8th Grade Math Study Guide ЁЯНО
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 тИй
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 partsтАФone 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 ax┬▓ + bx + c = 0, where a тЙа 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: тИЪ5, 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