7.6 n ormal f orm of a l inear e quation by the end of the section students will be able to write...
TRANSCRIPT
7.6 NORMAL FORM OF A LINEAR EQUATIONBy the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
WHAT IS A “NORMAL” LINE
We find an equation of a line using a point and a slope
Distance between two points comes from the Pythagorean theorem
Slope can be found using “Normal” mean perpendicular
Perpendicular slopes are OPPOSITE sign and RECIPROCAL
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
SIDES OF A TRIANGLE
- Greek letter can be pronounced either fee (as in a bank fee) or fi (rhymes with pie)
Thus, the sides of our triangle can be found using the angle and the length of the hypotenuse
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
𝜙𝑦
𝑥
𝑝
WHAT IS A “NORMAL” LINE
is the distance between the origin and the line (distance is measured
PERPENDICULARLY) is the angle made with the
positive x-axis
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
x
y
A
B
M
C
O
p
𝜙
𝜙
𝜙
We want the
equation of THIS line
WHAT IS A “NORMAL” LINE
How do we get the normal line?
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
x
y
A
B
M
C
O
p
𝜙
𝜙
𝜙𝑝 ∙ sin 𝜙
𝑝 ∙cos𝜙
(𝑝 ∙cos𝜙⏟𝑥1
,𝑝 ∙ sin 𝜙⏟𝑦1
)
𝑚=−cos𝜙sin 𝜙
𝑚=sin𝜙cos𝜙
We want the
equation of THIS line
HOW IS NORMAL FORM DIFFERENT THAN STANDARD FORM?
Normal Standard
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
Why do you have a problem with the last bullet points?
What is the difference between Normal and Standard?
Ratio coefficients
Positive leading coeff.Non ratio coefficients
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
EXAMPLE 1: WRITE THE NORMAL FORM OF THE EQUATION GIVEN BY THE LENGTH OF THE NORMAL SEGMENT AND THE ANGLE MADE WITH THE POSITIVE X AXIS
A.
B.
𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0
x
yA
x
yB
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
EXAMPLE 1: WRITE THE NORMAL FORM OF THE EQUATION GIVEN BY THE LENGTH OF THE NORMAL SEGMENT AND THE ANGLE MADE WITH THE POSITIVE X AXIS
C.
D.
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0
x
yC
x
yD
HOW DO WE CONVERT FROM STANDARD FORM TO NORMAL FORM?
Normal Standard
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
Divide everything by , use the opposite sign of the value for C
This is a sin value
This is a
cos value
Opposite,
y
Hypotenuse, p
𝜙𝑦
𝑥
𝑝
adjacent, x
𝜙𝐵
𝐴
√ 𝐴2+𝐵2
WHERE ARE ALL THE ANGLES??
Which quadrant is being described by each?
What is the measure of an angle? Values like are found on the unit circle, we can
give EXACT angles Ratios that we don’t recognize can still be found
using
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION AND IDENTIFY AND GIVEN THE NORMAL FORM
A. What do we divide by?
How do we know which sign to use?opposite of C
??? WHAT??? This is not in quadrant 3??The calculator gives you the PRINCIPAL value, you need to translate that to the appropriate quadrant
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
are not fractions
For the HW this is
acceptable, UNLESS it’s a
unit circle value
What quadrant is this angle in?
𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
What quadrant is this angle in? If you can identify
the angle at this
step from sin and
cos, you may do so.
EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION AND IDENTIFY AND GIVEN THE NORMAL FORM
B. What do we divide by?
How do we know which sign to use?opposite of C
Which of these will give use the quadrant we want?
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
are not fractions
𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0For the HW
this is acceptable,
UNLESS it’s a unit circle
value
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION AND IDENTIFY AND GIVEN THE NORMAL FORM
C. What do we divide by? How do we know which sign to use?opposite of C
This angle is in quadrant 4 so
What if our angle was in quadrant 2?Use as the reference angle for QII
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
are not fractions
For the HW this is
acceptable, UNLESS it’s a
unit circle value
What quadrant is this angle in?
𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION AND IDENTIFY AND GIVEN THE NORMAL FORM
What do we divide by?
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
are not fractions
10√2
=10 √22
=5√2
EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION AND IDENTIFY AND GIVEN THE NORMAL FORM
What do we divide by?
How do we know which sign to use?opposite of C+
Which of these will give use the quadrant we want?
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
are not fractions
10√2
=10 √22
=5√2
What quadrant is this angle in?
𝑥cos𝜙+𝑦 sin 𝜙−𝑝=0For the HW this is acceptable, UNLESS it’s a
unit circle value
SUMMARY
1. Write the equation in normal form.2. Write the standard form of the equation of a
line for which the length of the normal is and makes and angle of with the positive .
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
SUMMARY
1. Write the equation in normal form.
2. Write the standard form of the equation of a line for which the length of the normal is and makes and angle of with the positive .
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.