Linear+Relations+Study+Sheet

Definition: the relation between two variables resulting in a straight line. Each output variable is unique and proportional to another input variable.
 * __ LINEAR RELATIONS STUDY SHEET  __**

1. Linear Functions & Linear Systems 2. Interpolation & Extrapolation 3. Slopes & Intercepts 4. Inequalities
 * Key Concepts: **

The slope or gradient of a line describes the steepness or incline of a value.
 * Slopes: **

The slope of a line is basically the rate of change. In short, it is the difference of the x and y co-ordinates in comparison to the previous point in the line. This formula is best described as: Standard Slope Intercept Form: y= mx +b m represents the Slope y is the intercept form which in this case is the y-intercept form b is the y intercept

Example 1: Find the slope and y -intercept for the equation 3 y = -12 x + 30. First solve for y : 3y/3 = -12 x/3 + 30/3 y= -4x +10

Use the form: y = mx + b Answer: the slope ( m ) is -4 the y -intercept ( b ) is 10

Example 2: Find the slope and x-intercept for the equation 2x= -4y + 12 First solve for y: 2x/2= -4y/2 + 12/2 x= -2y + 6

Answer: -the slope (m) is -2 -the x-intercept (b) is 6

How to plot lines using the intercept format:
 * 1) Upon obtaining the y-intercept, use your number sense to apply to a graph. For example, if your intercept is y= 2x +1, you will start by plotting the first point at (1, 0) or the point where the graph “crosses” the y-axis.
 * 2) Use the slope (in this case, 2x) to plot the rise/run of the line or the difference of location in the points.
 * 3) Continuing this process until you have 3 points on the graph where you can now connect the lines and extend because 3 points is the magically number that deduces a pattern.
 * 4) Be sure to put an arrow at both ends which symbolizes that the line continues.

Point Slope Form:

This form is only used when you can obtain the x and y value through graphing the points. (A point on the line and the slope) Example 2: Given that the slope of a line is -3 and the line passes through the point (-2,4), write the equation of the line. The slope: //m// = -3 The point (//x//1 ,//y//1) = (-3,5) Use the form: //y - y//1 = //m// ( //x - x//1) //y// - 5 = -3 (//x// - (-3)) //y// - 5 = -3 ( //x// + 3) If asked to express the answer in "//y// =" form: //y// - 5 = -3//x// - 9 // y // = -3//x// – 4 Horizontal Line Form y=3 (or any number) Lines that are horizontal have “run” but no “rise”. The rise/run formula for slope always yields zero since rise=0. y=mx+b y=03+b y=3 In short, to obtain a horizontal line that is parallel to the y axis, simply have the rise/run of the equation equal 0 x=-2 (or any number) Lines that are vertical have no slope (it does not exist). They have “rise” but no “run”. The rise/run formula for slope always has a zero denominator and is undefined. Sources: @http://staff.argyll.epsb.ca/jreed/math9/strand2/2204.htm @http://www.mathwords.com/v/vertical_line_equation.htm @http://www.regentsprep.org/Regents/math/geometry/GCG1/EqLines.htm
 * Vertical Line Form **