Chapter+5


 * 5-1**

**Use the rate of change to solve problems**
**Slope-** the ratio of the y coordinates to corresponding x coordinates (Rise/Run- the change in y over the change of x)

Slope is determined by x and y coordinates of 2 points

Slope Formula= __y^2-y^1__ x^2-x^1

Plug 2 sets of ordered pairs into the equation (3,2) (2,3)
 * y || x ||
 * 3 || 2 ||
 * 2 || 3 ||
 * Ratio of the y-coordinate to corresponding x-coordinate

__Rise__ Run
 * How fast the line goes up/down

__Change of Y__ Change of X

__Y____ 2 – Y____ 1 __ X 2 – X 1 ^ main formula, equals //m//

Slope can be... (Read a graph from left to right) Positive- line goes up Negative- line goes down Zero (0)- horizontal Undefied Vertical- line goes straight up and down



Quiz:

A= Negative B= Zero C= Negative D= Undefined Verticle E= Positive F= Zero

**Rate of Change-** Tells an average how quantity changes over time

depend on the context. ||
 * || Every Day Definition || Math Definition ||
 * expression || Numbers, symbols and operators (such as + and ×) grouped together that show the value of something. || An infinite combination of symbols that is well-formed according to rules that
 * function || Special relationship between values: Each of its input values gives back exactly one output value. || Associates one quantity, the arguement of the function, also known as the //input//, with another quantity, the value of the function, also known as the //output//. ||
 * slope || how steep a line is || the //slope// or gradient of a line describes its steepness, incline, or grade. ||
 * intercept || the point that is on an axis || y intercept= (0,y) x intercept= (x,0) ||
 * parallel || when to lines will never cross each other || extending in the same direction and never converging or diverging ||

= 5-2 =

**Slope and Direct Variation**
What is a direct variation? ***//K// cannot equal 0** Variation || how much our y varies || (what everything else comes off) y=x || //The bigger number you multiply the steeper the line will get//
 * Write and Graph direct variation equations
 * Solve Problems using direct variation
 * when two variables are related so the ratio of the values is always the same
 * y=kx
 * x is always multiplied by the same number (k) to get the y value
 * K = m = slope
 * x || y ||
 * Direct Variation || y=kx ||
 * Constant of
 * Family of Graphs || graphs that have something in common ||
 * Parent Graphs || you main graph

**(Ex:** y = 2x ...translation... for every 1 you go over you go up 2**)**

***If the slope is the same, the lines will be parallel.** ***If the slope is NOT the same, the lines will intersect somewhere.** ***If the slope is an opposite multiplicative inverse, the lines will be perpendicular (90º inverse).**

= **5.3** =

**-Model real world data with an equation in slope-intercept form**
***Slope Intercept Form = y = mx + b

= 5.4 =

**Writing Equations in Slope Intercept Form**

 * ==== Write an equation of a line given the slope and one point on a line ====
 * ==== Write an equation of a line given to points on a line ====

__Steps:__

 * 1) ==== Plug in what we know ====
 * 2) Solve for //b//
 * 3) Rewrite equation (substitute only //m// and //b)//

Examples:

Slope=-2/3 Point= (10,5) y = mx + b 5 = -2/3 * 10 + b  5 = -20/3 + b  5/1 + 20/3 = b  15/3 + 20/3 = 35/3 = b  b = 11 2/3


 * * Way to find slope: **


 * (4,7) (9,12) **


 * 4 8 **
 * __- 9 12__ **
 * -5 -4 **


 * Slope= **
 * __-4__ //or// -4 / -5 = 0.8 **
 * -5 **

Vocab: Function ||  || points you know ||
 * Linear Extrapolation || Your prediction for what you don't know ||
 * Absolute Value
 * Classes of Functions ||  ||
 * Interpolation || What is between the

= 5.5 =

**Writing Equations in Point Slope Form**
__Point slope form: (y - y1) = m(x - x1)__ (best for when given slope and 1 point)

__Slope intercept form: y = mx + b__ (best for when given slope and y-intercept)

__Standard form: Ax + By = C__ (best for when given x- and y-intercepts)

1.) If the slope = 0, the line will be horizontal... SO... y = any number 2.) If the slope = undefined, the line will be vertical... SO... x = any number 3.) If you are given 2 points... -find slope  -use one point to find point slope form  -write in standard form

**__Point slope form: (y - y1) = m(x - x1)__** //y// = just y //y1// = y coordinate point //m// = slope //x// = just x //x1// = x coordinate point

**Parallel and Perpendicular Lines**
*For a **perpendicular slope**, find the multiplicative inverse of the slope.


 * __Parallel Lines__ **
 * Never intercept each other
 * All vertical lines are parallel

__Steps:__
 * 1) Convert the line to slope-intercept form
 * 2) Use this form to identify slope
 * 3) Put this slope and the given ordered pair into point-slope form
 * 4) Convert to slope-intercept form by simplifying each side of the equation. Move the constant to the right side by adding the equation

‍‍‍‍‍** __Perpendicular Lines__ **
 * 90 degree angles
 * All angles added together = 360 degrees
 * Slope= the opposite reciprocal
 * Add a different y

__Steps:__
 * 1) Convert the line to slope-intercept form
 * 2) Use this form to identify the slope
 * 3) Flip the slope upside down and change the sign to get the new slope
 * 4) Put the new slope and the given ordered pair into point-slope form
 * 5) Convert to slope-intercept form by simplifying each side of the equation, move the constant to the right side by adding the opposite

The scatterplot is obtained by plotting //y// against //x//, as shown below.



A line of best fit by eye is drawn through the scatterplot so that an equal number of points lie on either side of the line and/or the sum of the distances of the points above the line are roughly equal to the sum of the distances below the line.