Context for Mathematics: Algebra & Functions Flashcards
A combination of one or more values arrange in terms that are added together. Could be a single number, including zero.
Expressions
In a variable term, there is a variable and this real number value.
Coefficient
Numbers without a variable
Constants
Constant Terms
Expression that represents the sum of a single variable term, where the variable has no exponent, and a constant, which may be zero.
For example: ax + b
Linear expression
On a graph with two points, can be found with the formula (y2 - y1) / (x2 - x1). Represented by the variable, m, or the ratio, rise over run.
Slope
Equation that can be written as ax + b = 0, where a is not 0.
Linear Equation
Set of all solutions of an equation, can include one, multiple, or zero solutions.
Solution Set
A solution set for an equation with no true values.
Empty Set
Forms of linear equations.
Hint: there are 5 main types.
Standard Form
Slope Intercept Form
Point-Slope Form
Two-Point Form
Intercept Form
Ax + By = C
The slope is -A / B and the y-intercept is C / B.
Standard Form
y = mx + b
The slope is m, and b is the y-intercept.
Slope Intercept Form
y-y1 = m (x - x1)
The slope is m and (x1, y1) is a point on the line.
Point-Slope Form
y-y1 / x-x1 = y2 - y1 / x2 - x1
(x1, y1) and (x2, y2) are two points on the given line.
Two-Point Form
x / x1 + y / y1 = 1
(x1, 0) is the point at which the line intersects the x-axis, and (0, y1) is the point at which the same line intersects the y-axis.
Intercept Form
A set of simultaneous equations that use the same variables. A solution must be true for each equation.
System of Equations
Consistent systems have this many solutions.
At least one solution.
Inconsistent systems have this many solutions.
No solutions.
Strategies for solving systems of equations equations.
Hint: there are 3 main strategies.
Substitution
Elimination
Graphing
The following describes the strategy for finding the…
“x = -b / 2a to find the x-coordinate, then substitute that value back into the equation to find the corresponding y-coordinate;
“a” is the coefficient of x^2 and “b” is the coefficient of x in the standard form of the equation (y = ax^2 + bx + c).”
Vertex of a Parabola
The following describes the strategy for finding the…
“Using two points (x1, y1) and (x2, y2), average the x-coordinates, average the y-coordinates.
The formula = (x1 + x2) / 2 , (y1 + y2) / 2.
Midpoint of Two Points
The following describes the strategy for finding the…
“Using two points (x1, y1) and (x2, y2) to create the hypotenuse and lines parallel to the x & y axis to create a right triangle. Then use the Pythagorean theorem, a^2 + b^2 = c or c = sqrt (a^2 + b^2).
The formula = sqrt ( (x2 - x1)^2 + (y2 - y1)^2 ).
Distance between Two Points
Single constant, variable, or product of constants and variables, such as 7, x, 2x, or xy^3. There will never be addition or subtraction.
Monomial
Algebraic expressions that use addition and subtraction to combine two or more monomials.
Polynomials
The sum of the exponents of the variables.
Degree of a Monomial
The highest degree of any individual term.
Degree of a Polynomial
The following describes the strategy for…
“Find the sum of like terms. Terms have the same variable part. Add the coefficients, but keep the variable and exponent the same”
Adding Polynomials
The following describes the strategy for…
“Use distributive property to distribute the minus sign. Then, find the sum of like terms. Terms have the same variable part. Add the coefficients, but keep the variable and exponent the same”
Subtracting Polynomials
The following describe strategies for…
“FOIL method, Box method and the distributive property.”
Multiplying Polynomials
The following describe strategies for…
“Factoring. Long division, written as quotient + (remainder / divisor).”
Dividing Polynomials
The following is the ___________ theorem of algebra.
“Every non-constant, single-variable polynomial has exactly as many roots as the polynomial’s highest exponent. For example, if x^4 is hte largest exponent, the polynomial will have exactly 4 roots. Some may be real or complex numbers, or multiplicity (repeats).
Fundamental Theorem of Algebra
The following is the ___________ theorem of algebra.
“States that if a polynomial function f(x) is divided by a binomial (x - a), where a is a real number, the remainder of the division will be the value of f(a). If f(a) = 0, then a is the root of the polynomial.”
Remainder Theorem
The following is the ___________ theorem of algebra.
“Related to the remainder theorem, states that if f(a) = 0, then (x - a) is a factor of the function.”
Factor Theorem
The following is the ___________ theorem of algebra.
“States that any __________ of a polynomial equation with integer coefficients is in the form of (p / q), where p is a factor of the constant term and q is a factor of the leading coefficient.
For example: f(x) = 3x³ - 5x² + 4x + 2.
The constant term is 2 and its factors are ± 1 and ± 2. These would be the values of p.
The leading coefficient is 3 and its factors are ± 1 and ± 3. These would be the values of q.
Values of p/q when q = ± 1:
p/q = ± 1/ ±1, ±2 / ±1 = ± 1, ± 2.
Values of p/q when q = ± 3:
p/q = ± 1/ ±3, ±2 / ±3 = ± 1/3, ± 2/3
Answer: The possible rational zeros of f(x) are ± 1, ± 2, ± 1/3, and ± 2/3.”
Rational Root Theorem
The solutions that satisfy the equation when y=0; in other words, where the graph touches the x-axis. Can be found using the quadratic formula, factoring, completing the square, or graphing the function.
Roots
The following describes which strategy for solving a quadratic equation.
“Rewrite the equation in standard form, ax^2 + bx + c = 0, where a, b, and c, are coefficients. Substitute into x = (-b +/- sqrt (b^2 - 4ac) / (2a). Evaluate and simplify. Check by substitution.”
Quadratic Formula
In a quadratic formula, the portion of the formula under the radical (b^2 - 4ac).
Discriminate
In the quadratic formula, if the discriminate (the portion under the radical) is 0, there are _____ roots.
One Root
In the quadratic formula, if the discriminate (the portion under the radical) is positive, there are _____ __________ ______ roots.
Two Different Real Roots
In the quadratic formula, if the discriminate (the portion under the radical) is negative, there are _____ _________ _______ roots.
Two Complex Roots
(No Real Roots)
The following describes which strategy for solving a quadratic equation.
“Begin by rewriting the equation in standard form, ax^2 + bx + c = 0. The goal is to find f and g in (x + f) (x + g) format so that (f + g) = b, and fg = c. Determine the factors of c and look for pairs that could sum to b. Write these as two binomials, (x + f) (x + g), set each to 0, and solve for each x.”
Factoring
The following describes which strategy for solving a quadratic equation.
“Manipulate the equation into the form x^2 + bx + c +/- d = 0, so it can be rewritten using a perfect square, (x + k)^2 + d = 0 and solve for x in the resulting equations.”
Completing the Square
The following describes which strategy for solving a quadratic equation.
“Factor the equation and set each factor equal to zero”
Zero Product Property
