + (n-1)d$
• Sum of $n$ Terms ($S_n$): $S_n = \frac{n}{2}(2a + (n-1)d)$

Geometric Sequences

A sequence where each term is found by multiplying the previous term by a constant ratio ($r$).

• $n^{\text{th}}$ Term: $t_n = ar^{n-1}$
• Sum of $n$ Terms ($S_n$): $S_n = \frac{a(r^n – 1)}{r – 1}$ (where $r \neq 1$)

Worked Example 1: Find the $12^{\text{th}}$ term of the geometric sequence: $3, 6, 12, 24, \dots$

1. Identify $a$ (first term) and $r$ (common ratio): $a = 3$, $r = 2$.
2. Use the formula $t_n = ar^{n-1}$: $t_{12} = 3 \times 2^{(12-1)}$.
3. Calculate: $t_{12} = 3 \times 2^{11} = 3 \times 2048 = 6144$.

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1.2 Quadratic Functions and the Discriminant

Quadratic functions take the form $f(x) = ax^2 + bx + c$. In Methods, we are specifically interested in the nature of the roots (where the graph hits the $x$-axis).

The Discriminant ($\Delta$)

The part of the quadratic formula under the square root, $\Delta = b^2 – 4ac$, tells us how many $x$-intercepts exist:

• $\Delta > 0$: Two distinct real roots (The graph crosses the $x$-axis twice).
• $\Delta = 0$: One real root (The graph touches the $x$-axis at its turning point).
• $\Delta < 0$: No real roots (The graph never touches the $x$-axis).

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1.3 Function Notation, Domain, and Range

This is a critical conceptual shift. We stop writing “y =” and start using “$f(x) =$” to emphasize that the output depends on the input.

• Domain: The set of all possible $x$-values (inputs) for which the function is defined.
• Range: The set of all possible $y$-values (outputs) the function can produce.

Worked Example 2: State the domain and range for $f(x) = x^2 + 3$.

1. Domain: Since we can square any real number, the domain is $x \in \mathbb{R}$ (all real numbers).
2. Range: Since $x^2$ is always $\ge 0$, the smallest value $f(x)$ can take is $0 + 3 = 3$. Therefore, the range is $f(x) \ge 3$.

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1.4 Practice Problems

Part A: Sequences

1. Find the sum of the first 20 terms of the arithmetic sequence: $5, 11, 17, \dots$
2. A geometric sequence has $a = 10$ and $r = 0.5$. Find the $5^{\text{th}}$ term and the sum to infinity ($S_\infty = \frac{a}{1-r}$).

Part B: Quadratics and Algebra

3. Use the discriminant to determine the number of solutions for $2x^2 – 4x + 7 = 0$.
4. Solve for $x$ by factorising: $x^2 – 5x – 14 = 0$.
5. Challenge: Find the value of $k$ such that $x^2 + 6x + k = 0$ has exactly one real solution.

Part C: Functions

6. Given $f(x) = 3x^2 – 5x$, find $f(2)$ and $f(-1)$.
7. Determine the range of $g(x) = -x^2 + 10$.

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Solutions (Summary)

• 1. $S_{20} = 1240$
• 2. $t_5 = 0.625$, $S_\infty = 20$
• 3. $\Delta = -40$; No real solutions.
• 4. $(x-7)(x+2) = 0 \rightarrow x = 7, -2$
• 5. $\Delta = 36 – 4k = 0 \rightarrow k = 9$
• 6. $f(2) = 2$, $f(-1) = 8$
• 7. $g(x) \le 10$