Educational videos for Students in k-12 | Lumos Learning

Educational videos for Students in k-12




Transcript
00:0-1 Here's part two of all grown in math in 60
00:03 minutes or less . So hopefully you've watched Part one
00:05 . The algebra unit , um , got about 25
00:08 minutes left to get through the rest of grade nine
00:10 . Um , this section will be shorter than the
00:12 algebra section in the last section . Geometry over the
00:14 shortest section . Um , so this is part two
00:16 linear relations . Once again , I'm going to go
00:18 quickly through the concepts in just a few examples of
00:21 things . If you want more in depth explanations ,
00:23 go to Jensen math dot c A . There's videos
00:25 for each section there . Okay , so here we
00:27 go linear relations . So first thing you would have
00:30 looked at probably a scatter plot . Scatter plots is
00:33 a type of graph that shows the correlation between two
00:36 quantitative variables . So , for example , we have
00:38 two variables here year and movie attendance and millions .
00:42 And this is this is Canadian data . We have
00:44 two variables . It's important to understand difference between the
00:47 independent , dependent variable . If we have a table
00:49 of data and horizontal rows like this X , our
00:52 independent variable is usually the first row , and why
00:55 are dependent ? Variable is usually the bottom row .
00:58 It's important to know the difference between the two variables
01:00 , because that depends where that determines where we put
01:03 the variables on our graph . The independent variable goes
01:06 on the horizontal axis dependent on the vertical axis ,
01:10 Um , and also , um , so independent variable
01:13 is year in this one dependent variable is movie attendance
01:17 and another way to tell which variable is , which
01:21 is that . Attendance depends on the year , so
01:24 that makes our dependent variable . The year causes a
01:27 change in the attendance is another way to think about
01:29 it , so it makes it our independent variable .
01:31 If you're given data in vertical columns , remember that
01:35 X is usually given to us on the left ,
01:37 so independent on the left Y on the right .
01:39 And when we graph it , make sure you remember
01:41 that we are going to put our independent variable X
01:45 on the horizontal axis . So here's our year ,
01:47 and our dependent variable movie attendance is going to go
01:53 on a vertical axis . Dependent always goes on the
01:55 Y axis . The vertical axis independent always goes on
01:58 the x axis , the horizontal axis , and make
02:00 sure when you're choosing your scales , make sure you
02:03 choose a scale that goes up by an even amount
02:05 each time . So attendance notice that I should have
02:07 a break in here because I skipped some values .
02:11 But after that skipped values between zero and 80 it
02:13 goes up by a constant amount . Each time it
02:15 goes up by five . Each time , it's important
02:18 that your scale goes up by an even amount .
02:20 Each time and years , I started 1994 went up
02:22 by an even number of years , each time by
02:24 one year . Um , for each unit , Uh
02:28 , and why did I choose 80 to start out
02:29 for attendance while I needed my lowest value is 83.8
02:33 . My highest value is 1 25.6 . So I
02:36 needed a scale that goes at least between those ranges
02:39 . So I started at 80 and then I went
02:42 all the way up to 1 . 45 because I'm
02:44 going to be making some estimations later on this graph
02:47 and years , I started 94 I went up to
02:50 2000 and six because later we're going to making an
02:53 estimation for a year outside of our range . So
02:56 if you plot all of your points on this graph
02:59 . So this is going to represent a point on
03:00 the graph . This is going to represent a point
03:02 on the graph and so on . So in 1994
03:04 movie attendance was 83.8 million . So in 1994 83.8
03:09 would be roughly right about here . I put a
03:11 dot at that point right there , 1995 movie tens
03:15 was 87.3 million . So put a dot at about
03:19 87.3 , which is right about there and so on
03:22 . If we put a point for all of our
03:24 data values and then draw a line that shows the
03:26 trend in the data , you would have something that
03:28 looks like this have my break in my graph day
03:33 , and it would fall roughly a linear trend .
03:35 When you draw a trendline , make sure you follow
03:37 two rules . Make sure you're lying goes through as
03:39 many points as possible , and the points that it
03:42 doesn't go through like here , here , here ,
03:45 here , here , here are evenly distributed above and
03:51 below the line . So there's three above the line
03:53 three below the line . But make sure dryer line
03:55 that goes through as many as possible . The ones
03:57 that doesn't go through evenly distribute above and below ,
03:59 and you'll have a good trend line . The trend
04:01 line does not have to start at the origin .
04:03 It can start anywhere along the Y axis anywhere along
04:05 the X axis , just as long as it follows
04:07 the general trend in the data . Now we can
04:10 use that trend line to make estimations . But first
04:12 , let's describe the correlation between year and attendance .
04:16 I want to describe it in two ways . First
04:19 of all , since our line is going up to
04:22 the right , we say it's a positive correlation .
04:24 And since all the dots are fairly close to the
04:27 line , we say it's a strong correlation . So
04:29 it's a strong , positive correlation we have here .
04:31 So up to the right is positive and the dots
04:34 are close to line . So strong . So it's
04:35 a strong , positive linear correlation . And what that
04:38 tells us is that as the year increases , the
04:41 movie attendance is also increasing and now notice from our
04:45 data table there's no data for 2000 and one ,
04:47 so we want to make an estimation . I want
04:51 to make an estimation for the movie tenants using our
04:53 line of best fit . So in 2000 and one
04:56 , we go up to see where just 2000 and
04:57 1 m line of best fit it meets it right
05:00 here . We look across on ry access . That's
05:03 about right here . That's about 121 million would be
05:06 our estimation for the year 2000 and one . So
05:10 you make sure you're using your line of best fit
05:12 properly here . So 121 million And then we also
05:17 have to know , was this estimation something we call
05:20 an interpolation , or was it an extrapolation ? So
05:23 what's the difference between the two ? If we look
05:25 at the data were originally given , were given data
05:28 between 94 2000 and three ? That's the range of
05:31 our data . It goes all the way from 1994
05:34 to 2000 and three . If we make an estimation
05:37 for a year between our lowest and our highest year
05:40 , we call that an interpolation . If we make
05:42 an estimation that's outside of our range to either less
05:45 than 94 or bigger than 2000 and three , we
05:47 call that an extrapolation So we made an estimation for
05:50 2000 and one that's between 94 2000 and three .
05:53 So we call that an interpretation . Let's make an
05:59 estimation for 2000 and five . So if we look
06:02 at our line of best fit so 2000 and five
06:04 this year see where our line of best fit says
06:07 the attendance would be for 2000 and five . We
06:09 put a point on the line of Best fit at
06:11 2000 and five and then look across to the y
06:13 axis . It's about right here about 1 43 or
06:16 1 44 . I'm gonna say my estimation is about
06:20 143 . Remember , we're in millions here . Did
06:24 we use interpolation or extrapolation ? While 2000 and five
06:27 is outside of the range of data were originally given
06:29 , it's bigger than our highest value 2000 and three
06:31 . So we say that that is in fact ,
06:33 an extrapolation . Yeah , okay , next thing you
06:37 would have done and the distance time graphs . How
06:39 do we analyze the distance ? Time graphs describe motion
06:41 Well , here , have a distance . Time graphs
06:43 . If we read here , it says described the
06:45 following graph that represents a person's distance from home over
06:47 a period of time . So we have a period
06:50 of time seven minutes and the graph tells us the
06:53 distance the person is in meters from their house ,
06:56 Um , during that period of time . So at
06:58 the very beginning of the graph , when zero minutes
07:01 have elapsed there 0 m from the house , which
07:04 means they're at their house . But then two minutes
07:06 later will notice they are 100 m from their house
07:09 . So this rising line up to the right tells
07:13 us the person is moving away from their house ,
07:16 right . If they were 0 m away from their
07:18 house and now they're 100 m away from their house
07:19 , they must be moving away from their house .
07:21 So this rising line up to the right that tells
07:24 us movement away from the center or , in this
07:27 case , away from the person's house . So any
07:29 line moving up to the right is movement away .
07:31 And since it's a straight line with no curves ,
07:33 we say they're moving away at a constant rate ,
07:36 and I'm going to say this is a slow movement
07:38 compared to the next line segment because It takes two
07:42 minutes to travel 100 m in this line segment ,
07:45 but this line segment here from B to C two
07:48 minutes later , they've traveled 1 to 300 m .
07:52 They traveled 300 m in the same time it took
07:55 them to travel 100 m . So B C is
07:57 a faster movement . You can tell there between fast
07:59 and slow , because faster movement as a steeper line
08:02 , slower movement is a less steep line . So
08:05 let's describe the motion here of line segment A .
08:08 Be so up to the right , which means away
08:10 from the House . And we're going to say at
08:12 a slow , steady pace . Steady pace means that
08:16 the speed is constant . They're not changing . Speeds
08:18 are not accelerating or decelerating . NBC . The line
08:21 is still going up to the right , but it's
08:23 steeper . They're traveling a further distance in the same
08:25 amount of times they're moving faster , so we'll say
08:28 they're moving at a fast , steady pace away from
08:31 their house , so away from home at a fast
08:33 , steady pace . How about C D line segment
08:36 CD at the beginning , So at four minutes ,
08:40 there are 400 m away from their house and then
08:42 two minutes later , at six minutes , there's still
08:44 400 m away from their house . And why didn't
08:46 go up or down at all between those two ,
08:48 So the person must not have moved at all .
08:51 So for two minutes they didn't move . So any
08:53 horizontal line represents no movement . So in this case
08:58 , sorry . In this case , C d .
09:00 No movement D E is the only line segment going
09:03 down to the right . So they were 400 m
09:06 away from their house . But now they're only at
09:07 the end of the line segment there , only 200
09:09 m away from their house , they must moving towards
09:11 their house . Any line segment going down to the
09:13 right represents movement towards the sensor , in this case
09:16 towards the person's house . And it's a steep line
09:19 . So they're moving . They're moving fast towards their
09:21 house . Um , and there's no curves , so
09:24 it's at a steady rate , a constant speed .
09:27 So line segment D . E . We would say
09:30 the person is moving at a fast , steady pace
09:33 toward home and in distance time graphs . Be careful
09:37 to look for curves . If you have a curved
09:39 line , you have either an acceleration or deceleration .
09:42 You have an acceleration if the line starts off not
09:44 very steep than gets steeper and steeper and steeper and
09:47 steeper and steeper . That means their rate of movement
09:48 is increasing , so that be an acceleration . If
09:51 we have a line that starts off very steep and
09:54 then gets less deep and less deep and less deep
09:56 and less deep as we move from left to right
09:58 , that represents a deceleration . So be careful of
10:02 that when we're doing distance time graphs . Okay ,
10:05 the main section for linear relations would have looked at
10:07 Is the y equals MX plus B section . Basically
10:09 , if we have two variables that form a perfect
10:12 linear relationship , that happens when , um , as
10:15 one variable increases , Um , we have another variable
10:19 that increases at a constant rate , so one variable
10:22 is a constant multiple of the other variable . So
10:26 this equation , you would have learned all these variables
10:28 mean specifically would have learned the M represents the slope
10:31 or rate of change actually have that written right below
10:35 rate of change or called the constant variation . B
10:39 represents the Y intercept , graphically speaking or the initial
10:41 value , and we have Y and X right because
10:44 we have our two variables are have a relationship between
10:47 the two . Why being are dependent ? You know
10:49 that X being are independent . So if we have
10:51 two variables to form a linear relationship with one variable
10:53 to constant multiple , the other variable Um , we
10:57 say M is the rate of change between those two
10:59 variables . And if we have a table of data
11:01 , we can calculate them the rate of change by
11:03 figuring out what is why changing by divide that by
11:06 what X is changing by . If we have a
11:07 graph , we can figure out the rate of change
11:09 by counting our rise and dividing it by our run
11:12 and in an equation , you'll refer to them as
11:15 the slope a rate of change of constant variation .
11:18 So what we're gonna do in this section is be
11:21 able to work between the table graph and equation forms
11:24 of linear relationship . Before we do that , let
11:27 me just quickly show you what I'm talking about here
11:29 . So if we have a relationship here between distance
11:34 and time , so the distance traveled by bus varies
11:37 directly with time . So we have distance and times
11:39 are two variables distance depends on time . So distances
11:43 . Why time of exit , says the bus travels
11:45 240 kilometers in three hours or distance and our time
11:48 . Once again it travels 243 hours . But if
11:52 we want to figure out the rate of change or
11:54 the slope or constant variation where we want to call
11:56 it at the rate of change , we want to
11:57 figure out what is the distance changing by for every
12:00 one hour . So to calculate that we call it
12:02 our M . Our rate of change is m .
12:03 We do our change in Y , which is 240
12:07 in this case , kilometers and we divided by our
12:10 change in X , our independent variable which is times
12:12 of three hours . If we don't do this division
12:16 , it'll tell us how many kilometers the bus travels
12:19 for every one hour . And that's a rate of
12:20 change . We need to know what is why change
12:22 changing by everyone . Unit increase in x 024 divided
12:25 by three that gives us 80 kilometers per hour .
12:29 So the buses going 80 kilometers per hour . That's
12:31 RM . That's our rate of change . Another thing
12:34 you would have looked at before . I get into
12:37 more specific things about linear relations , as you would
12:39 have looked at between direct variation and partial variation relationships
12:43 . So there's two types of linear relationships . It
12:45 could be a direct variation relationship , where the initial
12:47 value is 00 for the relationship between the variables ,
12:53 meaning graphically . The line that represents the relationship between
12:57 two variables would pass through the origin in the equation
13:00 . You don't see anything added after the variable X
13:03 . There's nothing added off at the end Here .
13:05 You can write a plus zero if you want to
13:07 , but that's a direct variation relationship . Initial value
13:10 zero passes through the origin . The equation has no
13:13 nothing added . At the end , you don't see
13:14 a plus B because the B value is zero .
13:16 The Y intercept zero partial variation relationship has an initial
13:20 value that is not zero when x zero . Why
13:23 is something else in this case , I gave an
13:25 example of why being one when x zero , that
13:27 means graphically speaking , the line is not going to
13:29 pass through the origin . In this case , it's
13:30 going to pass through one in the equation . You're
13:33 going to see that Y intercept That initial value added
13:35 after the X and we call that a partial variation
13:38 relationship . So what we're gonna do now , we're
13:42 going to be able to given a linear relationship between
13:44 two variables . We're gonna be able to represent that
13:46 relationship between the variables in a table of values .
13:49 Um , we're going to be able to represent the
13:51 relationship using an equation , and we're going to represent
13:54 the relationship between the two variables using a graph .
13:57 So let me quickly show you those three things ,
13:59 and then we'll work at moving between those three things
14:01 . So costs to electoral work varies directly with time
14:03 . So it tells us it's a direct variation ,
14:06 and it's the relationship between cost and time . In
14:09 this case , cost depends on times that cost is
14:11 dependent . Time is independent . They charge $25 per
14:15 hour . It tells us how much it costs for
14:16 every one hour , so we know our rate of
14:18 change . We know our M value is 25 in
14:22 this case through electric work . If we want to
14:24 show this relationship in a table in a table ,
14:27 we put X on the left . Why on the
14:28 right and remember access time so they could do work
14:32 for 0123 And we can make this table as big
14:35 as we want to stop there . Those are ours
14:37 now . How much would it cost for each amount
14:39 of these hours For them to do work Well ,
14:40 if they don't work , it's not gonna cost anything
14:43 . If they work for one hour . Since they
14:45 charge $25 per hour , it's gonna cost $25 .
14:48 They work for two hours . It's going to cost
14:51 $50 right ? It's 25 bucks per hour . So
14:53 two times 25 is 50 . If they work for
14:55 three hours three times 25 75 it would cost $75
14:58 . And so on . This shows the relationship between
15:02 cost and time for this electrical company . We could
15:06 represent all these numbers who want equation . We can
15:08 make an equation that shows the relationship between X and
15:11 Y and indirect variation relationship . The equation is always
15:15 y equals MX . Any linear relationship is why equals
15:18 MX plus B , but you'll see why we don't
15:19 need to plus B for this equation , because the
15:21 initial value for this equation the cost from X zero
15:24 is zero . So we don't need the plus B
15:26 because B is zero in this case , the initial
15:28 value . The Y intercept is zero . So the
15:31 relationship between cost remember wise cost , excess time cost
15:36 is equal to $25 . That's a rate of change
15:39 times the amount of time that you work for .
15:41 And this equation shows us the relation between Y and
15:43 X for any value of X , right . If
15:45 we wanted to figure out how much it costs from
15:47 you three hours of work , you plug in three
15:49 into our equation for X £25.3 or 75 will get
15:52 cost equals $75 . We don't know how much it
15:55 costs to do 10 hours of work . You would
15:57 plug in 10 forex and we'd get cost equals $250
16:01 and so on . Now , graphically speaking , if
16:03 we were to plot these points on our graph ,
16:05 let's see what it would look like . Zero hours
16:08 of work cost $0.1 hour of work cost $25.2 hours
16:12 Costs $50 . 3 hours costs $75 . Notice how
16:15 this is going up by a constant amount each time
16:19 . That's what makes this a linear relationship , because
16:21 the rate of change is constant . And then we
16:23 would connect these points with a line to show the
16:26 relationship . And there's the graph of the relationship between
16:30 cost and time for these two variables , so we
16:34 could represent the relationship between costs and time in three
16:36 different ways . Here , in a graph using an
16:39 equation or in a table of values . All three
16:42 of those ways show the relationship between cost and time
16:44 for this electric company and notice . This is a
16:48 direct creation because it starts at 00 and goes up
16:51 by a constant amount each time . That makes it
16:53 a constant for sure . That makes it a direct
16:55 variation . Also , we have a plus zero here
16:57 that also shows us it's a direct variation , and
16:59 from a table we know it's a direct variation because
17:01 the initial value is zero Okay , let's look at
17:06 what if I gave you just a table ? We
17:08 don't know what X and y are here . We
17:10 know exes are independent wise , are different . We
17:12 don't know exactly what they are . We have a
17:13 table of values here What if I wanted to come
17:15 up in the equation to represent the relationship between X
17:18 and Y here ? What I'm going to need for
17:20 any linear relationship equation . It's always going to be
17:24 the form y equals MX plus B . I'm always
17:26 going to need to determine what is the value of
17:29 and what is the value of B to be able
17:31 to write the equation of the relationship between X and
17:33 Y . So what I'm going to do first is
17:36 I'm going to figure out the value of B .
17:38 That's the easiest thing to figure out the value of
17:40 B . That is our initial value or you'll hear
17:45 it referred to graphically as the Y intercept the Y
17:47 intercept . So that means , um , where does
17:52 it cross the Y axis ? So when x zero
17:55 , What is why ? So I'm just gonna write
17:57 that here When X equals zero . Why equals what
18:04 ? And we can get that from our table when
18:06 X zero y is four . So are y intercept
18:09 our B value . Our initial value B is equal
18:14 to four . So you just look in your table
18:16 when x zero . What's why that's your initial value
18:19 . That's your y intercept . If we were to
18:20 plot this 00.4 that would be right on the Y
18:22 Axis , which makes our line across the UAE except
18:25 four . So that's our B value . Mm .
18:27 How do we figure out m from the table of
18:30 values ? Remember , I showed you equations before your
18:33 rate of change , you're going to have to figure
18:34 out what is why changing by and divide that by
18:36 what X is changing by and that will tell you
18:39 for everyone . Unit increase in X . What is
18:41 ? Why changing by So to figure out the change
18:44 in y an organized way to do it is to
18:46 do your second Y value minus your first y value
18:49 . And divide that by two . Figure your change
18:51 in X to your second X value minus your first
18:54 X value . So y tu minus y one over
18:56 x two minus x one That will tell your change
18:58 and y divided by your change in X . Now
19:00 you have to take two points to be your 1st
19:02 and 2nd points . I'm just gonna choose the first
19:04 two points for it to be simple here . So
19:06 this is my 1st 20.4 So That's my first X
19:09 value . My first y value . I'll choose .
19:11 This is my second point . This is my second
19:13 X value , my second y value . If I
19:15 plug those values into my equation , I have eight
19:18 minus 4/2 minus zero , and I get 4/2 ,
19:23 which is two . So my M values to my
19:26 B values for I can write my equation by plugging
19:30 in m and B into ecos and exports . B
19:33 y equals two X plus four . This equation shows
19:37 the relationship between X and Y and notice this equation
19:41 works , right ? If I plugged in eight for
19:44 X into this equation , what should I get my
19:46 answer for ? Why to be It should be 22
19:49 times 8 to 16 plus four is 20 . It
19:51 shows the relationship between the two variables . Now ,
19:53 I just want to point out you didn't have to
19:55 choose . This is your first point . This is
19:57 your second point when you plugged into y two months
19:59 . Why ? Whenever x two , minus x one
20:02 , let's say we could have chosen any two points
20:04 . I could have chosen this one to my second
20:07 point x two y two . If I plugged into
20:09 my equation here , I would get 16 minus 4/6
20:13 , minus zero . And that would give me 12/6
20:15 , which is still the same answer of two .
20:19 So that's how to go from a table to the
20:21 equation . What if I give you an equation ?
20:23 I want you to draw the graph . Well ,
20:28 to be able to do that , we know two
20:30 things from our equation here . Em . We call
20:34 that graphically speaking . We call that our slope M
20:38 Equal Slope , which in this case is three and
20:42 any whole number is over one . And we're talking
20:44 about Slope . Remember , The top is our change
20:46 and why I'm going to call that rise . And
20:50 the bottom is our change in X . We call
20:52 that run , but where do I start from ?
20:56 Don't forget this year , that is our Y intercept
20:59 . That's our initial values . We know our graph
21:01 starts right here on the Y axis at negative two
21:04 . And then from there , I need to use
21:06 my slope of 3/1 plot more points . So that
21:09 means rise three . Run one . Remember , A
21:11 positive rise means up a positive run means right so
21:15 up . Three right one and then continue up .
21:17 Three . Right , one up , three , right
21:20 , one and two plot points on the other side
21:22 of the line . Do the opposite of up three
21:24 . Right home , which is down three . Left
21:25 one and we get more points on the same line
21:29 . Connect all the points should be on the line
21:32 if you've done it properly . And that's how to
21:34 go from the equations of the graph . Plot your
21:36 Y intercept to use your B value to plot an
21:38 intercept and use your slope . Use the rise and
21:41 the run . The numerator is the rise . And
21:42 then there's the run for rise . Counting up is
21:45 positive . Counting to the right is positive for run
21:48 . If we had a negative rise , we would
21:50 count down . If we had a negative run ,
21:52 we would count left and so on . What if
21:55 we wanted to go from a graph to the equation
21:57 ? Well , there's a couple ways we can do
21:59 this . We know our equation is why cause MX
22:02 plus B and we know we're going to need to
22:03 figure out the values of M and B , so
22:06 B is easier one to find , right ? The
22:08 is just the Y intercept . It tells us when
22:10 . X zero . What's why ? So where's across
22:12 the Y axis that crosses the Y axis right there
22:13 at two . Might be value is , too .
22:16 But to calculate the rate of change , we have
22:17 a couple options . We could choose to lattice points
22:20 , so points in the corners of the boxes and
22:22 we could count are rising , are run right M
22:25 equals rise over . Run Graphically speaking . So count
22:30 your rise . To get from one point to the
22:31 other , we would have to count up 1234 Up
22:36 is positive . So positive forces arise . And then
22:39 we'd have to go to the right one to going
22:41 to the right as a positive run . So four
22:43 or two , which is two . That's fine .
22:46 But what if you wanted to go from instead of
22:51 going from here to here ? What if we wanted
22:52 to go from here to here ? We would have
22:54 to count down 1234 So that would be negative for
22:58 as our rise . And then we would have to
23:00 go left to which may get negative . Two is
23:02 our run notice . Same answer negatively by negative is
23:05 positive . It doesn't matter whether at which point you
23:08 go from , you'll get the same answer either way
23:11 . Also , you could use the method we used
23:13 before using our change in y over change in X
23:19 , which is y tu minus y one over x
23:22 two minus X one that works perfectly fine as well
23:26 , changing wise the same as rise change . Next
23:28 is the same as Run . We could just choose
23:29 our two points here . Let's say this is our
23:31 first point . So that's our X one y one
23:33 . Here's our second point x two y two .
23:36 Plug into our equation . We get six minus 2/2
23:40 minus zero . Sorry , six minus 2/2 minus cereal
23:46 . That gives us 4/2 , which is two .
23:49 So either way you get the answer of two for
23:51 M . Plug into your equation . Y equals MX
23:53 plus B , and we get y equals two x
23:55 plus to this equation shows the relation between X and
23:59 Y . Um , for this graph , uh ,
24:04 we're going to do some equations of lines algebraic li
24:07 quickly , and then we'll be done this section .
24:09 So before we do that , you have to remember
24:10 parallel lines lines in the same plane that never meet
24:13 . Parallel lines have equivalent slopes , so you have
24:16 to remember that parallel lines have equivalent slopes . Perpendicular
24:22 lines , lines that meet at 90 degrees have slopes
24:24 that are negative , reciprocal of each other . So
24:27 you have to remember that negative reciprocal roles of each
24:31 other , and I'll explain that more in a second
24:34 . So determine the equation of a line algebraic given
24:37 different information I could give you Slope member Slope is
24:41 M and one point it passes through . So I
24:44 didn't give you the Y intercept . Find the equation
24:46 of a line you always need to things you always
24:49 need . Slope y intercept . I gave you slope
24:52 . I didn't give you y intercept you can use
24:56 at this point . Every point has an X and
24:59 A Y in the equation of a line will satisfy
25:01 X and Y for any point that's on the line
25:03 . So what we can do is use this information
25:06 m x and Y plug that in for M X
25:08 and Y in the equation and then solve for B
25:10 . So if we do that , if I plug
25:12 in five for Y negative number two for M negative
25:15 four for X and Y equals MX plus B .
25:18 So I'll have something . Looks like this . Negative
25:20 1/2 negative four plus b solve for B . I
25:24 will then know the B value . I can write
25:26 my equation by plugging in for M and B ,
25:28 so I'm just gonna solve this quickly . Negative .
25:30 One times negative four is four . So we have
25:32 4/2 . I know that four divided by two is
25:35 two , so I can just write that as two
25:38 plus be moved . The tour becomes a minus 25
25:41 minus two is three . So I have three equals
25:42 . B semi final equation . I plug in for
25:44 M and R B . I have negative 1/2 X
25:48 plus three . How about here ? What if I
25:51 give you a point and I tell you it's parallel
25:53 to this line . We know parallel lines of equivalent
25:55 slopes and the slope of this line . The number
25:57 in front of the X is a one . So
26:00 I have a parallel slope , is one . Those
26:03 two lines are parallel . That's a similar parallel .
26:06 The parallel slope is one , and I have a
26:08 point on the line X and y I can use
26:11 the X y and M plug into Why goes MX
26:14 plus B and solve for B Remember So like MX
26:19 plus B negative four is why one is m Xs
26:24 four . Solve for B . It's a negative .
26:27 Four equals four plus B . Move this for over
26:30 negative four . Minus four is negative . Eight .
26:33 So my final answer Y equals one x minus eight
26:37 . So we know parallel lines have equivalent slopes .
26:39 They don't have the same y intercept , so don't
26:41 use that at all . All we know para lines
26:44 have equivalent slope so I can use the slope from
26:46 that line along with the point we know is on
26:48 our parallel line . Plug it all in sulfur be
26:51 . There's the equation of our line . How about
26:53 perpendicular lines ? So we know our line goes through
26:57 this point and it's perpendicular to this line here .
27:01 So perpendicular lines of slopes , their negative reciprocal .
27:04 So this slope is negative . 2/5 , a perpendicular
27:07 slope . There's a symbol for perpendicular perpendicular slope .
27:10 We flip the slope , so instead of two or
27:13 five , we make it five or two and we
27:15 change the sign instead of it being negative , we
27:17 make it positive So we have a perpendicular slope .
27:20 We have a point . We don't use this y
27:21 intercept at all . All we know are perpendicular slope
27:24 . Sorry . Perpendicular lines have slopes or negative .
27:26 Reciprocal . So we use that perpendicular slope . At
27:29 that point , plug into y equals MX plus B
27:33 and solve for B . So why is zero And
27:36 we know it's 5/2 and X is five and we
27:42 don't know B If we solve this equation 25/2 plus
27:47 B and sorry , last step here and moved out
27:51 25 or two to the other side . We get
27:52 negative 25/2 equals beast and my equation of my perpendicular
27:57 line . I use my perpendicular slope 5/2 X minus
28:02 25/2 . That's the equation . Alignments perpendicular to this
28:06 line and goes through that point . The only other
28:09 way I could ask you to find the equation of
28:11 a line would be if I don't give you the
28:14 slope or the Y intercept right . We need both
28:16 mean m N b . But if I don't give
28:18 you either , but I do give you two points
28:21 , the line goes through . There's only one line
28:22 that goes through these two points and we'd have to
28:25 start by finding em by figuring out our change in
28:28 y So y tu minus y one over our change
28:31 in x x two minus x one . So if
28:34 this is our first point , that's our X one
28:36 y one . This is our second point . That's
28:38 our X two y two plug into our equation .
28:41 Be careful with your science with through four minus negative
28:44 too . Over six minus four . And if we
28:50 do this , we get 6/2 , which is three
28:53 . My slope is three . So I know em
28:56 . I'm going to now next step into y equals
28:58 MX plus B . I can plug in my M
29:01 and I can choose either of these points to be
29:03 my X and Y . Let's just choose the first
29:05 one here . This is my ex and my wife
29:07 . So why is negative ? Two m s three
29:10 and my ex is four and then we can solve
29:13 for B negative two equals 12 plus B and we
29:18 would get him in . The 12 were unable to
29:20 minus 12 is negative . 14 is be the equation
29:23 of my line y equals three x minus 14 ,
29:27 and I plug in M and B that I solved
29:29 for it . And you didn't have to choose this
29:31 point . You could have chosen this point to be
29:33 your X and y here . You would have gotten
29:35 the exact same answer . Um , the only other
29:40 thing from this section B horizontal vertical lines . Let
29:42 me just very , very quickly review this . If
29:44 you have an equation , looks like this . Why
29:46 equals the number ? You don't see an X at
29:48 all what that is . That's a horizontal line that
29:50 crosses through two on the Y axis . So you
29:53 would just draw a line perfectly horizontal that crosses through
29:56 two on the Y axis . That's the line y
29:59 equals to the slope of all horizontal lines is zero
30:02 . And the Y intercept in this case is to
30:05 Why is the equation y equals two ? So just
30:08 very briefly notice that every single point on this line
30:12 has a Y coordinate of two . It never goes
30:14 up or down . Right ? This is 20.62 This
30:18 is point negative . Eight to every single Y .
30:21 Coordinate of every point on this line is , too
30:23 . That's why we say the equation line is y
30:25 equals to have about an equation that says X equals
30:28 the number you don't see a y . Well ,
30:29 that's actually a vertical line that crosses through three on
30:32 the X axis . So we would just draw a
30:35 vertical line that crosses through three on the X axis
30:38 . And that is the line X equals three .
30:41 The slope of all vertical lines is undefined . And
30:45 the Y intercept while it never crosses the Y axis
30:48 , so we would say none . Why is the
30:51 equation X equals three for this will notice every single
30:53 point on this line . Right . This is 30.0.36
30:57 This is point negative six . Sorry . Uh ,
31:02 yeah , this is 60.3 . Negative six . Sorry
31:05 . This is 60.3 negative six . This is point
31:09 three . Negative two . Notice that the X coordinate
31:14 for every point on this line is three . So
31:16 we call the line X equals three X equals numbers
31:19 of vertical line y equals the number is a horizontal
31:21 line . Slope of horizontal lines . Zero slope of
31:23 vertical lines is undefined . All right , so that's
31:27 it for linear relations . Make sure you watch part
31:30 three geometry and then you will have done all of
31:32 grade nine math
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