Graph Types and Simultaneous Equations

Understanding different graph shapes is crucial for recognising functions at a glance. Linear graphs follow the pattern y = mx + c, where m is the gradient and c is the y-intercept. Quadratic graphs (containing x²) create U-shaped curves with a line of symmetry, whilst cubic graphs (containing x³) have distinctive S-shapes with turning points.

Exponential graphs show rapid growth or decay, and reciprocal graphs $1/x$ create hyperbolas that never touch the axes. Each graph type has its own unique characteristics that help you identify the underlying equation.

When solving simultaneous equations like 3x + y = 3 and 2x - 4y = 16, start by making either the x or y terms the same. Multiply the first equation by 2 to get matching x terms: 6x + 2y = 6 and 6x - 12y = 48. Then subtract to eliminate x and solve for the remaining variable.

Quick tip: Always check your answer by substituting both values back into the original equations!

Solving Methods and Number Properties

After eliminating one variable in simultaneous equations, substitute your answer back into either original equation. If you found y = -4, plug it into 3x + y = 3 to get 3x + (-4) = 3, giving you x = 2.8. This systematic approach works every time.

Finding the HCF (Highest Common Factor) of numbers like 84 and 140 involves drawing factor trees, then using a Venn diagram to identify common factors. The HCF is found by multiplying the numbers in the intersection - in this case, 28. For the LCM (Lowest Common Multiple), you'd multiply all the numbers instead.

Index laws might look scary, but they follow clear patterns. Remember that anything to the power of 0 equals 1, fractional powers mean roots $like 25^(1/2) = 5$, and negative powers create fractions $25^(-1/2) = 1/5$. For mixed powers like 25^(3/2), find the root first $square root of 25 = 5$, then apply the other power (5³ = 125).

Remember: Index laws are just shortcuts for repeated multiplication and division!

Statistics and Sequences

Calculating the estimated mean from grouped data requires finding midpoints for each interval. Multiply each midpoint by its frequency, add all these products together, then divide by the total frequency. For the height data shown, this gives 3936 ÷ 25 = 167.4 cm.

Combining ratios needs careful attention to make terms match. If a:b = 3:4 and b:c = 6:2, you need to make the 'b' values the same. Scale the ratios so both have b = 20, giving a:b:c = 15:20:8.

Sequences come in different types, each with distinct patterns. Linear sequences have constant differences $like 6, 10, 14, 18 with +4 each time$, giving the formula 4n + 2. Quadratic sequences have differences that change by a constant amount - look for the second difference to spot these. Fibonacci sequences add the two previous terms together (3, 1, 4, 5, 9...).

Pattern spotting: Always look at the differences between terms first - they reveal the sequence type!

Advanced Sequences and Proportion

Geometric sequences multiply or divide by the same value each time. In the sequence 5, 5√3, 45, 45√3, you're multiplying by √3 each time. The nth term formula is a × r^$n-1$, where 'a' is the first term and 'r' is the multiplier.

Direct proportion means as one variable increases, the other increases at the same rate (y ∝ x). The graph is a straight line through the origin. Inverse proportion means as one increases, the other decreases $y = k/x$, creating a curved graph.

To find proportion formulas, use the given values to work out the constant 'k'. If y is directly proportional to x, and when x = 8, y = 40, then y = kx becomes 40 = 8k, so k = 5. Therefore y = 5x.

For inverse proportion like d ∝ 1/w², use d = k/w². If w = 0.5 and d = 12, then 12 = k/0.25, giving k = 3. So d = 3/w².

Proportion tip: Direct proportion gives straight-line graphs, inverse proportion gives curves!

Data Analysis Techniques

Cumulative frequency tables build running totals by adding each frequency to the previous total. When plotting cumulative frequency graphs, always use the upper boundary of each class interval on the x-axis.

Finding quartiles from cumulative frequency is straightforward: the median is at ½ of the total frequency, the lower quartile (LQ) at ¼, and the upper quartile (UQ) at ¾. These values help you understand how data is spread out.

Histograms are different from bar charts because they show frequency density, not just frequency. Calculate frequency density using the formula: frequency ÷ class width. This ensures that areas of bars represent frequencies accurately, even when class widths vary.

The key to histogram success is remembering that area equals frequency. Wider bars need lower heights to represent the same frequency as narrower bars.

Data analysis: Always check whether you need frequency or frequency density - it makes a huge difference!

Angle Properties and Polygons

Angle facts form the foundation of geometry problem-solving. Triangle angles always add to 180°, whilst angles around a point total 360°. Quadrilateral angles also sum to 360°.

For polygons, the exterior angles always add up to 360°, regardless of the number of sides. The formula for total interior angles is $n-2$ × 180°, where n is the number of sides. Divide this by the number of sides to find each interior angle in a regular polygon.

Parallel line properties are essential for angle calculations. Corresponding angles are equal (in the same relative positions), alternate angles are equal (on opposite sides of the transversal), and co-interior angles add up to 180° (on the same side of the transversal).

Vertically opposite angles are always equal when two lines cross. These angle properties often combine in exam questions, so practise identifying multiple relationships in single diagrams.

Angle hunting: Look for parallel lines and transversals - they're goldmines for angle relationships!

Fraction Operations and Decimals

Adding and subtracting fractions requires a common denominator. Convert mixed numbers to improper fractions first $2⅔ becomes 8/3$, find the lowest common multiple of denominators, then add or subtract the numerators.

Multiplying fractions is simpler - just multiply numerators together and denominators together, then simplify. For division, use 'Keep, Change, Flip': keep the first fraction, change ÷ to ×, and flip the second fraction upside down.

Converting recurring decimals to fractions uses algebra. For 0.3̄ (0.333...), let x = 0.333..., then 10x = 3.333.... Subtract to get 9x = 3, so x = ⅓. For longer recurring parts like 0.123̄, multiply by 1000 and 10 to create the equation 990x = 122.

The key is choosing the right powers of 10 based on how many digits recur. One recurring digit needs ×10, two digits need ×100, and so on.

Fraction wisdom: Always simplify your final answer - it often makes the numbers much friendlier!

Completing the Square and Context Problems

Completing the square transforms quadratic expressions into a more useful form. Take x² + 8x + 6: halve the x-coefficient (8 ÷ 2 = 4), write as $x + 4$², then adjust for the constant. Since $x + 4$² = x² + 8x + 16, you need $x + 4$² - 10 to get back to the original expression.

The turning point of a parabola comes directly from the completed square form. From $x + 4$² - 10, the turning point is (-4, -10) - take the opposite sign of the first number and keep the second as is.

For expressions with coefficients of x², like 3x² + 18x - 1, factor out the coefficient first: 3$x² + 6x$ - 1, then complete the square inside the brackets.

Context problems involving direct and inverse proportion appear frequently. If 12 people take 5 hours to complete a job, that's 60 person-hours of work. With 15 people, the same job takes 60 ÷ 15 = 4 hours.

Proportion in context: Total work stays constant - more people means less time per person!

Advanced Algebra and Geometry

Algebraic fractions follow the same rules as number fractions but with algebra. Find common denominators for addition and subtraction, multiply straight across for multiplication, and use 'keep, change, flip' for division.

When solving equations with algebraic fractions, multiply through by the common denominator to eliminate fractions. For example, if 1/30x = 4, multiply both sides by 30x to get 1 = 120x, so x = 1/120.

Simplifying complex algebraic fractions often requires factorising first. Look for common factors in numerators and denominators that can cancel out.

Pythagoras' theorem $a² + b² = c²$ finds missing sides in right-angled triangles. Remember that 'c' is always the longest side (hypotenuse). Square the two known sides, add or subtract as needed, then square root your answer.

Ratios in context require careful reading. If the total of some parts equals a given amount, find the value of one part by dividing, then multiply to find other parts.

Algebraic confidence: Treat algebraic fractions just like number fractions - the rules are identical!