Slopes

Marked June 7th

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Definition: Slope is the steepness of a linear relation based on two co-ordinates on the line. media type="youtube" key="fCFaSvbwKe0" height="344" width="425"__ media type="youtube" key="REjcPZeypVg" height="344" width="425" **

Here are some lessons on SLOPE (ignore any of the homework (unless you have a MATH POWER 9 textbook) and random slides beacuse this was used in a classroom). And if the words are too small, go to the top of the screen and click the zoom button so see better. :)

There are TWO formulas that are more prominent than others for slope: **Formula b)** y=mx+b (this is the basic linear equation)
 * Formula a) ** m=y2-y1/x2-x1 (this is the basic slope formula)

Yint: -The y-intercept. -The point at which the line crosses the y-axis. Xint: -The x-intercept. -The point at which the line corsses the x-axis. When figuring out what the y/x intercept is...... If ur tring to figure out the Y-int, the X will be always 0 and when ur tring to figure out the X-int the Y will be 0 **__How to Determine the Equation of A Line From a Graph__** T o identify the linear equation of a line, observe the graph and try to identify the TWO main things.
 * 1) The "yint" (y-intercept)
 * 2) The slope (rise/run)

 This is the linear equation: y **=** **m** x + ** b **

=
//**B **// //**is the yint(the y-intercept). The yint is where the line crosses the y-axis. If you're in a situation which the line is not long enough to identify where the line crosses the y-axis, then follow either of these two procedures:** //======


 * 1) Perform the equation below: **//Finding B When Given Two Points,//** that shows the steps to find the "yint"
 * 2) Or just elongate the line with a ruler


 * M ** **is the slope. The slope is the rise over the run. Read** **Fi****nding "m" When Given Two Points** **to proceed with finding the slope.** 

To find the y-int or B when you have slope just fill in the equasion with x=0.


 * Formula a)** is to find the slope. To accomplish this, you must have two co-ordinate points. This is because you substitute the points with the variables within the formula. Here is the outline of finding "m", the slope:
 * __Finding 'm' when given two points:__ **

First of all you should know that '** m **' is ** slope ** or ** rate of change **. To find slope we must use a formula m=y2-y1/x2-x1. What this formula means is that m or the relation is equal to the second 'y' co-ordinate, subtract the first 'y' co-ordinate and then second 'x' co-ordinate subtract the first 'x' co-ordinate.

__Example__ This is what we would do If we are given the co-ordinates (2,4) and **(4,6)** and we are asked to find **'m'** (or the **slope**)

We would first write **y=mx+b** and then below that we would write __find m__ (we found this earsier to keep organised). Then we would find out which co-ordinates are 'Y2', 'Y1' , 'X2' and 'X1'. When given any co-ordinate we should know that the **x co-ordinate is the first number**, and **the y co-ordinate is the second number**. So for the co-ordinates (2,4) we now know that the **x co-ordinate is 2** and the **y co-ordinate is 4**. And for the second co-ordinate we know that the **x co-ordinate is 4** and the **y co-ordinate is 6**.

Now we know the **x** and **y** co-ordinates, how do we know which one is y2, y1, x2 or x1? We determine which co-ordinate is either **1** or **2** by simply looking at the question. **The first co-ordinates are the x1 and y1**, that being said we now know that
 * the second co-ordinates are x2 and y2**.

So if we have (2,4) and (4,6) we know that **2 is the x1 co-ordinate** and that **4 is the y1 co-ordinate**. Which means that **4 is the x2 co-ordinate** and **6 is the y2 co-ordinate**.

From this we can now solve m=y2-y1/x2-x1.

Our first line is __always__ the formula. following that would come our formula to find 'm'.
 * First line:** y=mx+b and then we would right __find m__
 * m=y2-y1/x2-x1**

With our second line we can start substituting our co-ordinates for numbers.
 * Second Line:** m=6-4/4-2

Our third line we subtract the given numbers which will give us our fraction.
 * Third Line:** m=2/2

Now that we are left with **2/2** that means that this fraction **is a whole number**.: 2/2 =1. We now know that m=1. We can substitute that into our formula. Using the formula '**y** **mx+b'** we can sub **'m' for 1.** Which gives us y=1x+b which is the same as y=x+b. (we don't need to write y=**1x**+b becuase '1x' is 'x' and a number multiplied by 1, is always the number being multiplied by 1.Therefore not needing a 1 infront of 'x').

So our fourth line is the formula, with all the numbers substituted in.
 * Fourth Line:** y=x+b

This is what your forumla being solved should look like on your page. __find m__ m=y2-y1/x2-x1 m=6-4/4-2 m=2/2 y=x+b** That is how we find 'm' when given two co-ordinates. :)
 * y=mx+b

** __Finding 'B' when given two points:__ ** We will now find 'b' using the co-ordinates we used for finding 'm' in the example above. This is what we would do If we are given the co-ordinates (2,4) and (4,6) and we are asked to find **'b'**
 * Formula b)** is used to find the y-intercept, or also called the "//yint//". To determine the yint, you have to find "b":

Again, the first thing we would do is write the formula **y=****mx+b**. We already know 'm' (it was 1). So our formula would know look like y=x+b.With the given co-ordinates (2,4) and (4,6) we know which numbers are the x and y co-ordinates. (**2 and 4 were the x co-ordinates** meaning that **4 and 6 were the y co-ordinates**).

__Example__

When we are finding 'b' and we use the formula 'y=mx+b' we substitute a pair of co-ordinates for 'x' and 'y'. We substitute 'x' for a 'x' co-ordinate from a pair of co-ordinates and then we substitute 'y' for the 'y' co-ordinate. (**Note:** the numbers you sub for 'x' and 'y' must be from the same co-ordinates given above. E.g. (use (2,4)) The x co-ordinate is 2. Which means we sub 2 for 'x'. Therefore the 'y' co-ordinate is 4. So we would sub 'y' for 4. Making our formula look like this **4=2+b**

From This we can now find 'b'

Our first line will consist of our formula y=x+b and then __find b__ __Find 'b'__
 * First Line:** Y=x+b

Our second line will have a pair of co-ordinates substituted in for 'x' and 'y'
 * Second Line:** 4=2+b

From here we still don't know 'b'. So for our third line we subtract 2 from 4, becuase this will leave us with a simple question wich is equal to 'b'
 * Third Line:** 4-2=b

At this point, we do the math to give us 'b'.
 * Fourth Line:** 2=b

Seeing as 2=b we can now sub 2 in for 'b' in our formula y=x+b. Doing this will give us our final formula. That being said, our final formula would be: y=x+**2**.

This is what your forumla being solved should look like on your page. Y=x+b __Find 'b'__ 4=2+b 4-2=b 2=b y=x+2.

That is how we find 'B' when given two co-ordinates. :)

__Rise over Run__
Slope = Rise/Run so if the slope is 2/3 the graph would look like so from the starting point, you rise 2 and over 3. In a y = mx+b the "m" is the slope and if the "m" is a whole number you can always change it to a fraction. for ex. 5 = 5/1

**__Types of Slopes:__** Slopes are individually categorized into different categories depending on the type of data pattern (ascending or descending). This is an example of a **positive slope**. The trend of the data displays a pattern that's increasing.

This is an example of a **negative slope**. The downward trend of the data results in the decreasing data which results the line to decline.

__There can be a **slope of ZERO**! This occurs when the line is horizontal.__

__This vertical line is an example of an **UNDEFINED** line. When a question asks what the formula for an undefined line is, just mention the x-intercept.__

=
Well, slope is everywhere. Almost everything has a slope from a house roof, to a slide. Another way that slope is found in our lives is in rate of change. Rate of change is the measure of something in the form of anything/time, like kilometers per hour, or revolutions per minute. We see this everywhere like on the inside of our car, on the side of the road, and even when you're at your computer(ex. megabits per second).
 * __SLOPE IS EVERYWHERE AND YOU DON'T EVEN KNOW IT!! [[image:slippery_slope.gif width="404" height="239"]][[image:NT166_superslide.jpg width="351" height="205"]][[image:images.jpg width="220" height="186"]][[image:hsd_ParkSlope.gif width="340" height="281"]][[image:http://magnust.files.wordpress.com/2008/06/3coc10.jpg]] [[image:http://bp1.blogger.com/_qu0xAdTmH4g/SCsIXdI7ESI/AAAAAAAAAlc/_X-ctayR0Vw/s320/downhill-mountain-bike3.jpg]][[image:http://www.wayfaring.info/wp-content/uploads/2007/04/downhill-skiing-1.jpg]]__**======
 * __Where Is Slope In Everyday Life?__**

Slope Jobs Many jobs have todo with slope seeing that slope is also rate of change so any job from a computer technition to a car maufacturer will deal with slope. But also jobs like downhill ski and bike designers use slope to determine how to build their facilities. There are countless jobs that are involved in slope, the possibilities are endless!!!

** Standard Form: ** The standard form of the equation y=mx+b is: **A,B and C being numebrs and y and x are the variables. Ax+By+C=0 BUT! There are 3 important rules you must use when using the standard form equation: **


 * 1) No fractions are to be present within this equation.
 * 2) All the terms must be on the left side.
 * 3) The first term is ALWAYS positive.

You will use this when asked to convert a y=mx+b equation into standard form **, and when a question asks you to **convert a standard form equation into a y=mx+b equation.

First Step Clarify which values are what. The 5X is AX, the Y is BY, and 1 is C. Second Step DON'T FORGET!! When a number crosses the equal sign, it changes to a positive if it was negative, or negative if it was positive. The answer is: 5X-Y+1=O
 * To convert a y=mx+b equation into standard form, visualize and logically seperate each variable without actually doing so. Pretend the "A", is the variable "M" in the y=Mx+b equation. Pretend the "C" in this equation is "B" from the y=mx+B equation. The "BY"'s product will equal the Y in the Y=mx+b equation. Here is an example: Convert y=5x+1 into standard form.

**To convert a standard form equation to a y=mx+b equation**, do the same thing as you would when converting a y=mx+B equation into standard form, but only this time, try to identify which paired values are the values in the y=mx+b equation. The AX is MX in this equation, BY is Y in this equation, and C is B in this equation.

Here is an example: Convert 4x-1+1=0 into y=mx+b form. **

First Step: 4X is MX, -1 is Y, 1 is B.**
 * Identify which numbers are y, mx, and b.

// Second Step: // 4X is MX. You're equation should look like this so far: y=-4x+b 1 is B. You're equation should now look like this: y=-4x-1 -1 is Y. You're equation should now look like this: -1=-4x-1 Notice how 4x and 1, both changes their operations because they CROSSED THE EQUAL SIGN!
 * Since you now know which values are which, just replace y=mx+b with actual numbers.

Standard form is used or mathematics further into high school but is also useful to know this foundation early for next year's preperation.**

How to Find the X-intercept and Y-intercept:
 * The x-intercept, (xint), is when the y-intercept is zero on a line, on a graph, and where the linear equation crosses the a-axis. The y-intercept ,(yint), is when the x-intercept is zero on a line, on a graph, and where the linear equation crosses the y-axis.

To find the xint when given an equation, just replace the y value with ZERO. The same application follows for when you're trying to find the yint when given an equation. Just replace the opposite value of what you're trying to solve, in this case you're trying to solve for y, so replace the x variable with 0. Here is an example: Find the xint and the yint of 4x+y-4=0

// First step // Write FIND XINT for a title. Then below that, write the formula down with the replaced value for y, which is 0. Doing this, the equation written on your paper should look like this:** Find Xint __**y=0 4x+0-4=0

**__//Second step// __Because all the unknown values are, known, the only variable left is x. This is great because we can now isolate it, solving it's value! Just proceed with solving the equation:__ Find Xint __**y=0 4x+0-4=0 4x+4=0 4x=-4 4x/4=-4/4 x=-1 Therefore, the xint of this linear equation is -1.(-1,0).

**//Third Step// Find Yint __**x=0 4(0)+y-4=0
 * Write FIND YINT as the title. Then below that, write the formula with the replaces value for x, which is always zero. Doing this, the equation written on your paper should look like this:**__

**//Fourth Step// Just proceed with solving the equation:**__ Find Yint __**x=0 4(0)+Y-4=0 y-4=0 y=4 Therefore, the yint of this linear equation is 4. (0,4)
 * Because all the unknown values are, known, the only variable left is y. Which is great because we can now isolate it.

**__ Word Problems:

__Word problems confuse SO many people because they do not know what to do with all the information. Here is an example of a slope word problem and how you could solve for it.

if you don't know what to do, write down y=mx+b and figure it out from there by subbing in what you know.

Ex 1. The line segment joining points A (x,3) and B(4,9) has a slope of 3. Find x.

To do this, you will first have to fill in what you know for the slope formula. M=Y2-Y1/X2-X1 (Since it tells us what the slope is we can fill that in first.) 3=Y2-Y1/X2-X1 -(And now you can fill in the other numbers you know using the two given points.) 3=9-3/4-x -(From this step, you can do either two things, cross multiply, or treat the slope as a fraction.) to.. **Cross multiply:** 3= 6/4-x --( Do the math that you can, and since we dont know the value of x, we just leave it.) 3(4-x)= 6/4-x ( The 4-x cross eachother out so then we will just be left with six on the right side of the equal sign.) 3(4-x) =6 12-3x= 6 -( Now you use the distributibe propetry and get rid of the brackets.) -3x= 6-12 (Re-arrange the equation and work it out) -3x/-3=-6/-3 x=2 Therefore x is 2.

OR.. treat it as a fraction.

3=6/4-x ( Do the math that you can, and since we dont know the value of x, we just leave it.) 3/1=6/4-x --( Then you have to see 3 as a fraction over one.) 3(2)/1(2)=6/4-x --( Then you do basic fraction work and see that to get to 6 from 3, you multiply it by 2. and do to the top you must do to the bottom.) 1(2)= 4-x --(The Two 6's on the top now cancel each other out and you solve for x.) 2=4-x 4-2=x 2=x Therefore x is 2. Both get the same answer, its just one way is easier and one way actually explains the math.__

How to Identify Perpendicular and Parallel Lines: When lines are parallel, they are lines that'll NEVER EVER touch. The two or more lines retain the same distance apart. When lines are perpendicular, they are two lines that cross at 90 degree angles. To determine the classification of a pair of lines, there are specific requirements. a negative recipricol is a number that's been flipped to become the denominator of a fraction, and the numerator switches its sign Examples of negative recipricols: 2/-3 and 3/2 because the numerator and denominator flipped and the numerator changed from a negative to a positive. -1/2 and -2/1 because the numerator and denominator flipped and the numerator switched its sign. 3/4 and 4/3 would NOT be negative recipricols becuase the sign did not change, even though the numeraor and demonerator switched.
 * 1) For a pair of lines to be parallel, they must have the same slope.
 * 2) For a pair of lines to be perpendicular, the slopes must be negative recipricols.

a) y= 5x+169 b) y= -4/2x+28 c) y= 5x+100049930493030003949949930049 d) y= 2/4x+1234567890 e) y=1/2x f) y=2/1x+3 b) and d) are perpendicular with eachother because there slopes are negative recipricols. e) and f) are NOT perpendicular because their signs did not swith. It doesn't matter what the y-intercepts are, you are only dealing with the slopes in this case.
 * Example:** Which lines are perpendicular or parallel with eachother?
 * Answer:** A and C are parallel with eachother beacause they have the same slope. (5=5)


 * Distance-Time Graph: **

when you are standing still the line will be straight, when you move away from the place you were standing the graph will go up and when you come back the line will drop.

A distance-time graph, graphs the relationship between distance and time. For excample if I walk away 2 meters in 2 seconds, stop for 2 seconds then run back 2 meters in 1 second it would look like this.

Slope is used in every day life even without you knowing. If your snowboarding and you hit a jump its all about the slope. It all matters about the rise over run for height. Or biking if there is a negative slope then its a breeze, if your going on a positive slope its hard, if you are on a 0 slope then its just normle biking, now if its a unidentified slope be prepaird for some broken bones.

A lot of time graphs can use a story to represent the data. For example. The story for this set of data could go something like this.

Shelly has to tie her shoe for 3 seconds. She then runs outside 8 meters in 8 seconds to wait for her bus. She waits for 5 seconds when she realized she forgot her book. She runs back to the kitchen 6 meters away in 2 seconds. She picks up her book, which takes 2 more seconds. She then has to go get her other book 2 meters away from her again. That takes her 5 seconds. And then she stops to pick the other book in 4 seconds.

media type="youtube" key="Ooa8nHKPZ5k" height="344" width="425"** media type="youtube" key="ssSYFZKKqsM" height="344" width="425" media type="youtube" key="RXx0K7jA1Gg" height="344" width="425" media type="youtube" key="hXP1Gv9IMBo" height="344" width="425"
 * Here are some videos on slope!

<span style="color: rgb(231,208,39); font-size: 150%;">Works cited:__


 * 1) Mr. Hendriks Knowledge Passed Down
 * 2) Negative, Positive, Zero, Undefined slopes jpg.
 * 3) Slope title jpg. made
 * 4) Hendriks, J.D. (2008). MPM 1D1 - Course Notes
 * 5) Hendriks, J.D. (2008). MPM 1D1 - Course Notes