Review for Unit 6 Test Module 11: Proportional Relationships Module 12: Nonproportional Relationships Module 13: Writing Linear Equations Module 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. 𝑆𝑙𝑜𝑝𝑒 & 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒= 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 = 𝑦₂−𝑦₁ 𝑥₂−𝑥₁ m= 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 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= 𝑦₂−𝑦₁ 𝑥₂−𝑥₁ = 10−2 8−4 = 5 4 𝑚𝑖/ℎ𝑜𝑢𝑟 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 Equations When 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.

Functions A 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 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)