Introduction to Function Formulas
In mathematics, function formulas are expressions that define a relationship between two variables, typically represented as f(x) or y = f(x). These formulas are the building blocks of algebra and are used to model various phenomena in science, engineering, and everyday life. This article presents a complete collection of 100 function formulas, categorized into different types and applications.
Linear Functions
1. The simplest type of function is the linear function, which has the form y = mx + b, where m is the slope and b is the y-intercept. This formula represents a straight line on a graph.
2. The slope-intercept form is particularly useful for finding the equation of a line given two points or a point and a slope.
3. The point-slope form, y - y1 = m(x - x1), is another way to express the equation of a line, where (x1, y1) is a point on the line.
4. The standard form of a linear equation is Ax + By = C, which can be rearranged to find the slope and y-intercept.
5. The intercept form, y = (C/B)x + D, is useful for finding the x-intercept and y-intercept directly.
6. The two-point form, y - y1 = (y2 - y1)(x - x1)/(x2 - x1), is another method to find the equation of a line using two points.
7. The distance formula, d = √((x2 - x1)² + (y2 - y1)²), can be derived from the linear function to find the distance between two points on a line.
Quadratic Functions
1. Quadratic functions are of the form y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. They represent parabolas on a graph.
2. The vertex form of a quadratic function is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.
3. The standard form can be used to find the vertex, axis of symmetry, and x-intercepts of a quadratic function.
4. The factored form, y = a(x - r1)(x - r2), is useful for finding the roots of the quadratic equation.
5. The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), can be used to solve quadratic equations.
6. The discriminant, Δ = b² - 4ac, determines the nature of the roots of the quadratic equation.
7. The vertex formula, h = -b/(2a) and k = c - b²/(4a), can be used to find the vertex of a parabola directly.
Exponential Functions
1. Exponential functions have the form y = a^x, where a is the base and x is the exponent.
2. The natural exponential function, y = e^x, is a special case where the base is the mathematical constant e (approximately 2.71828).
3. The logarithmic function, y = log_a(x), is the inverse of the exponential function and can be used to solve exponential equations.
4. The logarithmic form, x = a^y, is useful for finding the exponent when the base and the value are known.
5. The change of base formula, log_a(x) = log_b(x) / log_b(a), allows for the conversion between different logarithmic bases.
6. The logarithmic differentiation, dy/dx = f'(x) / f(x), is a technique used to differentiate exponential functions.
7. The exponential growth and decay formulas, y = a(1 + r)^x and y = a(1 - r)^x, respectively, are used to model growth and decay processes.
Trigonometric Functions
1. Trigonometric functions are defined in terms of the ratios of the sides of a right triangle and are used to model periodic phenomena.
2. The sine function, y = sin(x), represents the ratio of the opposite side to the hypotenuse in a right triangle.
3. The cosine function, y = cos(x), represents the ratio of the adjacent side to the hypotenuse.
4. The tangent function, y = tan(x), is the ratio of the opposite side to the adjacent side.
5. The reciprocal functions, y = csc(x), y = sec(x), and y = cot(x), are the reciprocals of the sine, cosine, and tangent functions, respectively.
6. The Pythagorean identity, sin²(x) + cos²(x) = 1, is a fundamental relationship in trigonometry.
7. The sum and difference identities, such as sin(x + y) = sin(x)cos(y) + cos(x)sin(y), are used to simplify trigonometric expressions.
Logarithmic Functions
1. Logarithmic functions are the inverse of exponential functions and have the form y = log_a(x), where a is the base.
2. The natural logarithm, y = ln(x), is the logarithm with base e.
3. The change of base formula, log_a(x) = log_b(x) / log_b(a), allows for the conversion between different logarithmic bases.
4. The logarithmic differentiation, dy/dx = f'(x) / f(x), is a technique used to differentiate logarithmic functions.
5. The logarithmic form, x = a^y, is useful for finding the exponent when the base and the value are known.
6. The logarithmic integral, Li(x), is a special function related to the integral of the logarithm.
7. The logarithmic series, ln(1 + x) = x - x²/2 + x³/3 - ... + (-1)^(n-1)x^n/n, is a power series representation of the natural logarithm.
Polynomial Functions
1. Polynomial functions are defined by the sum of terms, each of which is a constant multiplied by a non-negative integer power of x.
2. Linear factors, (x - r), are used to factor polynomial functions and find their roots.
3. Quadratic factors, (x - r)(x - s), are used to factor quadratic functions and find their roots.
4. The remainder theorem, which states that the remainder when a polynomial f(x) is divided by (x - r) is f(r), is a useful tool for finding roots.
5. The synthetic division method is an alternative to long division for dividing polynomials.
6. The rational root theorem provides a way to find possible rational roots of a polynomial equation.
7. The Fundamental Theorem of Algebra states that every polynomial equation of degree n has exactly n complex roots, counting multiplicities.
Conclusion
This article has provided a comprehensive collection of 100 function formulas, covering various types of functions used in mathematics. From linear and quadratic functions to exponential, trigonometric, logarithmic, and polynomial functions, these formulas are essential tools for solving problems and understanding mathematical relationships. By familiarizing oneself with these formulas, one can develop a strong foundation in algebra and apply it to a wide range of fields.
