Linear Equations.
Definitions and Examples.
Slope - The ratio of the change in the vertical distance to the change in the horizontal distance between any two points on a line. Examples. More examples. More examples. Interactive geoboard.
Linear equation - An equation whose solution is a straight line. An algebraic equation in which the highest degree term in the variable or variables is of the first degree. The graph of such an equation is a straight line if there are two variables. Examples.
Horizontal line - A line that has a slope of 0 and an equation of the form y = c. All the points on a horizontal line have the same y-coordinate, c. Example.
Vertical line - A line that has no slope and an equation of the form x = c. All the points on a vertical line have the same x-coordinate, c. Example.
Examples of horizontal lines and vertical lines.
Y-intercept - The y-coordinate of the point where a line intersects the y-axis. Examples.
Slope-intercept form of a line - The equation of a line with slope m and y-intercept b can be written in the form y = mx + b. Examples.
Examples of using slope and y-intercept to graph lines.
Examples of graphing linear equations using a t-table.
Point-slope form of a line - The equation of a line with slope m and ( x1, y1 ) is any point on the line and can be written in the form
y – y1 = m(x – x1). The point slope and slope intercept forms are easily interchangeable. Examples.
Parallel lines - Two lines that have equal slopes. Examples.
Perpendicular lines - Two lines that intersect and form 90 degree angles. Examples.
Intersecting lines - Two lines with different slopes that lie in the same plane always intersect in a single point. Examples. More examples.
Examples of intersecting lines and parallel lines. Finding the point of intersection with your graphing calculator.
Standard form of a line - The equation of a line can be written in the form Ax + By = C. Examples. More examples.
Function notation - A notation that describes a function. For a function ƒ, when x is a member of the domain, the symbol ƒ(x) denotes the corresponding member of the range [e.g., an equation of a function might be ƒ(x) = x+3]. Examples. More examples. Function notation with the graphing calculator. What is function notation?
Arithmetic operations on functions - Adding and subtracting functions. Examples.
Relation - The set of ordered pairs of the form (x, y). Examples.
Domain - The set of x values in ordered pairs. Examples.
Range - The set of y values in ordered pairs. Examples.
Function - A relation in which each element of the range and each element of the domain occurs in only one ordered pair. Examples.
Examples of relations and functions. More examples of relations and functions.
Additive inverse property - For every number a, a + (-a) = 0. Examples.
Multiplicative inverse property - For every nonzero number a/b, there is exactly one number b/a, such that a/b(b/a) = 1. Examples.
Commutative property - For any numbers a and b, a + b = b + a and a (b) = b (a). Examples.
Associative property - For any number a, b, and c, (a + b) + c = a + (b + c) and (ab)c = a(bc). Examples.
Examples of associative, commutative, and distributive properties.
Linear equation - An equation whose solution is a straight line. An algebraic equation in which the highest degree term in the variable or variables is of the first degree. The graph of such an equation is a straight line if there are two variables. Examples.
Horizontal line - A line that has a slope of 0 and an equation of the form y = c. All the points on a horizontal line have the same y-coordinate, c. Example.
Vertical line - A line that has no slope and an equation of the form x = c. All the points on a vertical line have the same x-coordinate, c. Example.
Examples of horizontal lines and vertical lines.
Y-intercept - The y-coordinate of the point where a line intersects the y-axis. Examples.
Slope-intercept form of a line - The equation of a line with slope m and y-intercept b can be written in the form y = mx + b. Examples.
Examples of using slope and y-intercept to graph lines.
Examples of graphing linear equations using a t-table.
Point-slope form of a line - The equation of a line with slope m and ( x1, y1 ) is any point on the line and can be written in the form
y – y1 = m(x – x1). The point slope and slope intercept forms are easily interchangeable. Examples.
Parallel lines - Two lines that have equal slopes. Examples.
Perpendicular lines - Two lines that intersect and form 90 degree angles. Examples.
Intersecting lines - Two lines with different slopes that lie in the same plane always intersect in a single point. Examples. More examples.
Examples of intersecting lines and parallel lines. Finding the point of intersection with your graphing calculator.
Standard form of a line - The equation of a line can be written in the form Ax + By = C. Examples. More examples.
Function notation - A notation that describes a function. For a function ƒ, when x is a member of the domain, the symbol ƒ(x) denotes the corresponding member of the range [e.g., an equation of a function might be ƒ(x) = x+3]. Examples. More examples. Function notation with the graphing calculator. What is function notation?
Arithmetic operations on functions - Adding and subtracting functions. Examples.
Relation - The set of ordered pairs of the form (x, y). Examples.
Domain - The set of x values in ordered pairs. Examples.
Range - The set of y values in ordered pairs. Examples.
Function - A relation in which each element of the range and each element of the domain occurs in only one ordered pair. Examples.
Examples of relations and functions. More examples of relations and functions.
Additive inverse property - For every number a, a + (-a) = 0. Examples.
Multiplicative inverse property - For every nonzero number a/b, there is exactly one number b/a, such that a/b(b/a) = 1. Examples.
Commutative property - For any numbers a and b, a + b = b + a and a (b) = b (a). Examples.
Associative property - For any number a, b, and c, (a + b) + c = a + (b + c) and (ab)c = a(bc). Examples.
Examples of associative, commutative, and distributive properties.