Sketching Quadratic Parabolas

Ever wondered how to tackle those "sketch the graph" questions without panicking? Quadratic functions like y = x² + 6x + 8 follow a simple pattern once you know what to look for.

Start by identifying whether your parabola opens upward (U-shaped) or downward (∩-shaped) - just check if the coefficient of x² is positive or negative. Next, find your y-intercept by substituting x = 0 into the equation.

The roots are where your graph crosses the x-axis $when y = 0$. Factor the quadratic to find these x-values quickly. Finally, the axis of symmetry sits exactly halfway between your roots, and that's where you'll find your turning point.

Quick tip: The turning point's x-coordinate is always the average of your two roots!

Completing the Square

Completing the square transforms messy quadratics into neat, workable forms that reveal everything about your parabola instantly. You're converting y = ax² + bx + c into the much more useful form y = a(x ± p)² + q.

Here's the magic formula: take half of your b coefficient, square it, then add and subtract this value cleverly. For x² + 8x + 3, you'd use $8/2$² = 16 to create the perfect square $x + 4$².

This method becomes incredibly powerful when finding roots of complex equations. Once you've completed the square, you can solve by taking square roots rather than struggling with factoring.

Remember: When the coefficient of x² isn't 1, factor it out first before completing the square!

Exact Trigonometric Values

Those special angles - 30°, 45°, and 60° - pop up everywhere in A-level maths, so memorising their exact values is absolutely crucial for your success.

The 45-45-90 triangle gives you equal sides of 1 and a hypotenuse of √2. This means sin 45° = cos 45° = 1/√2 (or √2/2 when rationalised).

The 30-60-90 triangle has sides in the ratio 1:√3:2. From this, sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2, and cos 60° = 1/2. These values are worth their weight in gold during exams!

Pro tip: Draw these triangles in your exam booklet if you forget - the side lengths will remind you of all the ratios instantly!

Distance Formula

The distance formula is your go-to tool for finding the length between any two points on a coordinate plane - think of it as Pythagoras' theorem in disguise.

Simply plug your coordinates (x₁, y₁) and (x₂, y₂) into √$(x₂ - x₁)² + (y₂ - y₁)²$. The formula calculates the hypotenuse of the right triangle formed by the horizontal and vertical distances between your points.

Quick check: Always ensure your final answer makes sense by roughly estimating the distance on a sketch!