8th Grade Math Study Guide 🍎

Exponential Function

A function that is defined by an exponential expression and written using function notation, and the variable is an exponent; example, f(x) = abx

Scatter Plot

A graph on which two different sets of data are plotted using coordinates

Line of Best Fit

A line drawn through the points on a scatter plot to summarize the relationship between the two sets of data

Square Roots

A number times itself equals the square under the radical sign

Coefficient

A number which multiplies a variable; example: 3x, 3 is the coefficient

Absolute Value

A number's distance from zero on a number line; examples: |-3| = 3 and |3| = 3; both numbers are 3 units away from zero

Perfect Square Trinomial

A perfect square trinomial is found in the expression where both the leading coefficients and the constant are both perfect squares. Ex: 16 is a perfect square of 4 times 4, and 9 is a perfect square of 3 times 3.

Trinomial

A polynomial with three terms that can be added or subtracted

Binomial

A polynomial with two terms

Combining Like Terms

A process of combining terms that have identical variable parts

Distributive Property

A property indicating a special way in which multiplication is applied to addition of two or more numbers in which each term inside a set of parentheses can be multiplied by a factor outside the parentheses, such as a(b + c) = ab + ac

Zero Exponent Property

A property that says any number (except 0) with an exponent of 0 is defined to equal 1; a0 = 1

Zero Product Property

A property that says that zero times anything equals zero. If ab = 0, then either a or b (or both) has to be zero. If a polynomial has factors multiplied together and the product is zero, then at least one of them has to be zero

Standard Form of a Quadratic Equation

A quadratic equation of the form y = ax^2 + bx + c, where each term is in decreasing order of degree

Relative Frequency

A ratio that compares the frequency of each category to the total.

Relation

A relationship that pairs input values with output values

Extraneous Solution

A solution of the simplified form of the equation that doesn't satisfy the original equation

Arithmetic Series

A sum of a list of numbers that are separated by a common difference

Variable

A symbol, usually a letter, which takes the place of an unknown value

Consistent system

A system of equations that has at least once solution; a pair of intersecting lines or same lines are consistent systems

Inconsistent System

A system of equations that has no solution is called an inconsistent system; parallel lines are an inconsistent system

Two-Way Frequency Table

A table where each cell in the table has the frequencies of two categories of data

Factor

A value that is multiplied by another value to get a product

Axis of Symmetry

A vertical line that passes through the vertex of the parabola

Box Plot

A visual that divides a list of data into four sections called quartiles

Histograms

A visual that shows data that appear in ranges (numerical data)

Dot Plot

A visual that shows the frequency of which something occurs; in a dot plot, the number of dots over a number line tells how many times that number occurs in the data

Point-Slope Form

A way of writing a linear equation using the slope and any point on the line y - y1 = m(x - x1) where m is the slope and (x1, y1) is a point on the line

Radical Equation

An equation that has a variable in the radicand

Literal Equations

An equation that has two or more variables

Function Rule

An equation that represents an output value in terms of an input value

Proportion

An equation that states two ratios are equal

Rational Exponent

An exponent expressed as a fraction

Prime Expression

An expression that does not have a common factor in the coefficients or the variables and cannot be factored in any other way

Linear Inequalities

An inequality that can be expressed as an equation and it graphed as a line with areas of inclusion and exclusion

Perfect Square

An integer that can be made by multiplying two of the same integers together

Sequence

An ordered list of numbers, objects, or events that often form a pattern

Law of Negative Exponents

Negative exponents in the numerator get moved to the denominator and become positive exponents. Negative exponents in the denominator get moved to the numerator and become positive exponents. Only move the negative exponents.

Term

One letter, number, or product of letters and numbers, also known as a monomial

Monomial

One letter, number, or product of letters and numbers, also known as a term

Order of Operations (PEMDAS)

Order in which you solve a math expression: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction

Function Notation

Notation that is used to represent a function; namely f(x), read f of x

Parallel Lines

Parallel lines have the same slope, are traveling in the same direction, and will never cross, but have different y-intercepts

Exponent

Tells how many times the base is used as a factor, or the number of time the base is multiplied by itself

Like Terms

Terms with the exact same variables with the exact same exponents for the variables

Mean Absolute Deviation

The average of the absolute values of differences between the mean and each value in a data set

Median

The number that falls in the middle when the data are ordered; when there is an odd number of terms, it is the middle term; when there is an even number of terms, it is the mean of the two middle terms

Run

The horizontal change of a line from one point to another

Intersection

The intersection of two or more sets is made up of the elements that the sets share; the term intersection is represented using an upside down symbol which looks like ∩

Greatest Common Factor (GCF)

The largest positive integer that divides without a remainder into two or more chosen integers

Boundary Lines

The line in the graph of an inequality that divides the plane into two parts—one side representing points that are solutions to the inequality and one side representing points that are not solutions to the inequality; the lines are dashed or solid based on the inequality symbol

Central Tendency

The measure of the "middle" of a set of data; common measures are the arithmetic mean, median, and mode

Interquartile Range (IQR)

The middle half of the data; it is a measure of how widely the middle half of the data is spread around the median

First Quartile

The midpoint of the lower half of the data when the data is listed in numerical order

Lower Quartile

The midpoint of the lower half of the data when the data is listed in numerical order; also known as the first quartile

Reciprocal

The multiplicative inverse of a number; for example, 2 divided by 3 is the reciprocal of 3 divided by 2

Negative Exponent Property

The negative sign on an exponent means the reciprocal. Think of it this way: just as a positive exponent means repeated multiplication by the base, a negative exponent means repeated division by the base

Area

The number of square units required to cover a surface Length x Width

Radicand

The number or expression contained under a radical sign

Base

The number or quantity being raised to a power

Exponential Decay

When the graph of an exponential function decreases from left to right; the base of the exponential expression must be between 0 and 1

Exponential Growth

When the graph of an exponential function increases from left to right; the base of the exponential expression must be greater than 1

Compound Interest

When the interest rate (or rate of growth) is applied to the principal and all other interests earned to that point

Algebraic Ratio

When the numerator, denominator, or both parts of a ratio contains a variable expression

Standard Form

When the polynomial is listed with each term in decreasing order of degree

Independent System

When two lines do not share all of the same points, such as parallel lines or intersecting lines

Intercept

Where the line you are graphing intersects the axes of the coordinate plane

Completing the Square

a process used to make a quadratic expression into a perfect square trinomial

Quadratic Equation

An equation of the form ax² + bx + c = 0, where a ≠ 0

Horizontal Lines

Lines that have a slope equal to zero

Vertical Lines

Lines that have a slope that is undefined

Rise

The vertical change of a line from one point to another

Perpendicular Lines

Two lines that intersect to form right angles

Compound Inequality

Two or more inequalities joined together by "and" or "or"

Difference of Two Squares

Two perfect squares being subtracted

PEMDAS

"Please Excuse My Dear Aunt Sally" is a mnemonic device reminding you that the order of operations is Parentheses, Exponents, Multiplication or Division, and finally Addition or Subtraction

Dividend

A number that is divided by another number.

Equal Sign

=; Tells you that what is on the left side of the equation equals what is on the right side of the equation

Parabola

A U-shaped curve, created by a quadratic equation

Statistics

A branch of mathematics that views numbers as data sets and teaches how to use a variety of methods to organize, interpret, and analyze these numbers to make informed decisions

Ratio

A comparison of two numbers by division

Composite Function

A composite function is generally a function that is written inside another function. Composition of a function is done by substituting one function into another function. For example, f [g (x)] is the composite function of f (x) and g (x).

Conditional Relative Frequency

A conditional relative frequency compares a frequency count to the marginal total that represents the condition of interest. For. example, the condition of interest in the first row is females. The row conditional relative frequency of females responding.

Constant Function

A constant function is a linear function of the form y = b, where b is a constant; it is also written as f(x) = b; the graph of a constant function is a horizontal line

Outlier

A data point that is significantly larger or smaller than the other values

Arithmetic Sequence

A list of terms that share a common difference; the difference between the consecutive terms is constant

Geometric Sequence

A list of terms that share a common ratio; each term is found by multiplying or dividing the same value from one term to the next

Polynomials

A mathematical expression that has: 1). One or more terms 2). Variables with only positive, whole number exponents 3). No variables in the denominators of each term

Function

A mathematical relationship in which each input value is paired with exactly one output value

Inequality

A mathematical sentence that contains less than, greater than, less than or equal to, greater than or equal to, or not equal

Equation

A mathematical statement containing an equal sign to show that two expressions are equal

Standard Deviation

A measure of spread to allow for comparisons to the overall data set; found by finding the sum of the square of the differences of each data point from the mean, dividing it by the number of data points, and square rooting that value

Sigma Notation

A method of writing a long series of numbers in an abbreviated format; also known as summation notation

Numerical Expression

A number or a group of numbers that are added, subtracted, multiplied, or divided; example: 4 + 6 is a numerical expression

Rational Number

A number that can be written as a fraction

Irrational Number

A number that cannot be expressed as a ratio (fraction) of two numbers; this includes decimals that never stop or repeat; examples: √5, 3.2516...

Slope-Intercept Form

A way of writing a linear equation where the y-variable is isolated on one side of the equation. Slope-intercept looks like y = mx + b, where m equals the slope and b equals the y-intercept

Whole Numbers

All positive whole numbers including zero; this category does not include any negative whole numbers; examples: 0, 3, 6

Natural Number

All positive whole numbers, not including zero; examples: 4, 7, 9

Real Number

All rational and irrational numbers

Absolute Value Inequalities

An absolute value inequality is an expression with absolute functions as well as inequality signs. For example, the expression |x + 3| > 1 is an absolute value inequality containing a greater than symbol. There are four different inequality symbols to choose from.

Algebraic Equation

An algebraic equation is a mathematical sentence that contains numbers, variables, and operations like addition, subtraction, etc. and also contains an equal sign that sets two algebraic expressions equal to each other; example: x + 4 = 9 is an algebraic equation Algebraic Equations contain an equal sign

Algebraic Expression

An algebraic expression is a mathematical sentence that contains numbers, variables, and operations like addition, subtraction, etc; example: x + 4 is an algebraic expression Algebraic Expressions don't contain an equal sign

Quotient

Answer to the division problem

Infinitely Many Solutions

Any ordered pair that satisfies one of the equations or inequalities will also satisfy the other equation or inequality; the equations describe the same line, or the inequality has the same border line

Standard Form of a Linear Equation

Ax + By = C, where A,B, and C are real numbers, and A and B are not both zero

System of Equations

Collection of two or more equations

Divisor

Denominator, or what is being divided by

Simplify

Distributing, combining like terms, etc.

Elimination Method

Eliminating one of the variables when combining the two equations

Degree of a Polynomial

Equal to the degree of the term with the highest degree

Cross Multiply

Finding the cross products of two ratios

Transitive Property

For any real numbers a, b, and c if a = b, and b = c, then a = c. Three parts: the first statement relates to the second, which relates to the third.

Inverse Functions

Functions whose compositions simplify to x; for example, if f(x) and g(x) are inverses of each other, then f(g(x)) = g(f(x)) = x

Associative Property of Addition or Multiplication

Grouping symbols do not affect the outcome. Numbers can be associated by using parentheses or brackets in an addition or multiplication problem and it does not change the solution

Factoring by Grouping

Grouping terms into pairs before factoring a GCF from each group

Accuracy

How closely a given set of values are compared to the expected value

Substitution

If two quantities are equal, one quantity may be put in the place of the other quantity and the value of the expression does not change

Addition Property of Equality

If you add the same number to each side of an equation, the two sides remain equal

Division Property of Equality

If you divide each side of an equation by the same nonzero number, the two sides remain equal

Multiplication Property of Equality

If you multiply each side of an equation by the same nonzero number, the two sides remain equal

Subtraction Property of Equality

If you subtract the same number from each side of an equation, the two sides remain equal

Leading Coefficient

In a polynomial in terms of x, the coefficient in front of the x term with the highest degree in the polynomial

Conjunction

Inequalities joined by the word "and"

Disjunction

Inequalities joined by the word "or"

Horizontal Asymptote

Invisible horizontal lines that a rational function approaches, yet does not cross, when graphed

Substitution Method

Isolating one variable in one of the equations and substituting it into the other equation

Negative Slope

Line falls from left to right

Opposite Operation

Performing the inverse operation to solve an equation; example: the inverse of addition is subtraction; the inverse of multiplication is division

Integer

Positive and negative whole numbers including zeros; this category does not include any fractions or decimals; examples: -2, 0, 4

Rational Exponent Property

Property that states that an expression raised to a rational number can be written as a rational expression

Grouping Symbols

Symbols that signify two or more things are grouped together; parentheses ( ) or brackets [ ]

Equivalent Systems

Systems of equations that have the same solutions

Constraints

Restrictions placed on a variable depending on a real-world situation

Joint Frequency

Shows the data for each two-category response and occurs in the body of the table; found by reading from the inner cells of a two-way frequency table

Parameter

The constant or the limits within the problem

Residuals

The difference between the actual values and the predicted values; normally used to determine the accuracy of the line of best fit created on a scatter plot

Range (of data set)

The difference between the largest and smallest values in a data set

Difference

The difference between two numbers (subtraction)

Rate of Change, or Slope

The difference of the y-values divided by the difference of the x-values

Slope

The difference of the y-values divided by the difference of the x-values

Commutative Property of Addition or Multiplication

The order in which you perform an operation does not affect the outcome. Numbers can move around or commute in an addition or multiplication problem and it does not change the solution

Parent Function

The original function or function you start with

X-Intercept

The point where a graph intersects the x-axis; the value of x when y = 0

Y-Intercept

The point where a line crosses the y-axis; the value of y when x = 0

Domain

The possible input values (x values)

Range

The possible output values (y-values)

Joint Relative Frequency

The proportional values found in the interior of a relative frequency table

Slope of a Line

The ratio of the vertical change to the horizontal change between two points, represented in equations by m

Correlation

The relationship between two groups of data

Sum

The result of adding

Equality

The same

Symmetric

The same on one side as it is on the other; a circle cut in half has two symmetrical sides

Positive Slope

The slope of a line going up to the right

Undefined Slope

The slope of a vertical line

Least Common Multiple (LCM)

The smallest (least) multiple that the two numbers have in common (the same). Example, the LCM of 2 and 3: Multiples of 2 are 2, 4, 6, 8, 10, 12, etc. Multiples of 3 are 3, 6, 9, 12, 15, etc. The smallest multiple they have in common is 6.

Quotient Property of Radicals

The square root of a quotient is equal to the quotient of the square roots

Mean

The sum of the data points divided by the number of data points

Degree of a Term

The sum of the exponents on the variables in the term

Closure Property

The sum or product of any two real numbers is a real number

Radical

The symbol for a square root

Vertex

The turning point (minimum or maximum) of a parabola

Solution

The value for the variable that makes the equation true

Power

The value of a number or quantity raised to some exponent

Average Rate of Change

The value that shows how much the range values have changed per unit in the domain for a specified interval in the domain

Marginal Frequency

The values calculated from the row and column totals in a two-way frequency table

Zeros of a Polynomial

The values of the input of a polynomial function that would make the function zero; appear as the x-intercepts on the graph of the polynomial

Marginal Relative Frequency

The values shown in the column and row totals of a relative frequency table

Maximum Point

The vertex of a parabola that opens down

Minimum Point

The vertex of a parabola that opens up

Upper Quartile (Q3)

This is the midpoint of the upper half of the data when the data are listed in numerical order; also known as the third quartile

Third Quartile

This is the midpoint of the upper half of the data when the data is listed in numerical order

Quotient of Powers Property

To divide powers with the same base, subtract their exponents

Expanded Form of an Exponent

To expand a power means to write the multiplication represented by the exponent

Power of a Power Property

To find a power of a power, multiply the exponents

Power of a Product Property

To find a power of a product, find the power of each factor and multiply

Power of a Quotient Property

To find the power of a quotient, find the power of the numerator and the power of the denominator and divide

Associate

To group with something else

Product of Powers Property

To multiply powers with the same base, add the exponents

Closed Circle

Used on number lines when graphing greater than or equal to or less than or equal to inequalities

Open Circle

Used on number lines when graphing greater than or less than inequalities

All Real Numbers

Used when solving an equation in which both sides of the equation are the same; all real numbers means that any real number would make the equation true

No Solution

Used when solving an equation or inequality in which the solution is false; no solution means that there is no possible value for x that would make the equation or inequality true

Function Composition

Using the output from one function call as the input to another

Prime

When a number can be evenly divided only by one and itself

Bar Graph

a diagram in which the numerical values of variables are represented by the height or length of lines or rectangles of equal width

Symmetric Property

if a=b, then b=a