Statistics and Polynomial Functions Quiz

The inverse trigonometric function of tangent, represented as tan⁻¹x, where x is a real number and the output, y, is restricted to the interval -π/2 < y < π/2.

A function is odd if f(-x) = -f(x).

A function is even if f(-x) = f(x).

The range of the inverse tangent function, tan⁻¹x, is limited to -π/2 < y < π/2. This means the output of the function always falls within this specific interval.

The sine function 'undoes' the inverse sine function, and vice versa. This relationship holds true for values of x within the valid domain of each function.

The cosine function 'undoes' the inverse cosine function, and vice versa. This relationship holds true for values of x within the valid domain of each function.

The tangent function 'undoes' the inverse tangent function, and vice versa. This relationship holds true for values of x within the valid domain of each function.

These properties relate the values of inverse trigonometric functions for positive and negative inputs. For example, sin⁻¹(-x) = -sin⁻¹(x) means the inverse sine of a negative value is the negative of the inverse sine of the positive value.

A polynomial in one variable, 𝑥, of degree 𝑛 with real coefficients is an expression of the form: 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 , where 𝑎𝑛 ≠ 0.

The highest power of the variable in a polynomial. Example: 4x³ + 2x - 1 has degree 3.

The coefficient of the term with the highest power of the variable in a polynomial. Example: In the polynomial 4x³ + 2x - 1, the leading coefficient is 4.

The term in a polynomial that doesn't contain any variables. Example: In the polynomial 4x³ + 2x - 1, the constant term is -1.

A process to divide a polynomial (the dividend) by another polynomial (the divisor) to get a quotient and remainder. The remainder's degree must be less than the divisor's degree. Example: 𝑃(𝑥) = 𝐴(𝑥).𝑄(𝑥) + 𝑅(𝑥)

A theorem stating that when a polynomial 𝑃(𝑥) is divided by (𝑥 − 𝑎), the remainder is 𝑃(𝑎).

A theorem stating that (𝑥 − 𝑎) is a factor of the polynomial 𝑃(𝑥) if and only if 𝑃(𝑎) = 0.

A root of a polynomial equation is a value of the variable that makes the equation true. The multiplicity of a root is how many times it appears in the factorization of the polynomial.

The sum and product of the roots of a polynomial equation are related to the coefficients of the equation. This relationship can be used to solve the equation.

The graph of a polynomial function can be used to determine the roots of a polynomial equation and their multiplicity.

A statistical tool that models events with two possible outcomes, like coin flips or success/failure in a medical trial. It helps predict the probability of a certain number of successes in a set number of trials.

A random variable that counts the number of successes in a fixed number of independent trials, each with two possible outcomes. Examples include the number of heads in 10 coin flips or the number of defective products in a batch.

The chance of getting a specific number of successes in a given number of trials using the Binomial Distribution. Calculated by considering all possible combinations of successes and failures.

A visual representation of the probabilities of all possible outcomes of a binomial random variable. It shows the probability of getting zero successes, one success, two successes, and so on, up to the total number of trials.

Uses the binomial distribution to analyze and interpret data. This can involve calculating probabilities, testing hypotheses, and making predictions about future events.

A Bernoulli random variable represents an event with two possible outcomes: success (1) or failure (0). The probability of success is denoted by 'p'.

A single trial with two possible outcomes, one labeled 'success' and the other 'failure'.

The probability distribution of a random variable that takes the value 1 with probability 'p' (success) and 0 with probability 'q' (failure), where q = 1 - p.

The coefficient of the term x^(n-r)y^r in the expanded form of (x + y)^n. It's calculated as n! / (r! * (n-r)!).

The expansion of a binomial expression raised to a power. It shows how the powers of the variables and coefficients are distributed.

An Indigenous Australian who identifies as and is accepted by the Aboriginal and/or Torres Strait Islander community.

The five cultural groups of Torres Strait Islander Peoples whose island territories were annexed by Queensland in 1879.

Describes motion without considering forces or masses. It focuses on describing motion in terms of displacement, velocity, acceleration, and time.

Motion in a straight line where the velocity changes at a constant rate. This means the object speeds up or slows down at the same rate over time.

The total change in position of an object. It's a vector quantity, meaning it has both magnitude and direction.

The speed of an object in a specific direction. It's also a vector quantity, meaning it has both magnitude and direction.

The rate of change of velocity over time. It's also a vector quantity, meaning it has both magnitude and direction.

The difference in velocity between two objects that are moving in the same direction, calculated by adding or subtracting their velocities.

A scientific model that uses mathematical equations and graphs to describe the relationship between time, distance, displacement, speed, velocity, and acceleration in straight-line motion.

The ability of science to explain why things happen and to predict future events. In the case of kinematics, it allows us to predict the motion of objects.

The refractive index (n) of a medium is a measure of how much light bends when it passes from one medium to another. It is defined as the ratio of the speed of light in a vacuum (c) to the speed of light in the medium (v).

Snell's Law describes the relationship between the angles of incidence and refraction when light passes from one medium to another. It states that the product of the refractive index of the first medium and the sine of the angle of incidence is equal to the product of the refractive index of the second medium and the sine of the angle of refraction.

Total internal reflection occurs when light traveling from a denser medium to a less dense medium (e.g., water to air) strikes the boundary at an angle greater than the critical angle. Instead of refracting out, the light is reflected back into the denser medium.

The dispersion of light is the phenomenon where white light is split into its constituent colors (red, orange, yellow, green, blue, indigo, violet) when it passes through a prism or a diffraction grating. This happens because different colors of light have different wavelengths and therefore bend at different angles.

The inverse square law states that the intensity of light is inversely proportional to the square of the distance from the source. This means that as you move farther away from a light source, the intensity of the light decreases rapidly.