CAIE AS Level Mathematics: Quadratics

CAIE AS Level Mathematics Summary Notes

• The notes are updated for the 2023-2025 syllabus.
• The notes are summarized from theory syllabus.
• The notes are for personal use only.

Quadratics

• Quadratic Formula: Used to find solutions (zeros) to quadratic equations of the form ax² + bx + c = 0. Formula: x = (-b ± √(b² - 4ac)) / 2a
• Completing the Square: A method to rewrite a quadratic in the form p(x + q)² + r. Useful for various applications.
• Factorisation: Expressing a polynomial as the product of simpler terms (e.g., x² - 2x + 2 = (x − 1)(x − 2)). Two common methods are splitting the middle term or using a calculator.
• Solving Quadratic Equations by Completing the Square: A technique to solve quadratic equations by transforming them into the form p(x - q)² + r = 0, to isolate x.
• Quadratic Inequalities: Solving inequalities involving quadratic equations. Involves factoring the inequality, sketching the quadratic, identifying intersections with the x-axis, and determining the region satisfying the inequality (e.g., y ≤ 0 or y ≥ 0), considering if the points of intersection must be included.
• Discriminant: The value (b² - 4ac) helps determine the nature of the roots of a quadratic equation (number of intersections with the x-axis): -Positive discriminant: Two distinct real roots (two intersections) -Zero discriminant: Two equal real roots (one intersection) -Negative discriminant: No real roots (no intersection)

Finding the Coordinates of the Vertex

• The vertex is the highest or lowest point on a quadratic curve.
• Formula for the x-coordinate of the vertex: x = -b / 2a

Simultaneous Equations

• A set of equations with unknown variables that satisfy a condition.
• Techniques include elimination and substitution.
• Elimination: Requires multiplying equations such that variables have the same coefficient to cancel and solve for one variable.
• Substitution: Make an unknown variable the subject of the equation and substitute that expression into the other equation to solve.
• Simultaneous equations may include quadratics and lines, requiring solving a quadratic equation which results in the points of intersection.

Types of Functions and Relations

• Functions: A relationship between x-values (input) and y-values (output) where each x-value is associated with only one y-value.
• One-to-One function: Each x-value maps to a unique y-value. The horizontal line test can be used to check.
• Many-to-One function: More than one x-value may map to the same y-value but each y-value is unique. The horizontal line test does not pass.
• Vertical Line Test: Used to see if a graph represents a function. A vertical line must not intersect more than one point on that curve for it to be a function.
• Domain: The set of x-values for which a function or relation is defined.
• Range: The set of y-values for which a function or relation is defined.

Domain and Range of a Quadratic Function

• Domain of quadratics is normally all real numbers unless specifically restricted.
• The range depends on whether the graph has a maximum or minimum y-value.

Inverse Functions

• Reverses the effect of a given function (denoted by f^-1(x)).
• Finding the Inverse: Swapping the x and y variables to get the new equation, and then making y the subject.
• Horizontal Line Test (for inverse functions): Ensure that the original function is one-to-one, only then will an inverse function exist.

Composite Functions

• Combining two or more functions (e.g., g(f(x)) or gf(x)). Input into one function is then used as input for a second function.
• The domain is restricted by the range of the inner function or functions.

Translations

• Transforming a function by shifting its graph to a new position.
  • x-translation (horizontal shift)
  • y-translation (vertical shift)

Stretches

• The resulting graph is transformed by scaling its x-values or y-values by certain factors.

Reflections

• Reflecting the graph of a function over either or both of the coordinate axes.

Order of Transformations

• Transformations occur in order from left to right from the base function equation.

Distance Formula

• The distance formula helps to calculate the distance between two points on a graph (A and B).
• A right angled triangle, using the x and y differences between points as the base and the height, is helpful.

Midpoint Formula

• Calculate the mid-point between two points (M) on a graph using the average of the x and y coordinates.

Gradient (or slope)

• The gradient of a line describes its inclination.
• It is calculated as the ratio rise over run.

Explore key methods and formulas related to quadratic equations in the CAIE AS Level Mathematics syllabus for 2023-2025. This summary includes the quadratic formula, completing the square, factorization techniques, and solving quadratic inequalities. Perfect for review and personal use.