The Complete AS Level / A Level Pure Math - High School Math

Explore quadratic functions defined as ax^2+bx+c with a ≠ 0, where x is real and b or c may be zero, as shown in caption examples.

Explore the quadratic graph, a parabola, with a minimum or maximum vertex. Determine the vertex type by the sign of a: a>0 yields a minimum, a<0 yields a maximum.

Learn how to complete the square by converting ax^2+bx+c to a(x+g)^2+h, with g=b/(2a) and h=c−b^2/(4a); compare coefficients or use the memorized formula, with worked examples.

Learn to find max and min of a quadratic by completing the square, rewriting ax^2+bx+c as a(x+g)^2+h. Min point is (-g,h) for a>0; max point is (-g,h) for a<0.

Learn to solve quadratic equations using factorization and the quadratic formula. Set one side to zero and identify a, b, c to find the roots.

Learn how the discriminant b^2 minus four ac determines quadratic roots: two real roots when >0, one repeated root when =0, and no real roots when <0.

Complete the square to locate the vertex of the parabola and determine the y-intercept. Use discriminant to decide x-intercepts and the graph’s orientation, including tangency when the discriminant is zero.

Learn to read and represent inequalities on a number line using <, ≤, >, and ≥. See how x ranges between A and B with open or closed endpoints.

Learn how to solve linear inequalities by isolating x, mirroring linear equations, and applying the rule to reverse the inequality sign when multiplying or dividing by a negative number.

learn to solve quadratic inequalities by turning one side to zero, solving the resulting equation, and using the parabola shape to identify intervals where the inequality holds, including endpoints.

Learn how to solve simultaneous equations using substitution and elimination, distinguishing linear from nonlinear cases and recognizing solution sets via the discriminant.

Learn to solve simultaneous equations using the substitution method, turning one variable into the subject, substituting into the other equation, and finding the intersection point for x and y.

Learn to solve simultaneous equations with one linear and one non-linear term by substitution, making a variable the subject, substituting, obtaining a quadratic, and solving for two solutions.

Solve the simultaneous equations to locate the points of intersection between a line and a curve; the number of solution sets equals the number of intersection points.

Explore the discriminant, b^2 - 4ac, and how it indicates real roots, a tangent, or no real roots, with graphs and x-intercepts; distinguish solving for x from solving for coefficients.

This lecture shows how to use the discriminant to analyze line–curve intersections, including tangent, two points, or none, by eliminating y and forming ax^2+bx+c=0 then applying b^2-4ac.

Learn how fixed and variable costs form a linear total cost function. The lecture models teddy bear production with f(x)=0.3x+9500, showing examples for 100 and 1000 bears.

Explore functions using a teddy bear production cost f(x)=0.3x+9500, with input x the monthly teddy bears, domain 0–100000, and output values of f(x) from 9500 to 39500.

Explore evaluating functions by substitution, using f(x)=x−3 and g(x)=3x−4, with examples showing that inputs can be numbers, other values, or even another function, and the idea of composite functions.

Explore domain and range concepts using set notation in real numbers, with curly bracket and interval forms, including infinity, inclusive and exclusive bounds, and unions of disjoint ranges.

Explain the concept of a one-to-one function and use the horizontal line test on f(x)=1/x to show it is one-to-one for all real x except zero.

Explore how inverse functions swap inputs and outputs using a teddy bear production cost example. Use f inverse to convert total cost back to production, showing 0.3 x + 9500.

Explore inverse functions and how the inverse maps total cost back to teddy bears. Learn the notation f minus one; note it is not the same as 1 over f.

Explore how restricting the domain makes a function one-to-one to allow an inverse, using f(x)=x^2+1 and restricting to x ≤ 0, with the largest valid m being zero.

Functions f and f inverse swap inputs and outputs, so domain and range swap between them; composing yields the original x, and their graphs reflect across y=x.

Learn to find inverse functions for f x = 3x − 2 and g x = x^2 − 2 with x>0, determine domain and range, and swap inputs and outputs.

Learn how to form composite functions, find the rule, domain, and range, and apply existence criterion that the right-hand range lies in the left-hand domain, with g∘h and h∘g examples.

Explore graph transformations: scale along the x-axis with y = f(ax) and scale the y-values with y = a f(x), then translate in x or y using a recommended order.

Explore graph transformations: reflect y=f(x) about the y-axis to get y=f(-x) by negating x coordinates, and reflect about the x-axis to obtain y=-f(x) by negating y coordinates.

Calculate the distance between two points using Pythagoras and the distance formula; find the midpoint by averaging coordinates, and determine the gradient with rise over run.

Explore the three straight line types—horizontal, vertical, and oblique—and their equations: y=k, x=k, and y=mx+c. Recognize m as the gradient and c as the y intercept, with example y=0.

Learn to derive oblique line equations in the form y = mx + c using three methods: m and c, m with a point, or two points.

Explore how the gradient in y = mx + c measures line steepness as the change in y over the change in x, and how parallel lines share gradients.

Solve simultaneous equations to find intersection points of two graphs; the number of solution sets equals the number of intersections, as shown by (-1,1) and (3,9).

Define the perpendicular bisector of AB as the line perpendicular to AB through its midpoint, where any point on it is equidistant from A and B, then derive its equation.

Explore circles in coordinate geometry by finding the center and radius from standard form (x−a)^2+(y−b)^2=r^2 and from the general form x^2+y^2+2gx+2fy+c=0. Sketch circles using the identified center and radius.

Explore two circles in coordinate geometry by comparing center distance D with radii R1 and R2 to classify external touch, internal touch, containment, or two points of intersection.

Determine if a point lies inside, on, or outside a circle by comparing its distance to the center with the circle's radius, using the distance formula.

Analyze how a line interacts with a circle by categorizing tangents, chords, and non intersections, using simultaneous equations and deriving tangents such as at (1,2).

Explore circle properties: angle in a semicircle is 90 degrees; tangents, radii perpendicular; central angles twice inscribed angles; angles in the same segment; alternate segment theorem; cyclic quadrilaterals.

Understand how the modulus function measures magnitude and yields nonnegative results. See how mod three stays three, while mod minus three becomes three and mod minus four becomes four.

Sketch y = |ax + b| by graphing y = ax + b, finding intercepts; then reflect portion below the x-axis about the x-axis to form the v-shaped graph.

Explore modulus properties: y=|x| equals x for x≥0 and −x for x<0, and that mod a = mod b implies a^2=b^2.

Learn to solve equations involving modulus by squaring both sides when both sides are nonnegative, using the property that mod A equals mod B implies A squared equals B squared.

Learn to solve equations with absolute value by splitting cases based on the inner expression, illustrated with |x-3|=2x+1 and |1-2x|=x, yielding x=1/2, and x=1/3 or 1.

Explains how to solve inequalities involving modulus by splitting into critical points and analyzing cases, using examples like |x+3|<5 and |1−x|≥4x+5 to illustrate the method.

Explore division with the 59 divided by eight example, using long division to identify the dividend, divisor, quotient, and remainder, and show 59 = 7 × 8 + 3.

Master polynomial division by long division and coefficient comparison to find quotients and remainders, while handling descending powers and missing terms, illustrated by a polynomial division example.

Apply the factor and remainder theorems to polynomials, find remainders with divisors like x-1, and verify factors such as 2x+1 by evaluating p at the divisor's zero.

Learn to convert between degrees and radians using pi, since 180 degrees equals pi radians and one radian equals 180 over pi degrees, and apply to common angles.

Learn how to compute arc length and sector area using the formulas arc length = r theta and area of sector = 1/2 r^2 theta, with theta in radians.

Compute the area and perimeter of a circle sector using radians, with radius five and angle 1.5 radians, yielding 18.75 cm squared and 17.5 cm.

Define progression as a set of numbers arranged by rules, such as adding two between terms, or identifying square sequences; define series as the sum of those progression terms.

Explore factorial, denoted by the exclamation mark, as multiplying a number down to one, with examples like 3! = 6, and using the x! function on a scientific calculator.

Learn to compute n choose r with the NCR formula, using examples such as 5 choose 3 equals 10, and remember n and r are integers with n ≥ r.

Master binomial expansion for positive integer powers, using n choose r and ascending powers of x, and apply to (1+x)^5 with coefficients 1,5,10,10,5,1.

Explore binomial expansion techniques with ascending powers of x, including the first four terms of (3+x)^10, the first three terms of (2-3x)^5, and the coefficient of x^2 in (1-x^2)(2-3x)^5.

Identify the constant difference in an arithmetic progression and apply the nth term formula, then compute sums with s_n = n/2 (2a + (n−1)d).

Identify a geometric progression by its constant ratio r, compute terms with u_n = a r^{n-1}, sum with S_n = a(1 - r^n)/(1 - r), and determine sum to infinity when |r|<1.