In a bell-shaped distribution, most of the data cluster near the center and fewer points are farther from the center. The dot plot or histogram for the data has the form of a bell.

This dot plot shows a bell-shaped distribution.

In a bimodal distribution, there are two very common data values. The dot plot or histogram for the data has two distinct peaks. 

This dot plot shows a bimodal distribution. The two common data values are 2 and 7.

Categorical data are data where the values are divided into groups, or categories. 

For example, the breeds of 10 different dogs are categorical data. Another example is the colors of 100 different flowers.

A categorical variable is a variable that takes on values that are divided into groups, or categories. 

For example, color is a categorical variable that can take on the values, red, blue, green, and so on.

In a causal relationship, a change in one of the variables causes a change in the other variable.

Completing the square is a method of rewriting a quadratic expression or equation. 

• A quadratic expression is rewritten in the form  \(a(x+p)^2-q\), where \(a\), \(p\), and \(q\) are constants and \(a\neq0.\)
• A quadratic equation is rewritten in  the form \(a(x+p)^2=q\).

A constraint is a limitation on the possible values of variables in a model. It is often expressed by an equation or inequality or by specifying that the value must be an integer.

For example, distance above the ground \(d\), in meters, might be constrained to be non-negative, expressed by \(d \ge 0\).

A correlation coefficient is a number between -1 and 1 that describes the strength and direction of a linear relationship between two numerical variables. 

• The sign of the correlation coefficient is the same as the sign of the slope of the best-fit line. 
• The closer the correlation coefficient is to 0, the weaker the linear relationship.
• When the correlation coefficient is closer to 1 or -1, the linear model fits the data better.

A dependent variable is a variable that represents the output of a function.

For example, the equation \(y = 6-x\) defines \(y\) as a function of \(x\).

• The variable \(x\) is called the independent variable because its value is chosen.
• The variable \(y\) is called the dependent variable because it depends on \(x\). Once a value is chosen for \(x\), the value of \(y\) is determined.

The distribution of a data set tells how many times each value occurs.

The domain of a function is the set of all of its possible input values.

Elimination is a method of solving a system of two equations in two variables. A multiple of one equation is added to or subtracted from another to get an equation with only one of the variables. (The other variable is eliminated.)

Equivalent equations are equations that have the exact same solutions.

Equivalent systems are systems that share the exact same solution set.

An exponential function is a function that has a constant growth factor. This means that it grows by equal factors over equal intervals.

For example, \(f(x)=2 \boldcdot 3^x\) defines an exponential function. Any time \(x\) increases by 1, \(f(x)\) increases by a factor of 3. 

A quadratic expression is in factored form when it is written as the product of a constant times two linear factors.

• \(2x^2 + 4x - 6\) written in factored form is \(2(x-1)(x+3)\).
• \(15x^2 + x - 2\) written in factored form is \((5x + 2)(3x-1)\).

The five-number summary is one way to describe the distribution of a data set. The five numbers are the minimum, the three quartiles, and the maximum. 

This box plot represents a data set with the following five-number summary: The minimum is 2, the three quartiles are 4, 4.5, and 6.5, and the maximum is 9.

A function is a rule that takes inputs from one set and assigns them to outputs from another set. Each input is assigned exactly one output.

Function notation is a way of writing the relationship between the inputs and outputs of a function.

For example, a function is named \(f\) and \(x\) is an input. Then \(f(x)\) denotes the corresponding output in function notation.

In an exponential function, the output is multiplied by the same factor every time the input increases by 1. This multiplier is called the growth factor.

In an exponential function, the growth rate is the fraction or percentage of the output that gets added every time the input is increased by 1. 

For example, if the growth rate is 20%, or 0.2, then the growth factor is 1.2.

A horizontal intercept of a graph is a point where the graph crosses the horizontal axis. If the axis is labeled with the variable \(x\), a horizontal intercept is also called an \(x\)-intercept. The term can also refer to only the \(x\)-coordinate of the point where the graph crosses the horizontal axis.

For example, the horizontal intercept of the graph of \(2x+4y=12\) is \((6,0)\), or just 6.

An independent variable is a variable that represents the input of a function.

For example, the equation \(y=6−x\) defines \(y\) as a function of \(x\). 

• The variable \(x\) is called the independent variable because its value is chosen. 
• The variable \(y\) is called the dependent variable because it depends on \(x\). Once a value is chosen for \(x\), the value of \(y\) is determined.

Two functions are inverses to each other if their input-output pairs are reversed.

• If one function takes \(a\) as input and gives \(b\) as an output, then the other function takes \(b\) as an input and gives \(a\) as an output.
• An inverse function can sometimes be found by reversing the processes that define the first function in order to define the second function.

For example, in the function \(w=52y\), the input is \(y\), the number of years, and the output is \(w\), the number of weeks. The inverse function, \(y=\frac{w}{52}\), is the result of reversing the process of multiplying by 52. For this function, the input is \(w\), the number of weeks, and the output is \(y\), the number of years.

An irrational number is a number that is not rational. This means it cannot be expressed as a positive fraction, a negative fraction, or zero. It cannot be written in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).

For example, the numbers \(\pi\) and \(\text{-}\sqrt{2}\) are irrational numbers.

A linear function is a function that has a constant rate of change. This means that it grows by equal differences over equal intervals. 

For example, \(f(x)=4x-3\) defines a linear function. Any time \(x\) increases by 1, \(f(x)\) increases by 4.

A linear term of an expression has a variable raised to the first power. 

• In the expression, \(5x+2\), the linear term is \(5x\).
• In the expression, \(20-x^2+x\), the linear term is \(x\).
• In the expression \(ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants, the linear term is \(bx\).

A maximum of a function is a value of the function that is greater than or equal to all the other values. The maximum of the function’s graph is the highest point on the graph.

A minimum of a function is a value of the function that is less than or equal to all the other values. The minimum of the function’s graph is the lowest point on the graph.

A model is a mathematical or statistical representation of information from science, technology, engineering, work, or everyday life, that is used to understand the situation and make decisions.

A non-statistical question is a question that can be answered by a specific measurement or procedure where no variability is expected. 

For example:

• How high is that building?
• If I run at 2 meters per second, how long will it take me to run 100 meters?
• Who was the first woman appointed to the Supreme Court?

Numerical data are data where the values are numbers, measurements, or quantities. Numerical data is also called measurement data or quantitative data.

For example, the weights of 10 different dogs are numerical data.

An outlier is a data value that is far from the other values in the data set. A value is considered an outlier if it is:

• More than 1.5 times the interquartile range greater than Q3. 
• More than 1.5 times the interquartile range less than Q1.

In this box plot, the minimum, 0, and the maximum, 44, are both outliers.

A perfect square is a number or an expression that is the result of multiplying a number or an expression to itself. In general, the multiplied number is rational and the multiplied expression has rational coefficients.

A piecewise function is a function defined using different expressions for different intervals in its domain.

A quadratic equation is an equation that is equivalent to one of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).

A quadratic expression is an expression that is equivalent to one of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).

The quadratic formula is \(x = {\text-b \pm \sqrt{b^2-4ac} \over 2a}\) and gives the solutions of the quadratic equation \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).

A quadratic function is a function where the output is given by a quadratic expression in the input.

For example, \(f(x) =ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants and \(a\ne0\), is a quadratic function.

The range of a function is the set of all of its possible output values.

A rational number is a number that can be written as a positive fraction, a negative fraction, or zero. It can be written in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). 

• 0.7 is a rational number because it can be written as \(\frac{7}{10}\).
• The numbers 0, 3, \(\text{-}\frac{3}{4}\), and 6.7 are all rational numbers. 

A relative frequency table is a version of a two-way table that shows how often data values occur in relation to a total. Each entry in the table shows the frequency of one response divided by the total number of responses in the entire table or by the total number of responses in a row or a column.

Each entry in this relative frequency table represents the proportion of all the textbooks that have the characteristics given by its row and column. For example, out of all 1,000 textbooks, the proportion of textbooks that are new and \\$10 or less is 0.025, or 2.5%.

In a skewed distribution, one side has more values farther from the bulk of the data than the other side. The mean is usually not equal to the median. The dot plot or histogram for the data shows only one peak leaning to one side.

This dot plot shows a skewed distribution. The data values on the left, such as 1, 2, and 3, are farther from the bulk of the data than the data values on the right.

The solutions to a system of inequalities are all the values for the variables that make all of the inequalities true.

This graph shows a system of two inequalities. The solutions of the system are all the coordinate pairs that make both inequalities true. On the graph, the solution is shown as all the points in the region where the graphs of the two inequalities overlap.  

A graph of two intersecting inequalities on a coordinate plane. The line x + y < 5 goes through the points 0 comma 5 and 5 comma 0. The region above the dashed line is shaded. The line 3 x - 2 y < 0 goes through the points 0 comma 0 and 4 comma 6. The region below the dashed line is shaded.

The standard deviation is a measure of the variability, or spread, of a distribution. It is calculated by a method similar to the method for calculating the MAD (mean absolute deviation). The exact method is studied in more advanced courses.

The standard form of a quadratic expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a\) \(\ne\) 0.

A statistic is a quantity that is calculated from sample data, such as mean, median, or MAD (mean absolute deviation).

A statistical question is a question that can only be answered by using data in which variability is expected.

For example:

• Who is the most popular musical artist at our school?
• When do students in our class typically eat dinner?
• Which classroom in our school has the most books?

Substitution is the action of replacing a variable with a number or expression it is equal to.

In a symmetric distribution, the data values on each side of the center mirror each other. The dot plot or histogram for the data has a vertical line of symmetry in the center, where the mean is equal to the median.

This dot plot shows a distribution that is symmetric about the data value 5.

Two or more equations that represent the constraints in the same situation form a system of equations. 

Two or more inequalities that represent the constraints in the same situation form a system of inequalities.

A two-way table is a way of organizing data from two categorical variables in order to investigate the association between them.

This two-way table can be used to study the relationship between age group and cell phone ownership.

	has a cell phone	does not have a cell phone
10–12 years old	25	35
13–15 years old	38	12
16–18 years old	52	8

In a uniform distribution, the data values are evenly spread out across the range of the data. The dot plot or histogram for the data shows no peaks.

This dot plot shows a uniform distribution. 

A variable is a characteristic of individuals in a population that can take on different values.

The vertex of the graph of a quadratic function or of an absolute value function is the point where the graph changes from increasing to decreasing, or vice versa. It is the highest or lowest point on the graph.

The vertex form of a quadratic expression is \(a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants and \(a \neq 0\). The vertex of the graph is at the point \((h,k)\).

A vertical intercept of a graph is a point where the graph crosses the vertical axis. If the axis is labeled with the variable \(y\), a vertical intercept is also called a \(y\)-intercept. The term can also refer to only the \(y\)-coordinate of the point where the graph crosses the vertical axis. 

For example, the vertical intercept of the graph of \(y=3x−5\) is \((0,\text{-}5)\), or just -5.

A zero of a function is an input that results in an output of 0. In other words, if \(f(a) = 0\), then \(a\) is a zero of \(f\).

The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.