Chapter 2: Linear Relations and Functions

2-1: Relations and Functions

ordered pairs: data represented as an input and output, (x, y)

Cartesian coordinate plane: composed of the x-axis (horizontal) and the y-axis (vertical), which meet at the origin (0, 0)

• divided into four quadrants

relation: a set of ordered pairs

domain: the set of all first coordinates (x-coordinates) from the ordered pairs

range: the set of all second coordinates (y-coordinates) from the ordered pairs

function: a special relation in which each domain value is paid with exactly **__ONE __**range value

mapping: shows how the members are paired

one-to-one function: a mapping in which each range value is paired with exactly ONE domain value

discrete: a relation in which the domain is a set of individual points

continuous: a relation that can be graphed with a line or a smooth curve

vertical line test: how to test is a graph is a function

1. if no vertical line intersects a graph in more than one point, the graph is a function
2. if a vertical line intersects a graph at 2 or more points, the graph is NOT a function

independent variable: the variable whose values make up the domain (usually x)

dependent variable: the variable whose values make up the range, the values that depend on the x (usually the y)

2-2 Linear Equations

linear equation: an equation that has no operations other than addition, subtraction, and multiplication of a variable by a constant

• variables cannot appear in the denominator
• cannot contain variables with exponents other than 1
• the graph is always a line

linear function: a function whose ordered pairs satisfy a linear equation

• can be written in the form: f(x) = mx + b, where m and b are real numbers

standard form: Ax + By = C, where A, B, and C are integers whose greatest common factor is 1

• example:
  • y = -2x + 3
  • 2x + y = 3
  • A = 2, B = 1, and C = 3

y-intercept: the y-coordinate of the point where a graph crosses the y-axis

x-intercept: the x-coordinate of the point where it crosses the x-axis

2-3 Slope

rate of change: measures how much a quantity changes relative to the change in another quantity

slope: the ratio of the change in y-coordinates to the corresponding change in x-coordinates

• the slope m of the line passing through (x1, y1) and (x2, y2) is:
  • m = (y2-y1) / (x2-x1), where x1 ≠ x2
• if the line rises to the right, the slope is positive
• if the line is horizontal the slope is zero
• if the line falls to the right, the slope is negative
• if the line is vertical, the slope is undefined

Parallel and Perpendicular Lines

family of graphs: a group of graphs that displays one or more similar characteristics

parent graph: the simplest of the graphs in the family

parallel lines: nonvertical lines with the same slope

• all vertical lines are parallel

perpendicular lines: two lines are perpendicular if and ONLY if the product of their slopes is -1

• any vertical line is perpendicular to any horizontal line

2-4 Writing Linear Equations

slope-intercept form: an equation written in the form: y = mx + b, where m is the slope and b is the y-intercept

point-slope form: an equation written in the form: y - y1 = m(x - x1), where x1 and y1 are coordinates of a point on the line

• used to find the equation of a line when given two points

2-5 Statistics: Using Scatter Plots

bivariate data: data with two variables

scatter plot: a set of bivariate data graphed as ordered pairs in the coordinate plane

positive correlation: positive slope in the scatter plot

negative correlation: negative slope in the scatter plot

no correlation: no visible trend, points are scattered and don’t have a positive or negative direction

Prediction Equations

line of fit: a line that approximates a set of data

prediction equation: the equation for a line of fit

• used to predict one of the variables when given the other

2-6 Special Functions

step function: a function that is not linear, consists of line segments or rays

greatest integer function: an example of a step function

• written as f(x) = [x']
• means the greatest integer less than or equal to x

constant function: when the slope is zero, every y-value is constant no matter what the x-value is

identity function: when the slope is 1, the x-values and y-values are equal

• passes through the origin

Absolute Value and Piecewise Functions

absolute value function: f(x) = |x|

piecewise: a function that is written using two or more expressions

2-7 Graphing Inequalities

boundary: a line that shows the points that satisfy the inequality

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