IDENTIFYING AND GRAPHINGMARCH, 2011

MS. ADLER

Parallel & Perpendicular Lines

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?

SLOPE IS RISE OVER RUN.

IT IS THE CHANGE IN Y DIVIDED BY THE

CHANGE IN X.

REVIEW: SLOPE

USING PENCIL AND PAPER, ANSWER THIS REVIEW QUESTION FROM LAST WEEK:

WHAT IS THE SLOPE BETWEEN THE POINTS (0 ,1) AND (2 ,7 )?

review

1/4 5 1/5 43

REMEMBER THAT SLOPE IS

RISE OVER RUN .

INCORRECT.

HINT

TRY THIS PROBLEM AGAIN.

IF YOU ARE STILL HAVING TROUBLE ,

REMEMBER THAT WE LEARNED TO USE THISFORMULA:

HINT

BACK TO REVIEW

QUESTION

GREAT JOB! YOU KNOW THAT SLOPE IS EQUAL TO

(Y 2-Y 1)/(X 2-X 1)

CORRECT!

Perpendicular LinesThis 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.

parallel

perpendicular

PERPENDICULAR LINES

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.

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

WHAT IS THE RECIPROCAL OF 7?

mini quiz!

-7 -1/7 14 1/141/7

TO FIND THE RECIPROCAL, DIVIDE 1 BY THE ORIGINAL NUMBER.

INCORRECT.

TRY THIS PROBLEM AGAIN

GREAT JOB! YOU KNOW THAT 1 /7 IS THE RECIPROCAL

OF 7 BECAUSE 1 /7×7=1

CORRECT!

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

WHICH OF THE FOLLOWING PAIRS OF SLOPES COULD BE THOSE OF A PAIR OF

PERPENDICULAR LINES?

mini quiz!

1/4 and 4 -3 and -3 2 and 1/2 -3/2 and -3/2 5 and -1/5

HINT:RECALL THAT PERPENDICULAR SLOPES

ARE NEGATIVE OPPOSITES OF ONE ANOTHER.

INCORRECT.

TRY THIS PROBLEM AGAIN.

HINT:REMEMBER THAT PERPENDICULAR

SLOPES ARE RECIPROCALS.

INCORRECT.

TRY THIS PROBLEM AGAIN!

GOOD JOB!YOU REMEMBERED THAT PARALLEL LINES

HAVE IDENTICAL SLOPES, BUT PERPENDICULAR LINES HAVE SLOPES

THAT ARE NEGATIVE RECIPROCALS OF ONE ANOTHER!

CORRECT!

Review of equation forms

STANDARD EQUATIONAx+By+C=0

SLOPE-INTERCEPT FORM

y=mx+b

POINT-SLOPE FORMy-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.

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.

mini quiz!

y=-2x-13/2y=-2x+13/2 2x+y+13/2=02x+y-13/2=0

HINT:STANDARD EQUATION FORM IS

AX+BY+C=O.SLOPE-INTERCEPT FORM IS Y=MX+B.

INCORRECT.

TRY THIS PROBLEM AGAIN.

HINT:WHEN STARTING THIS PROBLEM, TRY PUTTING IT INTO POINT-SLOPE FORM

FIRST.Y-Y 1 =M(X-X 1 )

INCORRECT.

TRY THIS PROBLEM AGAIN.

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.

CORRECT.

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:SlopeParallel linesReciprocalsPerpendicular linesEquation formsGraphing

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.

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.

QUIZ

BEGIN QUIZ

QUIZ

QUESTIONS:

9 10 11 12

654321

87

FINISHED

WHAT IS THE NEGATIVE RECIPROCAL OF 3 /2?

QUIZ QUESTION 1

2/3 -2/3 3 -3/23/2

TO FIND THE RECIPROCAL, DIVIDE 1 BY THE ORIGINAL NUMBER.

IN THIS PROBLEM, MAKE SURE YOU’RE FINDING THE NEGATIVE RECIPROCAL.

INCORRECT.

TRY THIS PROBLEM AGAIN

GREAT JOB! YOU CORRECTLY IDENTIFIED THE

NEGATIVE RECIPROCAL OF THE ORIGINAL NUMBER!

CORRECT!

BACK TO QUIZ

WHAT IS THE SLOPE OF THE LINE THAT RUNS THROUGH THE POINTS ( -1 ,6 ) AND (3 ,3)?

QUIZ QUESTION 2

-4/3 2/9 -1/4 -3/4 9/2

REMEMBER THAT SLOPE IS

RISE OVER RUN .

INCORRECT.

TRY THIS PROBLEM AGAIN.

GREAT JOB! YOU KNOW HOW TO CALCULATE SLOPE:

(Y 2-Y 1)/(X 2-X 1)

CORRECT!

BACK TO QUIZ

IDENTIFY TWO LINES THAT ARE PARALLEL:

QUIZ QUESTION 3

4x+y+7=0y=4x-2

y=3x+7(1/6)x+(1/2)y=0

5x+y-3=0y=(-1/5)x+4

y=-3x+16x+2y-1=0

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.)

INCORRECT.

TRY THIS PROBLEM AGAIN.

GREAT JOB!YOU KNOW THE QUALITIES OF SLOPES OF

PARALLEL LINES AND CORRECTLY IDENTIFIED THE PAIR OF PARALLEL

LINES!

CORRECT!

BACK TO QUIZ

IDENTIFY THE EQUATION IN STANDARD EQUATION FORM OF A LINE WITH SLOPE -

2 THAT PASSES THROUGH THE POINT ( -1 ,4 ) .

QUIZ QUESTION 4

2x+y-2=0

2x+y+6=0 -2x+y+2=0

y=-2x+2

MAKE SURE THAT THE LINE IS IN STANDARD EQUATION FORM.

IT IS MOST EFFECTIVE TO START WITH POINT-SLOPE FORM AND THEN

MANIPULATE.

INCORRECT.

TRY THIS PROBLEM AGAIN.

HINT

STANDARD EQUATION FORM:AX+BY+C=0

POINT-SLOPE FORM:Y-Y 1 =M(X-X 1 )

HINT

TRY THIS PROBLEM AGAIN.

GREAT JOB!YOU USED THE CORRECT EQUATION FORM

AND PROPERLY MANIPULATED THE INFORMATION!

CORRECT!

BACK TO QUIZ

IDENTIFY THE TWO SLOPES WHICH WOULD CREATE PERPENDICULAR LINES

WHEN GRAPHED.

QUIZ QUESTION 5

6 and 1/3

4 and -1/4 2 and 2 -1 and 1

1/7 and 7

RECALL THAT PERPENDICULAR LINES

HAVE SLOPES WHICH ARE NEGATIVE RECIPROCALS

OF ONE ANOTHER.

INCORRECT.

TRY THIS PROBLEM AGAIN.

GREAT JOB!YOU KNOW WHAT A RECIPROCAL IS AND

APPLIED YOUR KNOWLEDGE TO IDENTIFY THE PERPENDICULAR SLOPES!

CORRECT!

BACK TO QUIZ

IDENTIFY THE SLOPE OF THE LINE.

QUIZ QUESTION 6

0

1/2

1/4 2

4

REMEMBER THAT SLOPE IS

RISE OVER RUN .

INCORRECT.

TRY THIS PROBLEM AGAIN.

GREAT JOB! YOU KNOW HOW TO CALCULATE SLOPE:

(Y 2-Y 1)/(X 2-X 1)

CORRECT!

BACK TO QUIZ

IDENTIFY THE EQUATION OF THIS LINE.

QUIZ QUESTION 7

y=(-7/9)x+(1/3)

y=(-7/9)x+(39/9)

y=(-9/7)x-(13/7)

y=(-7/9)x-(1/3)

FIRST, USE THE TWO POINTS TO FIND THE SLOPE.

NEXT, USE THE SLOPE AND ONE OF THE POINTS TO PLUG INTO POINT-SLOPE FORM.

FINALLY, MANIPULATE THE EQUATION INTO ITS FINAL FORM.

POINT-SLOPE FORM:Y-Y 1 =M(X-X 1 )

INCORRECT.

TRY THIS PROBLEM AGAIN.

GREAT JOB!YOU CORRECTLY FOUND THE SLOPE

BETWEEN TWO POINTS ON THE LINE, THEN USED POINT-SLOPE FORM TO FIND THE CORRECT EQUATION OF THE LINE!

CORRECT!

BACK TO QUIZ

IDENTIFY THE PAIR OF POINTS THAT WOULD PRODUCE THE LINE Y=3X+1

QUIZ QUESTION 8

(0,1) and (-1,4)

(0,-1) and (1,2) (0,1) and (2,4)(1,4) and (-1,-

2)

ATTEMPT THIS PROBLEM BY TESTING THE PAIRS OF POINTS.

USE THE POINTS TO FIND THE SLOPE, THEN PLUG THE SLOPE AND ONE OF THE

POINTS INTO POINT SLOPE FORM.FINALLY, MANIPULATE TO THE FINAL

EQUATION, Y=3X+1

INCORRECT.

TRY THIS PROBLEM AGAIN.

GREAT JOB!YOU CORRECTLY IDENTIFIED THE PAIR OF

POINTS WHICH PRODUCE THE LINE Y=3X+1

CORRECT!

BACK TO QUIZ

IDENTIFY THE LINE THAT PASSES THROUGH THE POINT (O, -2) AND IS

PERPENDICULAR TO THE LINE Y=(-1 /6 )X+3

QUIZ QUESTION 9

y=6x+2

y=(-1/6)x-2 y=6x-2

y=6x+12

IT IS IMPORTANT TO USE POINT-SLOPE FORM

Y-Y 1=M(X-X 1)TO COMPLETE THIS

PROBLEM.

RECALL THAT PERPENDICULAR LINES

HAVE SLOPES WHICH ARE NEGATIVE RECIPROCALS.

INCORRECT.

TRY THIS PROBLEM AGAIN.

GREAT JOB!YOU CORRECTLY IDENTIFIED THE LINE THAT RUNS THROUGH THE POINT (0 , -2)

AND HAS SLOPE 6 .

CORRECT!

BACK TO QUIZ

IDENTIFY THE PAIR OF LINES THAT ARE PERPENDICULAR.

QUIZ QUESTION 10

y=-3x+13x+6y-1=0

y=0.5x-2-x+2y+4=0

y=(-1/3)x+126x+2y-12=0

y=-2x+3x-2y+3=0

THE BEST WAY TO APPROACH THIS PROBLEM IS TO MANIPULATE FROM ONE FORM OF THE EQUATION TO ANOTHER.

WATCH OUT FOR ANY SIMPLE MISTAKES IN ARITHMETIC!

RECALL THAT PERPENDICULAR LINES HAVE SLOPES WHICH ARE NEGATIVE

RECIPROCALS!

INCORRECT.

TRY THIS PROBLEM AGAIN.

GREAT JOB!YOU CORRECTLY IDENTIFIED THE TWO EQUATIONS THAT HAVE SLOPES WHICH

ARE NEGATIVE RECIPROCALS OF ONE ANOTHER.

CORRECT!

BACK TO QUIZ

IDENTIFY THE Y- INTERCEPT OF THE LINE THAT PASSES THROUGH THE POINTS (7 , -

2) AND (4 ,19) .

QUIZ QUESTION 11

(0,49) (0,-1) (O,-7) (0,47)

THE GOAL OF THIS PROBLEM IS TO FIND THE Y- INTERCEPT.

FIRST, USE THE POINTS TO FIND SLOPE.

THEN, PLUG THE SLOPE AND ONE OF THE POINTS INTO POINT-SLOPE FORM AND MANIPULATE INTO SLOPE -INTERCEPT

FORM. THIS WILL GIVE YOU THE INTERCEPT.

INCORRECT.

TRY THIS PROBLEM AGAIN.

GREAT JOB!YOU CORRECTLY USED THE TWO POINTS TO FIND THE Y- INTERCEPT OF THE LINE

BETWEEN THE POINTS!

CORRECT!

BACK TO QUIZ

FIND THE EQUATION OF A LINE PERPENDICULAR TO THIS LINE

THAT RUNS THROUGH THE POINT (2 ,2) .

QUIZ QUESTION 12

y=(-3/4)x+(7/2)

y=(3/4)x+(1/2) y=(3/4)x-(1/2)

y=(-4/3)x-(2/3)

THE EASIEST WAY TO DO THIS PROBLEM IS TO FIRST USE THE TWO POINTS ON THE

GRAPH TO FIND THE SLOPE OF THE ORIGINAL LINE.

THEN, TO FIND THE PERPENDICULAR LINE, USE POINT-SLOPE FORM.

USE THE POINT (2 ,2 ) AND THE NEGATIVE RECIPROCAL OF THE ORIGINAL SLOPE FOR THE SLOPE OF THE PERPENDICULAR LINE.

INCORRECT.

TRY THIS PROBLEM AGAIN.

GREAT JOB!YOU CORRECTLY USED THE TWO POINTS TO FIND THE SLOPE OF THE ORIGINAL

LINE, THEN USED WHAT YOU KNOW ABOUT PERPENDICULAR LINES TO FIND THE EQUATION OF THE PERPENDICULAR

LINE.

CORRECT!

BACK TO QUIZ

YOU ARE DONE WITH THE QUIZ!YOU SHOULD NOW BE ABLE TO

(OBJECTIVE) WITHOUT TOO MUCH TROUBLE!

PLEASE MAKE SURE THAT YOU HAVE SHOWN ALL OF YOUR WORK ON THE

PAPER THAT YOU WILL TURN IN TO THE SUB.

IF YOU NEED TO REVIEW QUIZ PROBLEMS TO SHOW WORK, USE THE BUTTON BELOW.

CONGRATS

BACK TO QUIZ!

What have you learned?

You also learned that parallel lines have identical slopes, and that perpendicular lines have negative reciprocal slopes.

What have you learned?

You also learned how to graph lines by plotting the y-intercept, and using slope to find other points on the

line.

What have you learned?

Standard Equation FormAx+By=C

Point-Slope Formy-y1=m(x-x1)

Slope-Intercept Formy=mx+b

You now know how to identify and graph

parallel and perpendicular lines!

PLEASE USE THE REST OF THE CLASS PERIOD REVIEWING CONCEPTS OR

STAYING QUIET OTHERWISE.

All Done!

Have a great day!

HIT THE ESC BUTTON TO

END PRESENTATIO

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