CAIE AS Level Mathematics: Quadratics

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Questions and Answers

What are the values of u that satisfy the equation (3u − 1)(2u + 1) = 0?

u = 1/3 and u = -1/2 (correct)

u = 2/3 and u = -1

u = 1/2 and u = -1/3

u = 1 and u = 0

Which of the following represents a function?

A relation where any x-value maps to one unique y-value (correct)

A relation with no defined outputs for any x-value

A relation where multiple x-values can map to the same y-value

A relation where one x-value gives two different y-values

Which graphical representation does NOT demonstrate a function?

A discrete set of points

A vertical line (correct)

A straight line with a slope

A parabola opening upwards

What is true about the outputs of a function?

A function assigns one y-value to each x-value without exception Signup and view all the answers

In the context of functions, which statement is correct?

A function can be represented by a formula, a table, or a graph Signup and view all the answers

If a relation gives one x-value input that produces two y-value outputs, how is this classified?

Not a function Signup and view all the answers

Which of the following examples depicts a quadratic equation?

y = x^2 - 5x + 6 Signup and view all the answers

How does one determine if a graph represents a function using the vertical line test?

By checking if vertical lines intersect the graph at most once Signup and view all the answers

What condition must be met for a function to be classified as One to One?

A horizontal line intersects the function at exactly one point. Signup and view all the answers

Which of the following best describes the range of a function?

The set of all possible outputs of the function. Signup and view all the answers

In the equation $y = p(x − q)^2 + r$, what does the parameter 'r' represent?

The vertical position or coordinate of the vertex. Signup and view all the answers

What does the vertical line test determine about a function?

If the function is a valid representation of a relation. Signup and view all the answers

If a quadratic function has an equation $y = p(x − q)^2 + r$ with $p < 0$, what is true about its vertex?

It is a maximum point. Signup and view all the answers

What defines the possible inputs (x-values) for a function?

The domain of the function. Signup and view all the answers

How is the range of the quadratic function determined?

By including the y-coordinate of the vertex and its direction. Signup and view all the answers

What is the implication of $4 < x < 10$ for a function?

The function outputs real values for x-values between 4 and 10. Signup and view all the answers

What determines the range of a function as described?

Where the function approaches a specific value from above or below Signup and view all the answers

In the example function f(x) = $rac{2}{3(3x - 1)} + 2$, what is the best way to find its range?

Evaluating the function at a large positive value of x Signup and view all the answers

For the function f(x) = ax + b, what must be true about the domain to ensure the function remains positive?

x must be greater than -b Signup and view all the answers

What characteristic of vertical lines can be used to determine if a relation is a function?

A vertical line cannot intersect the graph at more than one point Signup and view all the answers

For which of the following values of b would the function f(x) = ax + b be guaranteed to always remain positive?

b > 0 Signup and view all the answers

What does the term 'approaching a specific value a' from above or below imply about the function's behavior?

The function is asymptotic to the line y = a Signup and view all the answers

If a function approaches a value a from below, what inequality describes its range?

f(x) < a Signup and view all the answers

What is a necessary condition for a quadratic function to be treated as a function in regards to its vertical line test?

Every vertical line must intersect it at most once. Signup and view all the answers

Study Notes

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.

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Description

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.