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3.2 Lecture Guide: Special Forms of Linear Equations in Two Variables
Objective: Use the slope-intercept and point-slope forms of a linear equation.
Slope-Intercept Form
Algebraically y mx b is the equation of a linewith slope m and y-intercept 0,b .
Algebraic Example1
32
y x
Verbal Example
This line has slope 12
and a y-intercept of 0,3 .
Graphical Example
13
2y x
0,3
2,4
12
-5
5
-5 5
y
x
1. The line 2
45
y x has a
slope of ____________ and a y-intercept of ____________.
2. Graph the line 2
45
y x
using the slope and the y-intercept.
3. Complete the following table. This example stresses the fact that if we know the slope and the y-intercept, then we can immediately write the equation. Also, if we have the equation in slope-intercept form, we can immediately sketch the graph because we can quickly determine the slope and the y-intercept.
Slope y-intercept Equation
2
3 0,1
3 0, 4
57
3y x
15
3f x x
Point-Slope Form
1 1y y m x x Algebraically
is the equation of a line through 1 1,x y with slope m.
Algebraic Example
11 13
y x
Graphical Example
x
y
Verbal ExampleThis line passes through the point
1,1 with slope 13
.
1,1 4,2
13
4. Complete the following table. This example stresses the fact that the point-slope form of a line is useful when the slope and a point other than the y-intercept is given.
Slope Point Point-Slope Equation
2,14
3 2, 3
1
2 3,2
3, 4 2
5
2 3y x
34 1
2y x
General Form:
The general form, Ax By C , of an equation is useful forwriting linear equations without fractions.
Write each equation in general form.
5. 13 5
x y
Parallel and Perpendicular Lines:Recall that parallel lines have the same slope and that perpendicular lines have slopes that are opposite reciprocals.
Determine if the lines are parallel, perpendicular, or neither parallel nor perpendicular.
9. 32
4f x x and 3
14
f x x
Determine if the lines are parallel, perpendicular, or neither parallel nor perpendicular.
10. and 2 3y x 6 12 24x y
Determine if the lines are parallel, perpendicular, or neither parallel nor perpendicular.
11. and 3 2 8x y 6 4 12x y
Horizontal and Vertical Lines
Algebraically y b is the equation
0,b .
Example: 3y
Numerical Example
2 3
1 3
0 3
1 3
2 3
x y
Graphical Example
-4
4
-4 4
x
yof a horizontal line with y-intercept
Verbally This horizontal line has a y-intercept of 0,3and a slope of 0.
3y
-4
4
-4 4
x
y
Horizontal and Vertical Lines
Algebraically x a is the equation
,0a .
Example: 2x
Numerical Example
Graphical Example
of a vertical line with x-intercept
Verbally This vertical line has an x-intercept of 2,0and its slope is undefined.
2x
2 2
2 1
2 0
2 1
2 2
x y
12. All points on a horizontal line have the same _____-coordinate. This is the reason that the equation of a horizontal line is of the form _________________________. The slope of a horizontal line is _____.
13. All points on a vertical line have the same _____-coordinate. This is the reason that the equation of a vertical line is of the form _________________________. The slope of a vertical line is _________________________.
Graph each equation by completing a table of values and then give any intercepts. Can you check both of these on a graphing calculator? If not, why not?
Objective: Graph a line given one point and its slope.To graph a line or to give the equation of the line, it is sufficient to know any point on the line and the slope of the line. This is equivalent to knowing any two points on the line because we can calculate the slope given any two points on the line.
-5
5
-5 5
y
x
Complete the missing information. On all graphs, clearly label at least two points.
16. Equation:
Through:
Slope:
32
2y x
Graph:
-5
5
-5 5
y
x
Complete the missing information. On all graphs, clearly label at least two points.
17. Equation:
Through:
Slope:
2 3 6y x
Graph:
-5
5
-5 5
y
x
Complete the following table. On all graphs, clearly label at least two points.
18. Equation:
Through:
Slope:0m
3,4
Graph:
-5
5
-5 5
y
x
Complete the following table. On all graphs, clearly label at least two points.
19. Equation:
Through:
Slope:Undefined
4, 3
Graph:
-5
5
-5 5
y
x
Complete the following table. On all graphs, clearly label at least two points.
20. Equation:
Through:
Slope:
12 3
2y x
Graph:
-5
5
-5 5
y
x
Complete the missing information. On all graphs, clearly label at least two points.
21. Equation:
Through:
Slope:
3 2 1y x
Graph:
Use the point-slope form to write the equation passing through the given point with specified slope. Write the answer in slope-intercept form.
24. 4,2 , 34
m
Use the point-slope form to write the equation passing through the given point with specified slope. Write the answer in slope-intercept form.
25. , 4, 323
m
28. Write in slope-intercept form the equation of a line passing through 2, 3 and parallel to 5 2y x .
29. Write in slope-intercept form the equation of a line passing through 2, 3 and perpendicular to 5 2y x .
-5
5
-5 5
y
x
Complete the missing information.
34. Graph: Through:
Slope:
Point-Slope Equation:
Slope-Intercept Equation:
-5
5
-5 5
y
x
Complete the missing information.
35. Graph: Through:
Slope:
Point-Slope Equation:
Slope-Intercept Equation:
36. The given table displays the dollar cost of a collect phone call based on the length of the call in minutes.
Minutes Cost
3 $2.30
5 $3.00
10 $4.75
13 $5.80
x y (a) Determine the linear equation f x mx b for the line that contains
these data points.
(b) Determine the meaning of m and b in this application.