Inverse Functions and Graphing Circles

These flashcards cover key concepts in inverse functions, one-to-one functions, distance and midpoint formulas, and properties of polynomials.

What is the formula to find the distance between two points in the Cartesian plane?

The formula is d = √((x₂ - x₁)² + (y₂ - y₁)²).

One-to-One Function

A function where each input corresponds to a unique output, passing the horizontal line test.

Horizontal Line Test

A method to determine if a function is one-to-one by checking if horizontal lines intersect the graph more than once.

Domain

The set of possible input values (x-values) for a function.

Range

The set of possible output values (y-values) for a function.

Distance Formula

The formula used to find the distance between two points in the Cartesian plane: d = √((x₂ - x₁)² + (y₂ - y₁)²).

Midpoint Formula

The formula for finding the midpoint of a line segment joining two points: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Standard Form of a Circle

The equation of a circle in the form (x - h)² + (y - k)² = r², where (h,k) is the center and r is the radius.

Polynomial Functions

Functions that can be expressed in the form of a polynomial, with terms combining variables raised to whole number powers.

Degree of a Polynomial

The highest power of the variable in a polynomial expression.

Rational Zero Theorem

A theorem that provides a way to find possible rational zeros of a polynomial having integer coefficients.

Fundamental Theorem of Algebra

Every non-constant polynomial function has at least one complex root.

What is the formula for compound interest?

A = P(1 + r/n)^(nt), where A is the amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.

How is the standard deviation of a data set calculated?

Standard deviation = sqrt((Σ(xi - μ)²) / N), where μ is the mean, xi are the data points, and N is the number of points.

What is the formula for calculating the volume of a cylinder with radius r and height h?

Volume = πr²h.

What is the distance formula in three-dimensional space?

d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²).

What is the area of a triangle with base b and height h?

Area = (1/2) * b * h.

How do you find the factorial of a non-negative integer n, denoted as n!?

n! = n × (n-1) × (n-2) × … × 1, with 0! = 1.

What is the formula for the surface area of a sphere with radius r?

Surface Area = 4πr².

How do you calculate the expected value of a random variable X?

Expected value E(X) = Σ [x * P(x)], where x is a possible value of X and P(x) is the probability of x.

What is the formula for the geometric series sum?

Sum = a / (1 - r), where a is the first term and r is the common ratio, for |r| < 1.

What is the relationship between the sine, cosine, and tangent functions?

tan(θ) = sin(θ) / cos(θ), where θ is in radians.

How do you calculate the perimeter of a polygon?

Perimeter = Sum of the lengths of all sides.

What is the exponential function and its general form?

The exponential function is f(x) = a * b^x, where a is a constant, b is the base, and x is the exponent.

What is the formula for the probability of independent events A and B occurring?

P(A and B) = P(A) * P(B).

How is the median calculated for an even set of numbers?

For an even set, the median is the average of the two middle numbers after sorting the dataset.

What is the derivative of a function f(x) with respect to x?

The derivative f'(x) represents the rate of change of f(x) with respect to x.

What is the area of a rectangle with width w and height h?

Area = w * h.

What is the formula for the circumference of a circle with radius r?

Circumference = 2πr.

How do you calculate the average (mean) of a set of numbers?

Average = (Sum of all numbers) / (Total count of numbers).

What is the Pythagorean theorem?

a² + b² = c² (where c is the hypotenuse of a right triangle).

What is the formula for the slope of a line given two points (x₁, y₁) and (x₂, y₂)?

Slope (m) = (y₂ - y₁) / (x₂ - x₁).

How do you find the probability of an event occurring?

Probability = (Number of favorable outcomes) / (Total number of possible outcomes).

What is the formula for the sum of an arithmetic series?

Sum = n/2 (first term + last term), where n is the number of terms.

What is the quadratic formula for solving ax² + bx + c = 0?

x = (-b ± √(b² - 4ac)) / (2a).

How can you find the median of a set of numbers?

Arrange the numbers in order, and if there is an odd number of observations, the median is the middle number; if even, it is the average of the two middle numbers.

What is the binomial theorem?

(a + b)ⁿ = Σ (n choose k) a^(n-k) b^k, where k=0 to n.

What is the formula for the area of a circle with radius r?

Area = πr².

How do you calculate the volume of a sphere with radius r?

Volume = (4/3)πr³.

What is the formula for simple interest?

Simple Interest = P * r * t, where P is the principal, r is the rate, and t is the time in years.

How is a linear equation represented in slope-intercept form?

y = mx + b, where m is the slope and b is the y-intercept.

What is the formula for the surface area of a rectangular prism?

Surface Area = 2(lw + lh + wh), where l is length, w is width, and h is height.

What is the formula for the confidence interval of a population mean?

CI = x̄ ± z*(σ/√n), where x̄ is the sample mean, z is the z-score, σ is the population standard deviation, and n is the sample size.

What is the fundamental theorem of calculus?

It establishes the relationship between differentiation and integration, stating that if F is an antiderivative of f on [a, b], then ∫[a to b] f(x) dx = F(b) - F(a).

How do you convert a fraction to a decimal?

Divide the numerator by the denominator.

What is the formula for the median of a data set?

If the number of observations is odd, the median is the middle number; if even, the median is the average of the two middle numbers.

How do you calculate the range of a data set?

Range = Maximum value - Minimum value.