Thursday, April 5, 2007

Graphing - Parallel and Perpendicular Lines

How can you determine if two lines are perpendicular?  How can you determine if two lines are parallel?  If you have two lines on a graph, and you have determined their equations or slopes, you may be asked if the two lines are parallel or perpendicular to each other.  These are two favorite questions of teachers and you will undoubtedly have to answer them!  Keep reading to find out how to easily answer them, and please remember to click the +1 button at the end.

Parallel lines are at the same angle and will never cross... like two railroad tracks. It doesn't matter what direction the lines travel. As long as they are going the same way, they are parallel. In mathematical terms, two lines are said to be parallel if they have the exact same slope.

Remember our equation for a line, y = mx + b.  Two parallel lines, each defined by their own equation, will have the same value for m, the slope.  So, y = 3x + 5 and y = 3x + 200 are parallel lines (they differ in their y-intercepts, but they have the same slope m).  You can plot this out for yourself quickly to see that this is indeed the case.

The opposite of parallel lines are perpendicular lines.  But how can you determine if two lines are perpendicular?  Perpendicular lines have a bit of a twist to them. Two lines are perpendicular if they cross (remember, any two straight lines that are NOT parallel will cross at only a single point.  They cannot ever intersect again unless they curve back on themselves, in which case they are not straight!) and they form a 90 degree angle, or rather, a T-shape. For example: The x-axis and y-axis are perpendicular to each other. Mathematically, if line 1 has a slope of m1, then a perpendicular line 2 will have a slope m2=(-1/m1)... that is, it's slope will be the negative inverse of the first.

Try it out... y = 2x + 1 and y = (-1/2)x + 5... m1 = 2 and m2 = (-1/2). Check it out on the graph to see that they indeed form a 90 degree angle where they intersect.  If you don't believe that this is correct, go get a protractor and measure it for yourself!  You can then convince yourself that the relationship holds true.

example of perpendicular lines
Perpendicular graph

Also of interest to you: the symbol for "perpendicular" is ┴, an upside-down capital T, whereas the symbol for "parallel" is two vertical lines next to each other, like ||.  You would write AB┴CD to say that AB is perpendicular to CD, and similarly AB||CD to say that they are parallel.

So, I've shown you that to in order to determine if lines are parallel or perpendicular, all you need to know is their slopes!  If you know the equations of the lines, then this is only a matter of simply reading the m value, or at the worst, performing some basic mathematical rearrangements to find this info.  And once you have the numbers, you can easily see if either relationship applies.  This is much faster than manually creating a table of values and plotting out the graphs by hand!  Now, you should have no problem answering questions that ask you to quickly identify true parallel or perpendicular line relationships!  If this post helped solve your questions, please remember to do me a favor and click the +1 button below to help me get the word out!  Thanks!

Tuesday, April 3, 2007

Graphing - Equation of the Line

If you are given a line on a graph (or enough information to construct a line), you will likely also be asked to find the equation of the line. The equation of the line is unique to each line; that is, every line has a different equation. With it, you can readily tell the slope of the line, and you can calculate what the x-value is for any y-value on the line (and vice versa). It is very handy!

The most common and basic form of the equation of the line is:

y = mx + b

where m is the slope and b is the y-intercept (where the line crosses through the y-axis).

Another way to write it, which is more general is called the POINT-SLOPE FORMULA:

(y-y1) = m(x-x1)

where m is the slope, and x1 and y1 are the coordinates of a known point on the graph. (If you pay close attention, you can see that this way of writing it is really the same as the slope formula, but rearranged!) The version basically says 'give any point at all, and a slope, and you have enough information to draw the line."

Let's use the graph from the slope lesson to practice, using points (2,1) and (7,7). To get our final answer, we are going to have to do a couple of steps first. Let's use the y=mx+b equation. The steps we are going to do are:
1) Find the slope
2) Find the y-intercept
3) Write the equation of the line

Finding the slope is easy now, if you read the posting on slopes (I always leave numbers as fractions, instead of changing to decimals):

m = (y2-y1)/(x2-x1)
= (7-1)/(7-2)
= 6/5

The y-intercept is easy now... just plug numbers into y=mx+b, and solve for b. So, b=y-mx. We have our slope now, and for y and x, we just substitute in the coordinates of a single point (x and y MUST be from the same point!)

= 1-(6/5) x 2
= 1-12/5
= (-7/5) (you have to do some fraction math!)

So now we can write the equation of our line!

y = mx+b
y = (6/5)x - (7/5).... (if we leave it like this, we can read the values for slope and y-int right from the equation!)
y = (6x - 7)/5

If we work through using the other formula... (y-y1) = m(x-x1)... we get the same answer, and we don't have to explicitly solve for b! First, solve for slope. Second, substitute in values for slope and x1,y1, and then rearrange!

slope = 6/5 (same thing we did above)

so (y-y1) = m(x-x1)..... (sub in slope, and point (2,1) for (x1,y1))
(y-1) = (6/5)(x-2)...
y=(6/5)(x-2) + 1...
y = (6/5)x - (6/5)2 + 1...
y=(6/5)x -12/5 + 1...
y=(6/5)x -(7/5)... (same slope and y-int)

Same answer! That is the equation of the line. Now if you put in any value for x, you can say exactly what the value for y would be... if you wanted to know what y is when x is 1000, you can do that now! You will see that it will always be on the same straight line.

If we work through and do the same thing for the red line on the graph, with points (-6,2) and (-4,-5), we can get the equation for that line too! Let's try it this time using just the one formula.

(y-y1) = m(x-x1)
((-5)-2) = m((-4)-(-6))
(-7) = m(2)

Now put that back into the same formula along with a point, and rearrange:

(y-y1) = m(x-x1)
(y-2) = (-7/2)(x-(-6))
y-2 = (-7/2)(x+6)
y= (-7/2)x + (-7/2)(6) + 2
y = (-7/2)x -21 + 2
y = (-7/2)x -19

(see slope (-7/2) and y-int (-19)... look at the graph, and you will see that makes sense! Negative slope, very low y-int!)

That's all there is to it! Just remember that there are a few steps to follow, depending on the formula you are using... basically, remember to always find the slope first, then the y-intercept, and plug directly into y=mx+b... or use the point-slope formula to find the slope using 2 points, and then resubstitute it back in with a single point and rearrange. It's not that complicated once you practice and understand what you are doing! Either method is going to give you the same answer, so pick your favorite and stick with it!

Sunday, April 1, 2007

Graphing - Slopes

When you have a line on a graph, you will probably need to know its SLOPE. A slope on a graph means essentially the same thing as if you were talking about the slope of a driveway or the slope of a ski hill... it is a measure of how steep the driveway/hill/line is.

The minimum amount of information you need to find the slope of a line is the location of two points on the line. These could be endpoints for a line segment, or just points on a line that goes on forever. Since it's a single, straight line, ANY two different points that are on the line can be used and will give you the exact same slope as any other two points on the same line. Makes sense right? The slope is a property of the line, and all the points on the line make up the line.

The slope formula is easy to remember. The complicated way of saying it is 'the slope is the difference in height of two points on a line, divided by the difference in width of the same two points.' The much easier way is 'slope equals RISE OVER RUN.'

slope = rise / run

The rise (think rising = height) is the difference of the y-coordinates of two points on a line. The run (think about running on the street = horizontal) is the difference of the x-coordinates of the same two points.

Usually, slope is represented by the letter 'm.' So then, the slope equation can be written like:

m = (y2-y1) / (x2 - x1)

This appears to be a little more complex than the first one, but it means the same thing. The 1 and 2 are just names for the y's and x's. (They could be anything... A and B, or whatever.) The (y2-y1) means 'the difference between two y-coordinates, and the (x2-x1) means 'the difference between two x-coordinates.' IMPORTANT: Make sure that the point you use for y2 is the same point you use for x2, and likewise for y1 and x1. (Otherwise, you'll get the wrong slope.)

Another IMPORTANT thing to recognize: if the graph rises to the right, it is said to have a POSITIVE SLOPE. If it is falling towards the right, it has a NEGATIVE SLOPE. That means your slope value will have a positive or negative sign with its number. You can check that when you're done. (It's easy to make sign errors. It happens to everyone.)

This figure should hopefully clear things up.

Let's look at the black line first. Let's call the top point 'point 1' and the bottom point 'point 2'. So:

rise = (y2-y1) = (7-1) = 6
run = (x2-x1) = (7-2) = 5 **Notice that I didn't use (2-7)!

slope = rise/run = 6/5 or 1.2

That's it! Let's look at the red one now. It's a little trickier because of the negative signs, but you do the exact same thing. Point 2 on the left, point 1 on the right:

rise = (y2-y1) = (2-(-5)) = (2+5) = 7
run = (x2-x1) = ((-6)-(-4)) = ((-6)+4) = (-2)

slope = rise/run = 7/(-2) = (-7/2) or (-3.5)

Notice how the first graph had slope of (positive) 1.2 and was going up to the right, and the second one had slope (-3.5) and was falling to the right.

As long as you keep y2 and x2 coming from the same set of coordinates (and y1, x1), and you keep track of your signs, you'll get the right answer for your slope! And with the slope, you can do more complex things, like find an EQUATION that describes the line.

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