Notes#8: Section 2.3: Graphing lines, Section 2.4: Writing equations of lines
Forms of Lines: Lines can be written in either Slope-Intercept form (y = mx + b) or
Standard Form (Ax + By = C). You need to know how to convert from one to the other.
Converting to Slope-Intercept Form
Converting to Standard Form
Goal: y = mx + b
(where m and b are integers or fractions)
•
•
Goal: Ax + By = C
(where A, B, and C are integers and
where A is positive)
Get y alone
Reduce all fractions
•
•
•
Get x and y terms on the left side and
the constant term on the right side of
the equation
Multiply ALL terms by the common
denominator to eliminate the fractions
If necessary, change ALL signs so
that the x term is positive
1.) Convert to slope-intercept form:
2.) Convert to standard form:
4x – 12y = 8
y=
2
x−5
3
3.) Convert to both slope-intercept form and standard form:
a) y – 3 = -5(x + 4)
1
b.) y + 1 = − ( x − 3)
2
Section 2.3: A. Graphing Lines using the slope and y-intercept:
- Get y alone so the equation is in y = mx + b form (m = _________, b = _________)
- Graph b first. This point goes on the ____ axis.
- Use slope and count rise over run to the next point(s). When you have at least three points on your
graph, connect the points with a ruler to make a straight line.
- Label your graphed line with the original equation
Most common errors:
• Graphing b on the x-axis instead of the y-axis
• Graphing the slope in the wrong direction (e.g. forgetting a negative)
1
4.)
y=x
y
( ↑ I’m already in slope-intercept form!)
10
9
8
7
6
m = ___ (Å graph me second!)
b = ___ (Å graph me first! I go on the y-axis!)
5
4
3
2
1
x
5.)
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
y = − x−5
2
1 2 3 4 5 6 7 8 9 10
-2
-3
( ↑ I’m already in slope-intercept form!)
-4
-5
-6
-7
m = ___ (Å graph me second! Watch the negative!)
-8
-9
-10
b = ___ (Å graph me first! I go on the y-axis!)
Compare the graph to y = x
6.) x – 2y =2
( ↑ Get me in slope-intercept form first. Get y alone.)
(Graph for #4 and #5)
y
10
9
8
7
6
5
4
3
2
1
m = ______
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
b = ______
Compare the graph to
1 2 3 4 5 6 7 8 9 10
y=x
-4
-5
-6
-7
7.)
-8
-9
x + 3y = -6
-10
(Graph for #6 and #7)
m = ______
b = ______
Compare the graph to
y=x
B. Graphing Equations Using Intercepts
2
•
x-intercept is the x-coordinate of the point where a line crosses the __________. To find the x-
intercept, make y = 0 and solve for x.
•
y-intercept is the y-coordinate of the point where a line crosses the __________. To find the y-
intercept, make x = 0 and solve for y.
•
Remember: the intercepts are TWO different points!
Find the x- and y-intercepts.
8.) 3x + 4y = 8
x-intercept
(make y = 0)
9.) 4x − 9y = −12
y-intercept
(make x = 0)
(____, 0) and (0, ____)
Graphing Lines using the x- and y- intercepts. The intercepts are the point(s) where a line
intersects the axes of the coordinate plane.
- Find the x and y intercepts (by setting the opposite variable to zero)
- Write these answers as two different points
- Graph and connect these points to graph the line
- Label the graphed line with the original equation
Most common error:
Forgetting that the intercepts are two different points and graphing as just one
y
10.) x + 2y = 4
10
9
x-intercept
y-intercept
8
(set y = 0)
(set x = 0)
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
x-int: (
, 0)
y-int: (0,
)
1 2 3 4 5 6 7 8 9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
3
y
11.) 3x – y = 3
10
9
8
7
6
5
4
3
x-int: (
,
)
y-int: (
,
)
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1 2 3 4 5 6 7 8 9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
12.) 2x – 3y = 8
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1 2 3 4 5 6 7 8 9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
Special Cases: Graphing Horizontal and Vertical Lines
• Any line in the form x = ___ is a _______________ line because it intersects the ___ _________
• Any line in the form y = ___ is a _______________ line because it intersects the ___ _________
Use this pattern to graph these lines without a table of solutions.
13.) y = 3
14.) x = -2
15.) y = -4
4
Writing Equations of Lines:
A. Writing linear equations given the slope and y-intercept
- Find the slope (m) and y-intercept (b) [If the given information is a graph, then you will have to count by
hand to find these values.]
- Fill in m and b so you have an equation of the line in y = mx + b form. Convert to standard form if
necessary.
y = ________ x + ____________
( ↑ Put m here!)
( ↑ Put b here!)
17.) Find the equation of the
18.) Write the equation of a line
16.) Find the equation of the
given line in slope-intercept
that has the same slope as
line with slope of 5 and y4
form.
intercept of -2. Write in
y = x − 3 and has a y-intercept
standard form.
5
of 1. Write in standard form.
B. Writing linear equations given the slope and a point
• plug slope = m into y = mx + b
• name your point (x, y) and plug these values in for x and y
• solve for b
• plug m and b back into y = mx + b **(leave x and y as variables!**
• convert to standard form, if necessary
19.) Find the equation of the line with slope of -2
and going through (-1, 3) in slope-intercept form.
20.) Find the equation of the line with slope of
1
3
and going through (6, -2) in standard form.
5
C. Writing linear equations given two points
• find the slope using the slope formula (m = ___________ )
• pick one of your points to be x and y
• plug m, x, y into y = mx + b
• solve for b; plug m and b into y = mx + b (** Remember to leave x and y as variables! **)
• convert to standard form, if necessary
22.) Find the equation of the line with
21.) Find the equation of the line going
x-intercept 3 and y-intercept -2 in standard
through (-3, 1) and (4, 8) in slope-intercept
form.
form.
23.) Find the equation of the line going
through (5, 2) and (-1, 3) in standard form.
24.) Find the equation of the line with
x-intercept 5 and y-intercept -4 in slopeintercept form.
Writing an equation of a line given a parallel or perpendicular line
• Find m from the given line by getting y alone first
• If the line is parallel, __________________________
• If the line is perpendicular, ________________________________________
• Plug m, x, y into y = mx + b; solve for b
• Plug m and b into y = mx + b (leave x and y as variables!). Convert to standard form if
necessary.
25.) Write the equation of the line that passes
through (1, 2) and is parallel to y = 3x + 4.
Leave in slope-intercept form.
26.) Write the equation of the line that passes
through (3, -2) and is perpendicular to
y = 3x – 5. Leave your answer in standard
form.
6
27.) Write the equation of the line that passes
through (2, 4) and is parallel to 2x – y = 5.
Convert to standard form.
28.) Write the equation of the line that passes
through (-1, 5) and is perpendicular to
2x – 3y = 6. Leave in slope-intercept form.
29.) Write the equation of the line that passes
through (-3, 2) and parallel to x + 2y = 4.
Leave in slope-intercept form.
30.) Write the equation of the line that passes
through (-2, 0) and perpendicular to x = 2y – 1.
Leave in standard form.
HW#8:
Pg. 93: 3-54 x 3,
Pg. 101: 3-45 x 3
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