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May 25th

May 25th

# Graphs

1.3 Lines

1 Calculate and inter pret the slope of a Line (p. 20)
(1) Defintion

Let be two distinct points. If , the slope m of
the nonvertical line L containing P and Q is defined by the formula

If , L is a and the slope m of L is .

(2) Sketch the plot of the following equations, find its slope and de termine whether the line is
slants upward, slants downward, horizontal or vertical.

 Equations Sketch Slope Type m = 2 Slant upward

(3) Finding and Interpreting the Slope of a Line

2 Graph Lines Given a Point and the Slope (Slope is defined) *

(1) Graph the point (x, y) on the Cartesian coordinate system

(2) Write the slope as (Rise is integer and Run are positive integer )

(3) Graph another point (x + Run, y + Rise)

(4) Drawing the line through the two points

3 Find the Equation of a Line

(1) Equations of a line

 Equation of a Vertical Line (p. 24) A vertical line is given by an equation of the form Where a is the x-intercept. Equation of a Horizontal Line (p. 25) A horizontal line is given by an equation of the form Where b is the y-intercept. Point-Slope Form of an Equation of a Line (p. 24) An equation of a nonvertical line with slope m that contains the point is Slope- Intercept Form of an Equation of a Line (p. 25) An equation of a nonvertical line with slope m and y-intercept b is General Form (p. 27) The general form of the equation of a line is Where A, B and C are real numbers and A and B are not both 0.

(2) Find the Equation of a Vertical Line

(3) Find the Equation of a Horizontal Line

(4) Find the Equation of a General Line

a) Known a point and the slope, use the point-slope form
b) Known the slope and the y-intercept, use the slope-intercept form
c) Known two points, use the general form
d) Then substitute the known conditions into the relative form, and simplify the equation

(5) Identify the Slope and y-intercept of a Line from Its Equation
Write the equation of the line as the slope-intercept form: y = m x + b

(6) Graph the line with equation in different form
a) Known point-slope form, , see 2
b) Known slope-intercept form, y = mx + b, graph one point (b, 0) and the follow the steps
(2)-(4) in 2
c) Known the general form, Ax + By = C, then follow these steps
• substitute x=0 into the equation, and solve it for y, denoted as
• substitute y=0 into the equation, and solve it for x, denoted as
• graph two points and
• draw the line through two points

3 Parallel Lines

(1) Criterion for Parallel Lines
Two nonvertical lines are parallel if and only if their slopes are equal and they have different
y-intercepts.

(2) Equation of two parallel lines

(3) Showing that two lines are parallel

a) Find their slopes and y-intercepts
b) Compare their slopes: they are NOT parallel with different slopes
c) Compare their y-intercepts: they are NOT parallel with the same y-intercept

(4) Finding a line that is parallel to a given line

a) Write the equation of the given line into the slope-intercept form
b) Assume the equation of the parallel line is
c) Substitute the known equation into the equation, and solve for

4 Find Equations of Perpendicular Lines

(1) Definition
When two lines intersect at a right angle (90ยบ), they are said to be perpendicular.

(2) Criterion for Perpendicular Lines
Two nonvertical lines are perpendicular if and only if the product of their slopes is -1.

(3) Equation of two vertical lines

(5) Finding a line that is perpendicular to a given line

a) Write the equation of the given line into the slope-intercept form

b) Assume the equation of the parallel line is

c) Substitute the known equation into the equation, and solve for

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