1. PreCalculus 1st Semester
1.1 Lines in the Plane
PreCalculus
1st Semester

2. Objectives: Find the slopes of lines.
Find the distance between two points.
Write linear equations given points on lines and their slope.
Use slope-intercept form of linear equations to sketch lines.
Use slope to identify parallel and perpendicular lines.

3. Slope
The slope of a nonvertical line represents the number of units a line rises or falls vertically for each horizontal change from left to right.
The slope m of the nonvertical line through is

4. Example 1:
Find the slope of the line passing through each pair of points.
(a) (-2,0) and (3,1)
(b) (-1,2) and (2,2)

5. Distance
Length of the line between two points
Formula

6. Example 2:
Find the distance between the following points
(a.)
(b.)

7. Point – Slope Form
The point-slope form of the equation of the line that passes through the point and has a slope m is

8. Example 3:
Find an equation of the line that passes through the point (1,-2) and has a slope of 3.

9. Example 4:
During 2000, Nike’s net sales were over $9.0 billion, and in 2001 net sales were $9.5 billion. Write a linear equation giving the net sales y in terms of the year x. Then use the equation to predict the net sales for 2002.
Let x = 0 represent 2000 and (0,9.0) and (1,9.5) represent 2 points on the line representing the net sales.

10. Slope-Intercept Form:
The graph of the equation
y = mx + b
is a line whose slope is m
and whose y-intercept is (0,b)

11. Example 5:
Determine the slope and y-intercept of each linear equation. Then describe its graph.
(a) x + y = (b) y = 2
y = -x m = 0 b = (0,2)
m = -1 b = (0,2) horizontal line
Graph falls one unit
for every unit it moves
to the right.

12. General Form: Ax + By + C = 0 Where A and B are not both zero.
(Typically convert to this form from point-slope and slope-intercept form)

13. Summary of Equations of Lines:
1. General Form: Ax + By + C = 0
2. Vertical Line: x = a
3. Horizontal Line: y = b
4. Slope-intercept form: y = mx + b
5. Point-slope form:

14. Parallel and Perpendicular Lines:
Parallel Lines: Two distinct nonvertical lines are parallel if and only if their slopes are equal. That is,
Perpendicular Lines: Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is,

15. Example 6:
Find the slope-intercept form of the equation of the line that passes through the point (2,-1) and is parallel to the line 2x – 3y = 5

16. Example 7:
Find the slope-intercept form of the equation of the line that passes through the point (2,-1) and is perpendicular to the line 2x – 3y = 5.