Precalculus Formulas Flashcards

The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope (steepness of the line) and b represents the y-intercept (where the line crosses the y-axis).

The slope-point form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.

The slope of a line between two points can be calculated using the slope formula: **m = (y₂ - y₁) / (x₂ - x₁) **. This formula measures the steepness of the line between those two points.

If two lines are perpendicular, their slopes are negative reciprocals of each other. This means that if the slope of one line is m₁, the slope of the perpendicular line m₂ will be -1 / m₁.

Calculating the distance between two points follows the distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²] Applying this formula will provide the distance between the two points in a plane.

The midpoint formula is used to find the exact middle point between two given points. It is defined as (x₁ + x₂ / 2, y₁ + y₂ / 2). This formula will give you the coordinates of the midpoint.

The circle formula is represented as (x - h)² + (y - k)² = r² where (h, k) represents the coordinates of the center of the circle and r represents the radius.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

The difference quotient is a critical expression used in calculus to find the derivative of a function. It is calculated as (f(x + h) - f(x)) / h, where f(x) is the function and h is a small change in x.

The quadratic formula is used to solve for the roots of a quadratic equation: ax² + bx + c = 0. The formula provides the solutions for x: x = [-b ± √(b² - 4ac)] / (2a).

Direct variation occurs when two quantities are directly proportional, meaning that as one increases, the other increases by a constant factor. It is represented by the equation y = kx, where k is the constant of variation.

Inverse variation is a relationship between two quantities where as one increases, the other decreases by a factor that is inversely proportional. This is written as y = k / x, where k is the constant of variation.

Function addition involves combining two functions to create a new function. It is expressed as (f + g)(x) = f(x) + g(x), where f(x) and g(x) are the original functions.

Function subtraction involves determining the difference between two functions. It is represented by (f - g)(x) = f(x) - g(x), where f(x) and g(x) are the functions being subtracted.

Function multiplication involves finding the product of two functions, resulting in a new function. It is expressed as (fg)(x) = f(x) * g(x), where f(x) and g(x) are the original functions.

Function division involves dividing one function by another, resulting in a new function. It is represented by (f / g)(x) = f(x) / g(x), where f(x) is the numerator and g(x) is the denominator.

Function composition involves combining two functions by substituting one function into another. It is expressed as (f º g)(x) = f(g(x)), where the output of g(x) becomes the input of f(x).

A reflection across the x-axis transforms a function by flipping it vertically. This is represented by y = -f(x), where the function f(x) is reflected over the x-axis.

A reflection across the y-axis flips a function horizontally. This is represented by y = f(-x), where the function f(x) is reflected across the y-axis.

Even functions are symmetric about the y-axis. This means that for any input x, the function's output is the same as for the negative input -x. This is expressed as f(-x) = f(x).

Odd functions are symmetric about the origin. This means that the value of the function at -x is the opposite of the value at x. This is expressed as f(-x) = -f(x).

The logarithmic form log₁₀ x = y represents the base-10 logarithm of x, meaning 10 raised to the power of y equals x.

The exponent form 10^y = x is equivalent to the logarithmic form log₁₀ x = y. This form emphasizes that y is the exponent needed to raise the base 10 to get x.

The change of base formula log₅ M = log₁₀ M / log₁₀ 5 allows you to convert a logarithm from one base to another. This formula lets you calculate logarithms to different bases using base-10 logarithms.

For a 2x2 matrix, the determinant is calculated using the formula: |a d| |b c| = ad - bc. The determinant provides a scalar value that reveals important properties of the matrix.