Welcome to AS 1 Pure Mathematics: Algebra and Functions
Welcome to the core foundation of your CCEA A-Level Mathematics journey! Think of algebra as the universal toolkit of mathematics. Just like an engineer needs to master basic tools before building a bridge, mastering algebra and functions will unlock your ability to tackle calculus, coordinate geometry, mechanics, and statistics with confidence.
Don't worry if some of these topics felt daunting at GCSE. Here, we will break down every concept step-by-step with clear explanations, intuitive memory tricks, and worked examples so that you can score top marks on your exam.
1. Laws of Indices and Working with Surds
Laws of Indices
Indices (powers) tell us how many times a number is multiplied by itself. At A-Level, we frequently deal with negative and fractional indices. Let us review the essential rules:
• Multiplication rule: \(a^m \times a^n = a^{m+n}\)
• Division rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{mn}\)
• Zero power: \(a^0 = 1\) (where \(a \neq 0\))
• Negative power: \(a^{-n} = \frac{1}{a^n}\)
• Fractional power: \(a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\)
Memory Trick for Fractional Powers: Remember "Flower Power"! In a plant, the flower is on top (the power, \(m\)) and the root is underground at the bottom (the root, \(n\)). So in \(x^{\frac{2}{3}}\), the root is \(3\) (cube root) and the power is \(2\) (squared): \((\sqrt[3]{x})^2\).
Manipulating and Simplifying Surds
A surd is an irrational root, such as \(\sqrt{2}\) or \(\sqrt{7}\). The primary rules for surds are:
• Multiplication: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)
• Division: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)
• Addition/Subtraction: You can only combine like surds, e.g. \(3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}\).
Rationalising the Denominator
Mathematicians prefer not to have square roots on the bottom of a fraction. Rationalising means eliminating the surd from the denominator without changing the value of the expression.
Case 1: Single term in the denominator
Multiply numerator and denominator by the square root:
\(\frac{6}{\sqrt{3}} = \frac{6 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}\)
Case 2: Binomial in the denominator (e.g. \(a + \sqrt{b}\) or \(\sqrt{a} - \sqrt{b}\))
Multiply numerator and denominator by the conjugate (flip the middle sign):
\(\frac{4}{3 - \sqrt{5}} = \frac{4(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})} = \frac{4(3 + \sqrt{5})}{3^2 - (\sqrt{5})^2} = \frac{12 + 4\sqrt{5}}{9 - 5} = \frac{12 + 4\sqrt{5}}{4} = 3 + \sqrt{5}\)
Common Mistake to Avoid: \(\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}\). For instance, \(\sqrt{9 + 16} = \sqrt{25} = 5\), whereas \(\sqrt{9} + \sqrt{16} = 3 + 4 = 7\).
Key Takeaway: Always simplify surds by looking for square factor numbers (\(4, 9, 16, 25, 36, \dots\)) inside the root, and always multiply by the conjugate to clear bracketed roots in denominators.
2. Quadratic Equations, Completing the Square, and the Discriminant
Solving Quadratic Equations
A quadratic expression has the standard form \(ax^2 + bx + c = 0\) (where \(a \neq 0\)). You can solve quadratic equations by:
1. Factorising: Finding two brackets \((px + q)(rx + s) = 0\).
2. The Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
3. Completing the Square.
Completing the Square
Completing the square rewrites \(ax^2 + bx + c\) into the form \(a(x + p)^2 + q\). This form is exceptionally useful for finding the maximum or minimum value (vertex/turning point) of a parabola.
Step-by-Step for \(x^2 + bx + c\):
1. Halve the coefficient of \(x\): \(\frac{b}{2}\).
2. Write \(\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c\).
3. Simplify the constant numbers.
Example: Complete the square for \(x^2 - 6x + 13\):
\(\left(x - 3\right)^2 - (-3)^2 + 13 = (x - 3)^2 - 9 + 13 = (x - 3)^2 + 4\)
• Turning point (vertex): \((3, 4)\)
• Minimum value of the function: \(4\) (which occurs when \(x = 3\)).
If \(a \neq 1\), factor out \(a\) from the \(x^2\) and \(x\) terms first:
\(2x^2 + 8x + 3 = 2[x^2 + 4x] + 3 = 2[(x + 2)^2 - 4] + 3 = 2(x + 2)^2 - 8 + 3 = 2(x + 2)^2 - 5\).
The Discriminant
The expression under the square root in the quadratic formula is called the discriminant, denoted by \(\Delta = b^2 - 4ac\). It reveals the nature and number of real roots without solving the equation:
• If \(b^2 - 4ac > 0\): There are two distinct real roots (the curve crosses the \(x\)-axis twice).
• If \(b^2 - 4ac = 0\): There is one repeated real root / two equal real roots (the curve touches the \(x\)-axis at its turning point).
• If \(b^2 - 4ac < 0\): There are no real roots (the curve does not cross or touch the \(x\)-axis).
Exam Tip: If a question states that an equation has "real roots", this means they could be distinct or equal, so you must use the condition \(b^2 - 4ac \ge 0\).
Key Takeaway: Completing the square immediately provides the coordinate of the turning point \((-p, q)\) for \(y = a(x + p)^2 + q\), while the discriminant \(b^2 - 4ac\) tells you how many times the curve intersects the \(x\)-axis.
3. Simultaneous Equations and Inequalities
Simultaneous Equations (One Linear, One Non-Linear)
To find the points where a line intersects a curve (such as a circle or parabola), solve them simultaneously using the substitution method:
1. Rearrange the linear equation to make either \(x\) or \(y\) the subject.
2. Substitute this expression into the non-linear equation.
3. Solve the resulting quadratic equation to find the values of the first variable.
4. Substitute these values back into the linear equation to find the corresponding values of the second variable.
5. Pair your solutions clearly as coordinates \((x_1, y_1)\) and \((x_2, y_2)\).
Linear and Quadratic Inequalities
Linear Inequalities: Solve just like linear equations, but remember: if you multiply or divide by a negative number, you must reverse the inequality sign!
Quadratic Inequalities: Never treat a quadratic inequality like a simple linear equation! Follow this three-step process:
1. Rearrange so that zero is on one side, e.g. \(ax^2 + bx + c > 0\) or \(ax^2 + bx + c \le 0\).
2. Find the critical values by solving the equation \(ax^2 + bx + c = 0\).
3. Sketch the parabola:
• For \(> 0\) (or \(\ge 0\)), choose the regions above the \(x\)-axis: two separate intervals, e.g. \(x < x_1\) or \(x > x_2\).
• For \(< 0\) (or \(\le 0\)), choose the region below the \(x\)-axis: one combined interval, e.g. \(x_1 < x < x_2\).
Example: Solve \(x^2 - 5x - 6 > 0\)
Factorise: \((x - 6)(x + 1) = 0 \implies\) Critical values are \(x = -1\) and \(x = 6\).
Since the curve opens upwards (\(a = 1 > 0\)) and we want \(> 0\) (above the axis), the solution is:
\(x < -1\) or \(x > 6\).
Key Takeaway: Always sketch the parabola when solving quadratic inequalities. "Inside the roots" gives \(a < x < b\) for \(< 0\); "outside the roots" gives \(x < a\) or \(x > b\) for \(> 0\).
4. Polynomials: Factor and Remainder Theorems
Polynomial Division
A polynomial is an expression containing terms of the form \(a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0\). We can divide a higher-order polynomial by a linear factor \((x - a)\) using algebraic long division to obtain a quotient and remainder.
The Remainder Theorem
When a polynomial \(f(x)\) is divided by \((ax - b)\), the remainder is equal to \(f\left(\frac{b}{a}\right)\).
Example: Find the remainder when \(f(x) = 2x^3 - 3x^2 + 5x - 4\) is divided by \((x - 2)\):
Evaluate \(f(2) = 2(2)^3 - 3(2)^2 + 5(2) - 4 = 16 - 12 + 10 - 4 = 10\).
The remainder is \(10\).
The Factor Theorem
The Factor Theorem is a special case of the Remainder Theorem where the remainder is zero:
• If \(f(a) = 0\), then \((x - a)\) is a factor of \(f(x)\).
• Conversely, if \((x - a)\) is a factor of \(f(x)\), then \(f(a) = 0\).
• For \((ax - b)\), if \(f\left(\frac{b}{a}\right) = 0\), then \((ax - b)\) is a factor.
Step-by-Step: Factorising a Cubic \(f(x)\):
1. Use trial and error with factors of the constant term (try \(x = \pm 1, \pm 2, \pm 3\)) to find a value where \(f(a) = 0\).
2. Write down the linear factor \((x - a)\).
3. Use algebraic long division or inspection to find the quadratic quotient: \(f(x) = (x - a)(Ax^2 + Bx + C)\).
4. Factorise the remaining quadratic expression if possible.
Key Takeaway: Finding roots of a polynomial means finding values of \(x\) that make \(f(x) = 0\). The Factor Theorem turns roots into linear factors.
5. Graphs of Functions
Graph Shapes You Must Know
• Cubic Curves (\(y = ax^3 + bx^2 + cx + d\)):
If \(a > 0\): Starts bottom-left, goes up, turns down, then goes up to top-right (\(\nearrow\)).
If \(a < 0\): Starts top-left, goes down, turns up, then goes down to bottom-right (\(\searrow\)).
• Reciprocal Curves:
\(y = \frac{k}{x}\): Has asymptotes along the \(x\)-axis (\(y = 0\)) and \(y\)-axis (\(x = 0\)). Exists in quadrants 1 and 3 if \(k > 0\).
\(y = \frac{k}{x^2}\): Symmetrical about the \(y\)-axis. If \(k > 0\), both branches lie above the \(x\)-axis in quadrants 1 and 2.
Repeated Roots on Graphs
• A single linear factor \((x - a)\) crosses the \(x\)-axis cleanly at \(x = a\).
• A squared factor \((x - a)^2\) touches the \(x\)-axis and turns around at \(x = a\) (a turning point).
• A cubed factor \((x - a)^3\) creates a point of inflection on the \(x\)-axis at \(x = a\).
Key Takeaway: When sketching graphs, always clearly label all \(x\)-intercepts, \(y\)-intercepts, and any asymptotes.
6. Transformations of Functions
Understanding graph transformations allows you to sketch complex curves by modifying a known base curve \(y = f(x)\).
The Golden Rule of Transformations
• Outside the bracket: Affects the \(y\)-coordinates in the way you would expect.
• Inside the bracket: Affects the \(x\)-coordinates in the opposite way to what you would expect.
Summary of Function Transformations
• \(y = f(x) + a\): Translation parallel to the \(y\)-axis by vector \(\begin{pmatrix} 0 \\ a \end{pmatrix}\) (moves up by \(a\)).
• \(y = f(x + a)\): Translation parallel to the \(x\)-axis by vector \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\) (moves left by \(a\)).
• \(y = af(x)\): Vertical stretch by scale factor \(a\) parallel to the \(y\)-axis (multiply all \(y\)-coordinates by \(a\)).
• \(y = f(ax)\): Horizontal stretch by scale factor \(\frac{1}{a}\) parallel to the \(x\)-axis (divide all \(x\)-coordinates by \(a\)).
• \(y = -f(x)\): Reflection in the \(x\)-axis (all \(y\)-coordinates change sign).
• \(y = f(-x)\): Reflection in the \(y\)-axis (all \(x\)-coordinates change sign).
Example: The curve \(y = x^2\) has its minimum at \((0, 0)\).
For \(y = (x - 4)^2 + 3\), the graph is translated by \(4\) units right and \(3\) units up, moving the vertex to \((4, 3)\).
Key Takeaway: Check whether the modification is happening to the input (inside \(f\dots\)) or the output (outside \(f\dots\)). Inside changes \(x\) inversely; outside changes \(y\) directly.
Quick Chapter Checklist
Before sitting your AS 1 exam, make sure you can confidently:
• Simplify powers and rationalise surd denominators using conjugates.
• Complete the square for any quadratic \(ax^2 + bx + c\) and state the vertex.
• Apply the discriminant \(b^2 - 4ac\) to determine the number of real roots.
• Solve simultaneous equations combining linear and quadratic forms.
• Find critical values and sketch graphs to solve quadratic inequalities.
• Factorise cubics using the Factor Theorem and algebraic division.
• Sketch and transform polynomial and reciprocal graphs using translation, reflection, and stretches.