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7.6 Normal Form of a Linear Equation

7.6 Normal Form of a Linear Equation. 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.

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7.6 Normal Form of a Linear Equation

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  1. 7.6 Normal Form of a Linear Equation 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.

  2. 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 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

  3. 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

  4. 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 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 We want the equation of THIS line

  5. 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 is a “Normal” Line • How do we get the normal line? We want the equation of THIS line

  6. 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. How is Normal Form Different than Standard form? Positive leading coeff. Non ratio coefficients Normal Standard Ratio coefficients Why do you have a problem with the last bullet points? What is the difference between Normal and Standard?

  7. 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

  8. 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. 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

  9. 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. How do we convert from standard form to normal form? This is a cos value Normal Standard This is a sin value Divide everything by , use the opposite sign of the value for C Opposite, y Hypotenuse, p adjacent, x

  10. 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. 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

  11. 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. 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 are not fractions 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 What quadrant is this angle in? For the HW this is acceptable, UNLESS it’s a unit circle value

  12. 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. 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 are not fractions 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? What quadrant is this angle in? If you can identify the angle at this step from sin and cos, you may do so. For the HW this is acceptable, UNLESS it’s a unit circle value

  13. 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. 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 are not fractions 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 What quadrant is this angle in? For the HW this is acceptable, UNLESS it’s a unit circle value

  14. 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 are not fractions What do we divide by?

  15. 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 are not fractions 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? What quadrant is this angle in? For the HW this is acceptable, UNLESS it’s a unit circle value

  16. 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 • Write the equation in normal form. • 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 .

  17. 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 • Write the equation in normal form. • 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 .

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