2-1 Relations and Functions Objectives Students will be able to: Analyze and graph relations Find functional values Note: You cannot spell function without “fun”

Ordered pair: a pair of coordinates, written in the form (x, y), used to locate any point on a coordinate plane Relation: a set of ordered pairs Domain: the set of all x-coordinates of the ordered pairs of a relation Range: the set of all y-coordinates of the ordered pairs of a relation Terminology

A function is a special type of relation in which each element of the domain is paired with exactly one element of the range. One-to-one function: a function where each element of the range is paired with exactly one element of the domain Functions

A mapping is a way of showing how each member of the domain is paired with each member of the range. Mappings

Mappings

State the relation shown in the graph. Then list the domain and range State the relation shown in the graph. Then list the domain and range. Is the relation a function? Relation: Domain: Range: Function??? Example 1

You try Relation: Domain: Range: Function??? State the relation shown in the graph. Then list the domain and range. Is the relation a function? Relation: Domain: Range: Function??? {(-4, -2), (-2, 3), (2, -3), (2, 1)} {-4, -2, 2} {-3, -2, 1, 3} No! The x value of 2 repeats You try

When given a graph of a relation, one can perform a vertical line test to determine whether a relation is a function. If a vertical line, does not intersect the graph in more than one point, then the relation is a function. If they do intersect the graph in more than one point, then the relation is not a function. Vertical Line Test

Vertical Line Test

Example 2: Vertical line Test Yes, is a function Not a function Example 2: Vertical line Test

Yes, is a function Not a function Try these:

Function Notation

Example 3

You Try!

2-2 Linear Equations Objective Students will be able to identify and graph linear equations and functions

Linear Equations A linear equation is the equation of a straight line. The only operations that exist in linear equations are addition, subtraction, and multiplication of a variable by a constant. Linear equations are often written in slope-intercept form (y=mx + b). Linear functions can be written in the form f(x)=mx + b. What linear equations would not be linear functions? Linear Equations

Graphing w/ Intercpets One way to graph a linear equation is by finding its x-intercept and y-intercept. The x-intercept is the point at which the graph crosses the x-axis. At this point, the y value will be 0. The ordered pair will be (x, 0). The y-intercept is the point at which the graph crosses the y-axis. At this point, the x value will be 0. The ordered pair will be (0, y). Graphing w/ Intercpets

Find the x-intercept and the y-intercept for each equation Find the x-intercept and the y-intercept for each equation. Then use the intercepts to graph the equation. x-intercept: y-intercept: Example 1:

x-intercept: y-intercept: You try.

Problems w/ Intercepts! NOTE: When finding intercepts, there are times when you will not attain two ordered pairs. Remember, to graph a linear equation, you need at least two ordered pairs. Times you will not attain two ordered pairs occur when: The equation is vertical x=constant The equation is horizontal y=constant Both intercepts occur at (0, 0) Let’s look at an example… Problems w/ Intercepts!

x-intercept: y-intercept: Graph?!

Other Graphing Methods When you do not attain two ordered pairs via the intercept method, you have a few options. You can create a table of x and y values. This is a way of attaining a few ordered pairs to help you graph the line. If the equation is in slope-intercept form, use the y-intercept and slope to graph the line. If it is not in slope-intercept form, get it in slope-intercept form! Other Graphing Methods

2-3 Slope Objectives Students will be able to: Find and use the slope of a line Graph linear equations using slope-intercept form

The slope of a line is the ratio of the change in y-coordinates to the corresponding change in x-coordinates Slope is also referred to as rate of change. Slope…

Four types of slope

Find the slope of the line that passes through each pair of points. (-1, 4) and (1, -2) (1, 3) and (-2, -3) (6, 4) and (-3, 4) Example 1:

Find the slope of the line that passes through each pair of points Find the slope of the line that passes through each pair of points. d) (-6, -3) and (6, 7) e) (5, 8) and (5, 0) You Try

Slope-Intercept Form y = mx + b Why is it so useful? The equation gives us two pieces of information we need to graph a linear equation: it’s slope, and it’s y-intercept. If we have these pieces of information we can graph any linear equation. Slope-Intercept Form

Example 2: Graph each equation.

You Try

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