Parallel & Perpendicular Lines Identifying and graphing March, 2011 Ms. Adler BEGIN
Review: Parallel Lines Parallel lines are two lines that run in the same direction; the two lines lie in the same plane and NEVER intersect. What exactly makes them parallel?
REVIEW: SLOPE Two lines are parallel because they have equal slopes. What is slope?
REVIEW: SLOPE Slope is rise over run. It is the change in y divided by the change in x.
review Using pencil and paper, answer this review question from last week: What is the slope between the points (0,1) and (2,7)? 3 1/4 5 1/5 4
INCORRECT. Remember that slope is Rise over run. TRY THIS PROBLEM AGAIN. HINT
HINT If you are still having trouble, remember that we learned to use this formula: BACK TO REVIEW QUESTION
CORRECT! Great job! You know that slope is equal to (y2-Y1)/(x2-x1)
Perpendicular Lines parallel perpendicular This is new material, so pay close attention! Perpendicular lines are the opposite of parallel lines. While parallel lines never cross, perpendicular lines cross and create a right angle at their intersection.
Take a look at the perpendicular lines shown here. What makes them different from the parallel lines we studied earlier? Pay close attention to the differences in the slope. You can see this in the slope-intercept form. PERPENDICULAR LINES
Reciprocals A reciprocal is the “upside down” version of a number. It is the number that you multiply by to get 1. For example, let’s find the reciprocal of 3: 3 can be re-written as 3/1. Now flip it. It’s reciprocal would be 1/3! 3/1 × 1/3 = 1
Reciprocals, cont’d A reciprocal is the number you multiply by to get 1. Another way to think of this is the number you get when you divide 1 by the original number. Ex) 1=3r (r is for reciprocal) 1/3=r Ex) 1=1/4×r 4=r
mini quiz! What is the reciprocal of 7? -7 1/7 -1/7 14 1/14
INCORRECT. To find the reciprocal, divide 1 by the original number. TRY THIS PROBLEM AGAIN
CORRECT! Great job! You know that 1/7 is the reciprocal of 7 because 1/7×7=1
Slopes Recall that parallel lines have the same slope. Perpendicular lines have slopes which are NEGATIVE reciprocals of one another. Ex) parallel slopes: 2 and 2 perpendicular slopes: 2 and -1/2
mini quiz! Which of the following pairs of slopes could be those of a pair of perpendicular lines? 1/4 and 4 -3 and -3 2 and 1/2 -3/2 and -3/2 5 and -1/5
INCORRECT. Hint: Recall that perpendicular slopes are negative opposites of one another. try this problem again.
INCORRECT. Hint: Remember that perpendicular slopes are reciprocals. Try this problem again!
CORRECT! Good job! You remembered that parallel lines have identical slopes, but perpendicular lines have slopes that are negative reciprocals of one another!
Review of equation forms STANDARD EQUATION Ax+By+C=0 SLOPE-INTERCEPT FORM y=mx+b POINT-SLOPE FORM y-y1=m(x-x1) Example of each form. All of these examples are equal: -4x+y-2=0 y=4x+2 y-2=4(x-0) With pencil and paper, practice manipulating these equations and discovering how they relate before moving on to the next slide.
mini quiz! Find the equation of the line which has slope m=-2 and includes the point (3,1/2). Put the final answer in STANDARD equation form. 2x+y-13/2=0 y=-2x+13/2 y=-2x-13/2 2x+y+13/2=0
INCORRECT. Hint: STANDARD equation form is ax+by+c=o. Slope-intercept form is y=mx+b. TRY THIS PROBLEM AGAIN.
INCORRECT. Hint: When starting this problem, try putting it into point-slope form first. Y-y1=m(x-x1) TRY THIS PROBLEM AGAIN.
CORRECT. GREAT WORK! YOU KNEW TO PLUG THE SLOPE AND X AND Y-VALUES INTO POINT-SLOPE FORM AND MANIPULATE INTO STANDARD EQUATION FORM, AX+BY+C=0.
How to Graph We have learned in class how to graph equations, but let’s review and apply it to our current lesson. First, manipulate the equation into point slope form.
How to Graph When you have the graph in slope intercept form (y=mx+b), plot the y-intercept. From that point on the y-axis, apply the slope to determine the next point. Once you have the two points, you can use the slope to find the next point, and draw the line.
Graphing an Actual Problem For instance, the line y=2x+2 has a y-intercept value of 2, which means that the line crosses the y-axis at point (0,2). Then, since the slope is 2, and slope is equal to the change in x over the change in y (rise over run), and 2 is equal to 2/1, we can “rise” 2 units and “run” 1 unit, which gives us the point (1,4). Now that we have two points, we can draw a straight line through them, giving us our final graph.
Think It Over What have you learned so far? This presentation has addressed the following: Slope Parallel lines Reciprocals Perpendicular lines Equation forms Graphing
Evaluation It is important that you apply the information that you have learned to more complex problems. The quiz that follows will be not be taken for a grade. However, I want you to work out every problem on looseleaf paper and turn your work in to the sub at the end of class.
QUIZ Make sure that you have pencil and paper. Put your name and date on the sheet you turn in. You may not talk to your neighbors. BEGIN QUIZ
QUIZ Questions: 1 2 3 4 5 6 7 8 9 10 11 12 FINISHED
QUIZ QUESTION 1 What is the negative reciprocal of 3/2? 2/3 -2/3 3/2 3 -3/2
INCORRECT. To find the reciprocal, divide 1 by the original number. In this problem, make sure you’re finding the negative reciprocal. TRY THIS PROBLEM AGAIN
CORRECT! Great job! you correctly identified the negative reciprocal of the original number! BACK TO QUIZ
QUIZ QUESTION 2 WHAT IS THE SLOPE OF THE LINE THAT RUNS THROUGH THE POINTS (-1,6) and (3,3)? -4/3 2/9 -1/4 -3/4 9/2
INCORRECT. Remember that slope is Rise over run. try this problem again.
CORRECT! Great job! You know how to calculate slope: (y2-Y1)/(x2-x1) BACK TO QUIZ
QUIZ QUESTION 3 Identify two lines that are parallel: 4x+y+7=0 y=4x-2 y=3x+7 (1/6)x+(1/2)y=0 y=-3x+1 6x+2y-1=0 5x+y-3=0 y=(-1/5)x+4
INCORRECT. Remember that parallel lines have equal slopes, and perpendicular lines have negative reciprocal slopes. (You most likely missed this question because of arithmetic errors, so double check your work.) try this problem again.
CORRECT! Great job! you know the qualities of slopes of parallel lines and correctly identified the pair of parallel lines! BACK TO QUIZ
QUIZ QUESTION 4 Identify the equation in STANDARD equation form of a line with slope -2 that passes through the point (-1,4). 2x+y-2=0 y=-2x+2 2x+y+6=0 -2x+y+2=0
INCORRECT. Make sure that the line is in STANDARD equation form. It is most effective to start with point-slope form and then manipulate. HINT try this problem again.
HINT STANDARD equation form: Ax+by+c=0 Point-slope form: Y-y1=m(x-x1) try this problem again.
CORRECT! Great job! You used the correct equation form and properly manipulated the information! BACK TO QUIZ
QUIZ QUESTION 5 Identify the two slopes which would create perpendicular lines when graphed. 4 and -1/4 2 and 2 -1 and 1 6 and 1/3 1/7 and 7
INCORRECT. Recall that perpendicular lines have slopes which are negative reciprocals of one another. try this problem again.
CORRECT! Great job! You know what a reciprocal is and applied your knowledge to identify the perpendicular slopes! BACK TO QUIZ
QUIZ QUESTION 6 Identify the slope of the line. 0 4 1/4 2 1/2
INCORRECT. Remember that slope is Rise over run. try this problem again.