Review for Unit 6 Test

Module 11: Proportional RelationshipsModule 12: Nonproportional RelationshipsModule 13: Writing Linear EquationsModule 14: Functions

Linear and Nonlinear Relationships

A linear equation is an equation whose solutions are ordered pairs in the form of a line when graphed on a coordinate plane.Linear equations are written in the form y=mx+b.A non-linear equation is not written in the form y=mx+b because it doesn’t form a straight line.If a relationship is nonlinear, then it is non-proportional. If it is linear, it can be either proportional or non-proportional.

Proportional & Non-proportional Relationships

▪ A proportional relationship is a relationship between 2 quantities in which the ratio of one quantity to the other quantity is constant. In the form y=mx+b, b=0

▪ Constant of proportionality describes a proportional relationship where in the equation y=kx, k is a constant number.

▪ A non-proportional relationship will NOT go through the origin (0,0). In the form y=mx+b, b≠0.

Proportional Relationships using a graph

A relationship is a proportional relationship if the graph is a straight line through the origin. Examples below.

Non-proportional Relationships using a graph

A relationship is a non-proportional relationship if the graph does not go through the origin. Examples below.

Non-linear Linear (+ slope) Linear (- slope)

RATE OF CHANGE AND SLOPE

• A rate of change is the ratio of the amount of change in the dependent variable (output) to the change in the independent variable (input).

• The slope of a line is the ratio of the change in y-values(rise) for a segment of the graph to the corresponding change in the x values.

• = • m=• Remember that a unit rate has UNITS!!

(y per x).

Unit Rate

A unit rate is a rate in which the second quantity in the comparison is one unit.

Remember that a unit rate has UNITS!!

Example: Find the Unit Rate

Slope=Unit Rate= = =

Time (Hours)

4 8 12 16

Distance (mi) 5 10 15 20

Slope-Intercept Form

A linear equation is written in the form of y=mx+b, which is the slope-intercept form of an equation.– m represents the lines slope– b represents the lines y-intercept

• The y-intercept is the y-coordinate of the point where the graph interests the y-axis. The x-coordinate of this point is ALWAYS 0. (0,b)

Writing Linear EquationsWhen using a Graph:– Step 1: Choose 2 points on the graph (x₁,y₁) and (x₂,y₂) and calculate the

slope– Step 2: Read the y-intercept when x=0– Step 3: Use your slope and y-intercept and plug them into the equation.

y=mx+b.

Situations:– Step 1: Identify the input (x) and Output (y) variables– Step 2: Write the informations as ordered pairs– Step 3: Find the slope– Step 4: Find the y-intercept by plugging in the slope and choosing one

ordered pair (x,y) into y=mx+b – Step 5: Use your slope and y-intercept and substitute them into the

equation. y=mx+b.

Writing Linear Equations cont.

When using a:Table:– Step 1: Choose 2 points on the table (x₁,y₁) and (x₂,y₂) and

calculate the slope.– Step 2: Read the y-intercept when x=0 or find the y-

intercept by plugging in the slope and choosing one ordered pair (x,y) into y=mx+b

– Step 3: Use your slope and y-intercept and substitute them into the equation. y=mx+b (slope-intercept form).

Bivariate Data

Bivariate data is a set of data that is made up of two paired variables.– If the relationship between the variables is

linear then the rate of change (slope) is constant.

– If the graph shows a non-linear relationship, then the rate of change varies between pairs of points.

FunctionsA function assigns exactly one output to each input.A linear function is a graph of a non- vertical line.– Input- the value that is put into a function.– Output- the result

Vertical Line Test

Examples of functions & non-functions

Functions Not Functions

Comparing Functions

To compare a function written as an equation and another function represented by a table, find the equations for the function in the table.

ALSO for your test

For your Unit 6 test you will also need to know how to:– Analyze functions– Graph a line– Write an equation for a line– Compare data from a graph– Match a situation with a graph

(function)