# Algebra 1 Glossary

Absolute value: The distance a number is from $0$.

Algebra: The study of mathematical symbols and the rules for manipulating those symbols.

Algebra grid: A grid used to illustrate values of algebraic expressions.

Arithmetic sequence: A sequence of numbers in which each number can be computed by adding the same amount to the previous number.

Associative law of addition: For any three numbers $a$, $b$, and $c$, it is always true that $(a+b)+c = a+(b+c)$.

Associative law of multiplication: For any three numbers $a$, $b$, and $c$, it is always true that $(a(b))(c) = a(b(c))$.

Axis of symmetry: A line that you can flip (or reflect) a graph across that results in the same graph.

Base: A number that is raised to a power.

Best fit line: When the points on a grid are not all on a straight line, but seem to have a somewhat linear pattern, you can find a line that is the “best fit” (closest) to the points.

Box plot: A box with “whiskers” showing the median, quartiles, and extremes (least and greatest values) of a collection of data values.

Break even: Have a profit of zero (that is, make exactly as much money as you spend).

Ceiling function: $\ceiling(x)$ is the closest integer which is greater than or equal to $x$.

Clearing denominators: Multiplying both sides of an equation by some nonzero number that turns all the fractions in the equation into integers.

Coefficient: A constant that a variable or expression is multiplied by.

Combining like terms: Using the distributive law to add any two multiples of an expression such as $x$. For example, you can simplify $4x+5x$ into $9x$.

Common difference: In an arithmetic sequence, the amount that can be added to each number to get the next one.

Common ratio: In a geometric sequence, the amount that each number can be multiplied by to get the next number.

Commutative law of addition: For any two numbers $a$ and $b$, $a + b = b + a$.

Commutative law of multiplication: For two numbers $a$ and $b$, $a(b) = b(a)$.

Completing the square: Rewriting the equation $x^2+2mx=n$ as $(x+m)^2 = n + m^2$ so that it can be more easily solved.

Composition: The composition of two functions $f$ and $g$ is the function $f ∘ g$ that transforms $x$ into $f(g(x))$.

Conditional relative frequency: A joint frequency divided by the total of its row or column in a two-way frequency table.

Constant: A single fixed number (unlike a variable, whose value can vary).

Constant coefficient: A constant term, thought of as a coefficient of $1$.

Constant term: A term that is a constant. For a polynomial in $x$, it’s the term without an $x$.

Coordinates: A point on a 2-dimensional plane is described by a pair $(x, y)$. The coordinate $x$ is given by the labels below the grid, and the coordinate $y$ is given by the labels to the left of the grid.

Coordinate plane: A 2-dimensional flat surface used for plotting points, lines, curves, and regions. It contains an $x$ and a $y$ axis which intersect at the origin.

Coordinate grid: A grid of lines on a coordinate plane that makes it easy to see $(x, y)$ coordinates of locations in that plane.

Correlation coefficient: A number between $1$ and $-1$ that indicates how much increasing one variable will tend to increase or decrease the other variable. If the best fit linear model for $x$ and $y$ is $y=mx+b$, the correlation coefficient $r$ of $x$ and $y$ satisfies $$r^2=1-(\text"root-mean-square error of model")^2 / (\text"standard deviation of " y \text" data")^2$$. The correlation coefficient $r$ has the same sign (positive, negative, or 0) as $m$. If $x$ and $y$ have means $m_x$ and $m_y$ and standard deviations $σ_x$ and $σ_y$, it is also true that the correlation coefficient is the mean of $(x-m_x)(y-m_y)$ divided by $σ_x σ_y$.

Cost: In economics, how much money a company spends to produce a product.

Cube root: The cube root of $a$, written $√^3 a$, is the number whose cube is $a$. That is, $(√^3 a)^3 = a$.

Data: A collection of related measurements.

Decimal: A fractional quantity written with a decimal point (like $0.5$).

Denominator: The bottom number or expression in a fraction.

Difference: The distance between two quantities, or the answer to a subtraction problem.

Dilation: A dilation by a positive number $r$ about a point $A$ is a transformation that moves each other point $B$ along the ray from $A$ that passes through $B$, and multiplies distances from $A$ by $r$. $A$ is called the center of the dilation.

Discriminant: The discriminant of the equation $ax^2+bx+c=0$ is the quantity $b^2-4ac$. A quadratic equation has two solutions if its discriminant is positive, one solution if its discriminant is zero, and no real solutions if its discriminant is negative.

Distributive law of multiplication over addition: For any three numbers $a$, $b$, and $c$, $a(b+c) = a(b)+a(c)$, and $(b+c)(a) = b(a)+c(a)$.

Distributive law of multiplication over subtraction: For any three numbers $a$, $b$, and $c$, $a(b−c) = a(b)−a(c)$, and $(b−c)(a) = b(a)−c(a)$.

Domain: The set of inputs ($x$-coordinates) of a relation or function.

Dot plot: A diagram showing data values as dots above a number line.

Equation: A mathematical sentence with an equals sign (like $3x+5=11$).

Equivalent: Two fractions are equivalent if they have the same numerical value. Two equations or inequalities are equivalent if they have the same solution set.

Even function: A function $f$ with $f(x)=f(-x)$ for all $x$. $f(x) = x^n$ is an even function if $n$ is an even integer. A function is even if and only if its graph has the $y$-axis as an axis of symmetry.

Expanding an expression: Using the distributive law to turn expressions which need parentheses (like $3(x+2)$) into expressions which do not (like $3x+6$).

Exponent: In a power, the number of times the base is multiplied by itself.

Exponential: Using exponents, especially using variables in exponents.

Exponential decay: Decreasing toward $0$ due to a variable in an exponent, such as in $y=2^{-x}$.

Exponential growth: Increasing rapidly due to a variable in an exponent, such as in $y=2^x$.

Expression: A combination of variables and numbers using arithmetic (like $6-x$).

Factor: An expression that is multiplied by another expression, or that can be multiplied by another expression to produce a specified result.

Factoring: Rewriting an expression as a product.

Fibonacci sequence: The sequence 1, 1, 2, 3, 5, ... with remaining terms $F(n)$ given by $F(n) = F(n-1) + F(n-2)$ for $n > 2$.

First quartile: For $n$ data values, the median of the $$n/2$$ smallest values if $n$ is even, and of the $${n-1}/2$$ smallest values if $n$ is odd.

Floor function: $\floor(x)$ is the closest integer which is less than or equal to $x$.

Formula: An expression that is used to compute a value.

Fraction: A numerator divided by a denominator (like $$1/2$$). Usually we require the numerator and denominator to both be integers.

Frequency: In statistics, the number of times something occurs, or is observed.

Function: A relation in which no $x$-coordinate appears in more than one $(x, y)$ ordered pair. This means you can think of a function as a transformation that takes each $x$-coordinate to its single corresponding $y$-coordinate.

Fundamental theorem of arithmetic: Any integer greater than $1$ can be written as a product of prime numbers, ordered smallest to largest, in exactly one way.

Geometric sequence: A sequence of numbers in which each number can be computed by multiplying the previous number by the same amount.

Graph: An image formed by plotting the solutions to an equation or inequality, or some other set of pairs of numbers, on a coordinate plane. To graph an expression containing the variable $x$, set $y$ equal to that expression.

Histogram: Rectangles of equal width above a number line, where each rectangles’s height shows the number of data values in that portion of the number line.

Horizontal: Going from side to side, like the horizon.

Improper fraction: A fraction in which the numerator is larger than the denominator (like $$3/2$$).

Inequality: A mathematical sentence that uses one of the symbols $<$, $>$, $≤$, or $≥$.

Infinite: More than any finite (real) number.

Input: A number that can be “put into” a relation to produce one or more “outputs.” If a relation is given by a two column table of rows $(x, y)$, you “look up” the input $x$ value in the first column, and the output(s) are given by the $y$ values in those matching row(s).

Integer: A whole number or the negative of a whole number. For instance, $37$ and $0$ and $-5$ are integers, but $2.7$ and $$-3/2$$ are not.

Interquartile range: The third quartile minus the first quartile.

Inverse functions: Functions $f$ and $g$ such that ${g ∘ f}(x)=x$ for every $x$ in the domain of $f$, and ${f ∘ g}(y)=y$ for every $y$ in the domain of $g$.

Irrational number: A number that cannot be written as a fraction $$m/n$$ where $m$ and $n$ are integers.

Isolate: Make a variable appear alone on one side of an equation or inequality, and not occur in the other side of the equation or inequality.

Joint frequency: The number of events that satisfy both of two specified criteria.

Joint relative frequency: A joint frequency divided by the total number of events.

Laws of exponents: $$a^{c + d} = a^c a^d$$, $$(a b)^d = a^d b^d$$, and $$(a^c)^d = a^{cd}$$. These are always true when $c$ and $d$ are positive integers. If $a$ and $b$ are nonzero, then they are true for any integers $c$ and $d$, as is $$a^{c - d} = a^c / a^d$$. If $a$ and $b$ are positive, then all four laws are true for any $c$ and $d$.

Linear: A straight line, or an equation or expression whose graph is a straight line. If $m$ and $b$ are constants, then $mx+b$ is a linear expression, and a function $f$ defined by $f(x)=mx+b$ is a linear function.

Linear coefficient: For a polynomial in $x$, the number that $x$ (without an exponent) is multiplied by.

Linear model: An estimate for a variable using a linear expression in another variable.

Linear optimization: Maximizing or minimizing a linear goal or cost expression, while remaining within some constraints given by linear inequalities.

Marginal frequency: The total of a row or column in a two-way frequency table.

Marginal relative frequency: A marginal frequency divided by the total number of events.

Mean: The average of a collection of data values. This can be computed by adding all the values and then dividing by the number of values.

Mean absolute deviation: The mean distance of data values from some central value, such as the mean, median, or mode of the collection. In other words, the mean of $|x-m|$ where $x$ is each data value and $m$ is the mean, median, or mode of all the data values.

Mean squared deviation: The mean squared distance of data values from some central value, such as the mean, median, or mode of the collection. In other words, the mean of $(x-m)^2$ where $x$ is each data value and $m$ is the mean, median, or mode of all the data values.

Median: The middle number in an ordered list of data values. If there are an even number of values, the median is halfway between the two middle numbers in the list.

Mode: The most common value in a collection, or “modes” if more than one are tied.

Monic: A polynomial whose leading (first) coefficient is $1$.

Monomial: A product of variables and numbers, like $3x$ or $5x^2$. A monomial is also sometimes called a term.

Negate: Take the opposite of a number, by multiplying it by $-1$.

Negative number: A value less than zero (like $-3$).

$n$th root: An $n$th root of $a$ is a number $b$ whose $n$th power is $a$. That is, $b^n = a$. If $a ≥ 0$ and $n$ is an integer and $n > 0$, then “the” $nth root of$a$, written$√^n{a}$, is the$n$th root of$a$that is positive or zero. Numerator: The top number or expression in a fraction. Odd function: A function$f$with$-f(x)=f(-x)$for all$x$.$f(x) = x^n$is an odd function if$n$is an odd integer. A function is odd if and only if its graph has the point$(0, 0)$as a point of symmetry. One-to-one: A function$f$for which$f(x)$has a different value for every distinct (different) value of$x$. Origin: The point on a coordinate plane where the$x$-axis and$y$-axis intersect. It is represented by the coordinates$(0, 0)$. Outlier: A value that “lies outside” (is much smaller or larger than) most of the other values in a collection. Output: A number produced by applying a relation or function to an input. Parabola: The shape of the graph of$y=x^2$. Parallel: Two lines are parallel if they always have the same distance between them, so they never intersect. If two lines are parallel, they have the same slope. Perfect square: A number that is the square of a rational number. For example,$1$,$4$, $$25/16$$, and$0$are perfect squares. An integer is a perfect square only if it is the square of an integer, which can be proven using the fundamental theorem of arithmetic. Period: For a periodic function, the amount of time before it repeats. That is, if$f$is a periodic function, its period is the smallest possible positive$h$where$f(x+h)=f(x)$for every$x$. Periodic function: A function that repeats after a certain period$h$with$h > 0$, so that$f(x+h)=f(x)$for every$x$. Perpendicular: Two lines are perpendicular if they create a 90-degree angle. If two lines are perpendicular and the slope of one of them is$m$, then the slope of the other line is $$-1/m$$. Piecewise-defined function: A function that is defined by different formulas at different inputs. Point: A location in the coordinate plane. A point has coordinates$(x,y)$, where$x$is given by the labels below a coordinate grid, and$y$is given by the labels to the left of a coordinate grid. Point of symmetry: A point that you can rotate a graph around by 180° that results in the same graph. Point-slope form: If a line contains the point$(x_1,y_1)$and has slope$m$, then its equation can be written as$y−y_1=m(x−x_1)$. An equation in the form$y−y_1=m(x−x_1)$is said to be in point-slope form. Polynomial: A sum of monomials. Usually terms with higher powers are written first. Positive number: A value greater than zero (like 3). Power: An expression of the form$a^d$.$a$is called the base,$d$is called the exponent, and$a^d$is called “the$d$th power of$a$”. If$d$is a positive integer,$a^d$means$a$multiplied by itself$d$times. Prime number: An integer greater than$1$that can only be written as a product of two whole numbers in one way: as itself multiplied by$1$. Product: The answer to a multiplication problem. Profit: Revenue minus cost. Quadrant: Each of the four sections of a coordinate plane made by the intersecting$x$- and$y$-axes. The four quadrants are labeled I, II, III, and IV, counterclockwise from the top right. Quadratic: An expression or equation in which the highest power of a variable has exponent$2$. Quadratic coefficient: For a polynomial in$x$, the number that$x^2$is multiplied by. Quartiles: The first quartile, median, and third quartile are values which divide a data collection into four roughly equal parts. Quotient: The answer to a division problem. Range: The set of outputs ($y$-coordinates) of a relation or function. Rate of change: The speed at which a variable changes over a period of time. This is given by the change in the variable divided by the change in (amount of) time. Rational number: A number that can be written as a fraction $$m/n$$ where$m$and$n$are integers. Reflection: Rigid motion across a fixed line$AB$in a plane, like a mirror image. Relation: A set of ordered pairs$(x, y)$. Relative frequency: A frequency divided by the total number of events, often expressed as a percentage. Residual: An observed value minus its estimated value. Restriction: A function$g$is a restriction of the function$f$if$g(x) = f(x)$for every$x$in the domain of$g$, but that domain may be smaller than the domain of$f$. Revenue: How much money a company receives in sales. Rigid motion: A motion that preserves distances and angle measures, with no stretching, shrinking, or bending. A rigid motion in the plane is a sequence of one or more translations, rotations, and/or reflections. Root-mean-square error: A number that tells you how far away a line or curve is from a set of points (a smaller number means the line is a better “fit” to the points). More precisely, it is the square root of the mean of the squared residuals (differences) between observed and estimated values. Roots: The values of$x$where a polynomial is zero. These are the$x$-coordinates of the$x$-intercepts of the polynomial’s graph. Rotation: Rigid motion around a fixed center$A$, with turning but no reflection. Scatter plot: Dots in the coordinate plane representing pairs of linked measurements, such as heights and weights for a group of people. Sequence: A list of numbers that may be generated by some rule. Set: An unordered collection of numbers or other mathematical objects, without repetitions. Similar: Two geometric figures are similar if they have the same shape but possibly different sizes, with corresponding lengths differing by a single common scale factor. Simplify: To rewrite an expression in a way that means the same thing but is simpler (or shorter). You can simplify$3x - x + 6$into$2x + 6$. Slope: A number that measures how steep a line is. It shows the amount of change in the height of the line as you go 1 unit to the right. The slope of the line$y=mx+b$is$m$. Slope-intercept form: The form$y=mx+b$for a linear equation, where$m$and$b$are constants. The numbers$m$and$b$give the slope and$y$-intercept of the line that is the graph of that equation. Solution: In an equation or inequality, a number that can be substituted for the variable to make that equation or inequality true. If the equation or inequality has more than one variable, a solution is a list of numbers that when substituted for the list of variables makes the equation or inequality true. For a system of more than one equation or inequality, a solution must make all of the equations or inequalities true. In chemistry, a solution is a liquid mixture. Solution set: All solutions to an equation, inequality, or system. Solve: Find the solutions to an equation, inequality, or system. Square root: A square root of$a$is a number$b$whose square is$a$. That is,$b^2 = a$. If$b$is a square root of$a$, then so is$- b$. If$a ≥ 0$, “the” square root of$a$, written$√a$, is the square root of$a$that is positive or zero. Standard deviation: The square root of the variance. Standard form: For a linear equation, the form$Ax+By=C$where$A$,$B$, and$C$are constants. For a quadratic equation, either the form$y=ax^2+bx+c$or$ax^2+bx+c=0$, where$a$,$b$, and$c$are constants. Statistic: A number used to describe or summarize data. Statistics: The study of data, and the methods used to describe or summarize data. Step function: A piecewise-defined function where each piece’s formula is a constant (doesn’t change with$x$). A step function’s graph looks like stair steps. Substitution: In an expression or equation, eliminating a variable by replacing it with another expression that it is equal to. Sum: The answer to an addition problem. Symmetry: Repeating pattern or shape. System: For equations or inequalities, two or more equations or inequalities that are all required to be true. Table: In mathematics, a rectangular arrangement of rows and columns. Term: Element in a sum, difference, or sequence. The quadratic formula: The formula $$x = {-b ± √{b^2-4ac}} / {2a}$$, which gives the solutions to any equation in the form$ax^2+bx+c=0$with$a ≠ 0$. The equation has two solutions when$b^2-4ac > 0$; it has one solution when$b^2-4ac=0$; and it has no real solutions when$b^2-4ac < 0$. Third quartile: For$n$data values, the median of the $$n/2$$ largest values if$n$is even, and of the $${n-1}/2$$ largest values if$n$is odd. Translation: Rigid motion by a constant distance in a single direction, with no rotation or reflection. Two-way frequency table: For events that can be divided into categories two different ways, a table of joint frequencies, using rows of the table to group the events one way, and columns of the table to group the events the other way. Unit: A standard measurement, such as a meter or an hour. Value: A number that a variable or expression can equal. Variable: A letter (like$x$) that we can use to mean different numbers at different times. Variance: The mean squared distance of data values from their mean$m$. This can be computed by adding$(x-m)^2$for each data value$x$, and then dividing by the number of data values$n$. When measuring a sample from a population, for instance heights of people, the variance of the sample is usually different than the variance of the entire population. To estimate the population’s variance, it is usually better to divide by$n-1$instead of$n$. Vertex: The point where a parabola crosses its axis of symmetry, or an end of a side of a polygon, or the corner point of an angle. Vertex form: A quadratic equation in the form$y=a(x-h)^2+k$. Vertical: Going up and down. Whole number: One of the numbers 0, 1, 2, 3, ... .$x$-axis: The horizontal line running through the origin on a coordinate plane.$x$-coordinate: The horizontal value in a coordinate pair. It tells how far to the left or right the point is. The$x$-coordinate is always written first in the coordinate pair.$x$-intercept: A point where a curve meets the horizontal axis (the$x$-axis).$y$-axis: The vertical line running through the origin on a coordinate plane.$y$-coordinate: The vertical value in a coordinate pair. It tells how far up or down the point is. The$y$-coordinate is always written last in the coordinate pair.$y$-intercept: A point where a line or curve meets the vertical axis (the$y$-axis). The$y$-intercept of the line$y=mx+b$is the point$(0,b)$. Zeros: The values of$x$where an expression is zero. These are the$x$-coordinates of the$x\$-intercepts of the expression’s graph. For a polynomial expression, these are usually called roots.