ALGEBRA AND FUNCTIONS

graphical representation

gives a picture of the function

(b^2 - 4ac = 0)

If the discriminant is equal to zero, the roots are real and equal.

Inconsistent

A system of equations that has no solution.

Logarithm Rules

Rule 1. log a + log b = log (a • b) Rule 2. log a - log b = log (a/b) Rule 3. log a^b = b • log a

sense

Two inequalities are said to have the same sense if their signs of inequality point in the same direction. The sense of an inequality remains the same if both sides are multiplied or divided by the same positive real number. If each side of an inequality is multiplied or divided by the same negative real number, however, the sense of an inequality becomes opposite.

domain

the first set of numbers in an ordered pair (x); if a member is repeated, it is listed once

logarithms (logs)

the inverse of an exponential function. When you find an inverse, you interchange x and y. So if y = b^x, the inverse is x = b^y. The use of logarithms helps us to solve exponential functions for y, since the notation log_b_y = x is the same as b^x = y. The base b cannot be negative. When you are asked to find a logarithm, call it x. Then write the expression as an exponential and solve it using the technique above.

symbolic representation

the most common way to represent a function (using a special function notation); this notation is either in the form of "y =" or "f(x) =". In the f(x) notation, we are stating a rule to find y given a value of x. Using this notation, it is easy to evaluate the function—plugging in a value of x to find y.

range

the set of second numbers of the ordered pairs in a relation (y); if a member is repeated, it is listed once

slope

the steepness of a line on a graph, equal to its vertical change divided by its horizontal change; given two points (x1, y1) and (x2, y2), the slope of the line passing through the points can be written as: m = rise/run = y2 - y1/x2 - x1 Lines that go up to the right have a positive slope, while lines that go up to the left have a negative slope.

simultaneous equations

two or more equations of the form ax + by = c, where a, b, c are constants and a, b do not equal 0 (also called linear equations with two unknown variables)

equivalent equations

two or more equations with the same solution

Quadratic Formula

x = -b ± √(b² - 4ac)/2a; if the quadratic equation does not have obvious factors, the roots of the equation can always be determined by the quadratic formula in terms of the coefficients a, b, and c, where (b^2 - 4ac) is called the discriminant of the quadratic equation.

slope-intercept form

y = mx + b, where m is the slope of the line and b is the y-intercept; used to find the point of intersection of the graphs of linear equations

If f(x) = {x^2 - 3, x > or = 0 {2x + 1, x < 0, then find: (A) f(4) (B) f(3) - f(-1) (C) f(f(0))

(A) f(4) = 16 - 3 = 13 (B) f(3) - f(-1) = (9 - 3) - (-2 + 1) = 6 - (-1) = 7 (C) f(0) = 0 - 3 = -3 f(-3) = -6 + 1 = -5

reciprocal

(mathematics) one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2

advanced algebraic theorem

If a polynomial equation f(x) = 0 with real coefficients has a root a + bi, then the conjugate of this complex number a - bi is also a root of f(x) = 0.

parallel lines

lines in the same plane that never intersect; given two linear equations in x, y, their graphs are parallel lines if their slopes are equal; if the lines are parallel, they have no simultaneous solution.

decay curve

occurs if 0 < b < 1

growth curve

occurs when b > 1, and the larger b is, the steeper the curve is.

exponential function

A function that contains a variable in an exponent; y = b^x, where the constant b is called the base and b is a positive number. An exponential graph tends to increase or decrease rapidly because x is in the exponent. All exponential functions in the form of y = b^x pass through the point (0, 1) as any base b raised to the zero power is one. Solving basic exponential equations can be accomplished by using the fact if b^x = b^y, then x = y. If the bases are the same, the powers must also be equal.

ordered pair

A pair of numbers that can be used to locate a point on a coordinate plane; it is in the form of (x,y) where x and y are real numbers.

function

A relation, a set of points (x,y), such that for every x, there is one and only one y. In short, in a function, the x-values cannot repeat while the y-values can.

inconsistent

A sentence is inconsistent if it is always false when its variables assume allowable values.

Consistent

A system of equations that has at least one ordered pair that satisfies both equations

Solving linear equations

A. If the equation has unknowns on both sides of the equality, it is convenient to put similar terms on the same sides. B. If the equation appears in fractional form, it is necessary to transform it using cross-multiplication, and then repeat the same procedure as in (A).

absolute inequality

An absolute inequality for the set of real numbers means that for any real value for the variable, x, the sentence is always true.

a^m/n = n✔️a^m (if m and n are positive integers)

An exponent can also be a fraction.

a^0 = 1

An exponent that is 0 gives a result of 1, assuming that the base itself is not equal to 0.

general form

Ax + By + C = 0 where A, B, and C are integers. This is the least useful form of a linear equation but it has no fractions in it. An equation in this form often has to be put into the other forms, described below, to get more information about the line, such as its slope.

Remainder Theorem

If a is any constant and the polynomial p(x) is divided by (x - a), the remainder is p(a).

Fundamental Theorem of Algebra

Every polynomial equation f(x) = 0 of degree greater than zero has at least one root either real or complex.

advanced algebraic theorem

Every polynomial equation of degree n has exactly n roots.

(a^p)^q = a^p•q

Ex: (2^3)^2 = 2^6 = 64

(a•b)^p = a^p•b^p

Ex: (3 X 2)^2 = 3^2 X 2^2 = (9) (4) = 36

(a/b)^p = a^p/b^p, and b does not equal 0

Ex: (4/5)^2 = 4^2/5^2 = 16/25

a^p/a^q = a^p-q, bases must be the same, a does not equal 0

Ex: 3^6/3^2 = 3^6-2 = 3^4 = 81

a^p•a^q = a^p+q, bases must be the same

Ex: 4^2•4^3 = 4^2+3 = 4^5 = 1,024

exponential growth/decay

Exponential equations and functions are used to model situations in which the rate of change of y increases faster over the same time step or change in x. When the rate of change is positive, it is called exponential growth and when the rate of change is negative, it is called exponential decay. The graph of an exponential growth or decay model will be an exponential curve. The equation is in the form of f(x) = a • b^x and the values of a and b are generally given.

numerical representation

Functions are given by the set of points: F = {(-1, 1), (0, 2), (1, 3), (2, 4), (3, 5)} is an example of a function given numerically.

advanced algebraic theorem

If a + the square root of b is a root of polynomial equation f(x) = 0 with rational coefficients, then a - the square root of b is also a root, where a and b are rational and the square root of b is irrational.

rules for inequalities

If a > b and a, b, and n are positive real numbers, then a^n > b^n and a^-n < b^-n. If x > y and q > p, then x + q > y + p. If x > y > 0 and q > p > 0, then xq > yp.

(b^2 - 4ac > 0)

If the discriminant is greater than zero, then the roots are real and unequal. The roots are rational if and only if a and b are rational and (b^2 - 4ac) is a perfect square; otherwise, the roots are irrational.

(b^2 - 4ac < 0)

If the discriminant is less than zero, the roots are complex numbers, since the discriminant appears under a radical and square roots of negatives are imaginary numbers. A real number added to an imaginary number yields a complex number.

composition of functions

If we have two functions f and g, we can find f(g(a)) or g(f(a)), which are different than f(a) • g(a). To find a composition of functions: plug a value into one function, determine an answer, and plug that answer into a second function. f(g(x)) can also be written as (f o g)(x).

Factor Theorem

If x = c is a solution of the equation f(x) = 0, then (x - c) is a factor of f(x).

linear growth/decay

Linear equations and functions are used to model situations in which a quantity y grows or declines the same amount over the same time step or change in x. The graph of a linear growth/decay model will be a straight line. You will usually be given the slope m and the y-intercept b and the linear function is given by f(x) = mx + b.

horizontal lines

Lines that are horizontal in the form: y = constant. Horizontal lines have zero slope. In the formula for slope, y2 - y1 = 0, so the slope is always zero.

vertical lines

Lines that are vertical are said to have no slope (it is so steep that we cannot give a value to its steepness). In the formula for slope, x2 - x1 = 0, and division by zero is impossible, making the slope undefined.

perpendicular lines

Lines that intersect to form right angles; if the slopes of the graphs of two lines are negative reciprocals of each other, the lines are perpendicular to each other.

Solving Linear Equations with two variables

Method 1: Substitution — Find the value of one unknown in terms of the other. Substitute this value in the other equation and solve. Method 2: Addition or Subtraction — If necessary, multiply the equations by numbers that will make the coefficients of one unknown in the resulting equations numerically equal. If the signs of equal coefficients are the same, subtract the equations; otherwise add. The result is one equation with one unknown; we solve it and substitute the value into the other equations to find the unknown that we first eliminated. Method 3: Graph — Graph both equations. The point of intersection of the drawn lines is a simultaneous solution for the equations, and its coordinates correspond to the answer that would be found by substitution or addition/subtraction.

Absolute value

The absolute value of a number is the distance the number is from the zero point on the number line. The absolute value of a number or an expression is always greater than or equal to zero (I.e., nonnegative). The absolute value of a, denoted | a |, is defined as: | a | = a when a > 0, | a | = -a when a < 0, | a | = 0 when a = 0.

point-slope form

The equation of a line with slope m and passing through the point (x1, y1) is given by y - y1 = m(x - x1). If we are given two points, this is the easiest way to find the equation of the line. Find the slope first, then plug in either point into the point-slope formula.

intercept form

The equation of a line with x-intercept a and y-intercept b is given by x/a + y/b = 1. If we know these intercepts, we can immediately write the equation of the line.

Exponents

When a number is multiplied by itself a specific number of times, it is said to be raised to a power. The way this is written is a^n = b, where a is the number or base, n is the exponent or power that indicates the number of times the base appears when multiplied by itself, and b is the product of this multiplication. An exponent can be either positive or negative. A negative exponent implies a fraction, such that if n is a negative integer a^-n = 1/a^n where a does not equal 0.

piecewise function

a function that computes different expressions based on its input; there are two or more rules, based on the value of x

common logarithm

a logarithm with base 10. The notations log_10_x and log x are the same. If the base is not specified, it is assumed to be 10.

Equation

a mathematical statement that two expressions are equal

vertical line test

a method to determine if a graph is a function or not; if it is possible for a vertical line to intersect a graph at more than one point, then the graph is not a function.

Complex Number

a number of the form a+bi where a and b are real numbers and i is the square root of -1; the number a is the real part, and the number bi is the imaginary part of the complex number. i is defined to be a number with a property that i^2 = -1. Obviously, i is not a real number since the square of a real number cannot be negative. C = {a + bi | a and b are real numbers}

relation

a set of ordered pairs (points)

solution of an equation

a value for a variable that makes an equation true when it is substituted for the variable

quadratic equation

an equation in which the highest power of an unknown quantity is a square; ax^2 + bx + c = 0, where a does not equal 0. General factor format: (x + ___)(x + ___) = 0. To solve the quadratic equation, set each factor equal to 0 to yield the solution set for x.

linear equation

an equation whose graph is a straight line; can be put into the form ax + b = 0, where a and b are constants, and a does not equal 0. To solve a linear equation means to transform it into the form x = -b/a.

empty set (null set)

an equation without a solution, represented by ø

conditional inequality

an inequality whose validity depends on the values of the variables in the sentence. That is, certain values of the variables will make the sentence true, and others will make it false.

dependent equations

equations that represent the same line; therefore, every point on the line of a dependent equation represents a solution.

equivalent inequalities

inequalities that have the same set of solutions