Transcript Lesson 6.2b Writing Equations of Parallel and

LESSON 6.1B
WRITING EQUATIONS OF
PARALLEL AND
PERPENDICULAR LINES
CONCEPT: WRITING EQUATIONS OF LINES
EQ: HOW DO WE WRITE THE EQUATION OF A
LINE THAT IS PARALLEL OR PERPENDICULAR
TO A GIVEN LINE? (G.GPE.5)
VOCABULARY: SLOPE, PARALLEL, PERPENDICULAR,
RECIPROCAL
FIRST WORD
• Create an acrostic about
everything you remember from
the previous lesson using the word.
•S
L
O
P
E
INTRODUCTION
• Parallel lines are lines which never touch or
intersect.
• Perpendicular lines are lines which meet or
intersect and create a 90° angle aka a right
angle.
• We can find the equation of two lines which
are parallel or perpendicular and prove
whether the lines are parallel or
perpendicular.
PARALLEL LINES
• Lines which are parallel always
have the same slope or value for
m, but a different y-intercept or
value for b.
• Ex. 𝑓 𝑥 = 2𝑥 + 5 𝑎𝑛𝑑 𝑓 𝑥 = 2𝑥 − 11
are parallel lines because they have
the same slope but different yintercepts.
EXAMPLE 1
• Two lines, 𝑙1 𝑎𝑛𝑑 𝑙2 , are linear equation. Line 𝑙1 has the
1
1
equation 𝑦 = 𝑥 + 4. Line 𝑙2 has the equation 𝑦 = 𝑥 − 2.
4
4
Are they parallel?
EXAMPLE 2
• The two lines in the
graph to the right,
m1 and m2 , are
shown. Are they
parallel?
𝑚1 𝑠𝑙𝑜𝑝𝑒: ___________
𝑚2 𝑠𝑙𝑜𝑝𝑒: ___________
YOU TRY 1
• You are the given two equations of two different
lines, 𝑦 = 3𝑥 − 5 𝑎𝑛𝑑 𝑦 = 3𝑥 + 7. Are these lines
parallel?
YOU TRY 2
• Are these lines
parallel? Explain
𝐿𝑖𝑛𝑒 1 𝑠𝑙𝑜𝑝𝑒: ___________
𝐿𝑖𝑛𝑒 2 𝑠𝑙𝑜𝑝𝑒: ___________
EXAMPLE 3
• A line which contains the point (2,5) is parallel to the
line 𝑓 𝑥 = 3𝑥 − 7. Find the equation to this line.
1. State the point-slope formula.
2. Identify m, 𝑥1 , 𝑎𝑛𝑑 𝑦1 .
3. Substitute the values into the
formula.
4. Simplify into slope-intercept form.
EXAMPLE 4
• Using the graph to the right,
find the equation of a
parallel line which passes
through the point which is
not on the given line.
1. Find the slope of the given line.
2. Identify m, 𝑥1 , 𝑎𝑛𝑑 𝑦1 .
3. Substitute the values into the
point-slope formula.
4. Simplify into slope-intercept form.
YOU TRY 3
• A line which contains the point (0,-11) is parallel to
the line 𝑓 𝑥 = 5𝑥 + 3. Find the equation to this line.
1. State the point-slope formula.
2. Identify m, 𝑥1 , 𝑎𝑛𝑑 𝑦1 .
3. Substitute the values into the
formula.
4. Simplify into slope-intercept form.
PERPENDICULAR LINES
• Lines are always perpendicular if their slopes are a
negative reciprocal of one another. Perpendicular
lines can have the same y-intercepts though.
1
• Ex. 𝑓 𝑥 = 2𝑥 + 5 𝑎𝑛𝑑 𝑓 𝑥 = − 𝑥 + 5 are perpendicular lines
2
with the same y-intercepts. Another example would be
1
𝑓 𝑥 = 2𝑥 + 5 𝑎𝑛𝑑 𝑓 𝑥 = − 𝑥 − 11. These two lines are
2
perpendicular because of their slopes but have different yintercepts.
EXAMPLE 5
1
• Two lines are given, 𝑓 𝑥 = 2𝑥 − 3 𝑎𝑛𝑑 𝑓 𝑥 = − 𝑥 − 3.
2
Are they perpendicular?
EXAMPLE 6
• The two lines in the
graph to the right,
m1 and m2 , are shown.
We need to determine
if the two lines are
perpendicular.
𝑚1 𝑠𝑙𝑜𝑝𝑒: ___________
𝑚2 𝑠𝑙𝑜𝑝𝑒: ___________
YOU TRY 4
• You are given two equations, 𝑦 =
𝑦=−
5
𝑥. Are
11
11
𝑥
5
+ 5 𝑎𝑛𝑑
these two lines perpendicular?
YOU TRY 5
• Are these lines
perpendicular?
Explain
EXAMPLE 7
• A line which contains the point (2,5) is perpendicular to the
line 𝑓 𝑥 = 3𝑥 − 7. Find the equation to this line.
1. State the point-slope formula.
2. Identify m, 𝑥1 , 𝑎𝑛𝑑 𝑦1 . Find the
negative reciprocal of m.
3. Substitute the values into the
4. Simplify into slope-intercept form.
formula. Use the new slope you
found for m.
EXAMPLE 8
• Using the graph to the right,
find the equation of a
parallel line which passes
through the point which is
not on the given line.
1. Find the slope of the given line.
2. Identify m, 𝑥1 , 𝑎𝑛𝑑 𝑦1 . Find the
negative reciprocal of m.
3. Substitute the values into the
4. Simplify into slope-intercept form.
point-slope formula. Use the new
slope you found for m.
YOU TRY 6
• A line which contains the point (0,5) is perpendicular to the
2
line 𝑓 𝑥 = − 𝑥 − 5. Find the equation to this line.
5
1. State the point-slope formula.
2. Identify m, 𝑥1 , 𝑎𝑛𝑑 𝑦1 . Find the
negative reciprocal of m.
3. Substitute the values into the
4. Simplify into slope-intercept form.
formula. Use the new slope you
found for m.
3-2-1
• List three important facts you learned about today.
• List two concepts you already knew.
• List one task you still need to work on.