Functions - Domain, Range, Vertical Line Test

Functions may seem like a totally foreign and incomprehensible concept at first, but they're not really the monsters that many students think they are. The notation may seem a little weird from what we're used to, making them look difficult, but when you understand the concepts of what is being presented, you will see that they're not really doing anything that doesn't make sense.

To give a very simple definition, a function is basically just a way of relating two variables... although their notation makes it look otherwise. But we've already seen something like this before! If we take a regular X-Y graph and draw a straight line on it, the equation of the line is the way of relating the x variable with the y variable. However, the standard way of writing a function is to write f(x) instead of y. We say it like f of x. So for example, taking a line, we can do the following:

y = 3x + 5
f(x) = 3x + 5

The function notation says that "the whole expression with x on the right" evaluates to some number (the equivalent of y). The math is still the same, no tricks with what you can do with the expression. It's just a different way of describing things. But we also now have a way of saying "tell me what that thing evaluates to when x=10" and we can write that out like this:

f(10) = 3(10) + 5 = 30 + 5 = 35
f(10) = 35

Again, it's essentially the same thing as saying 'solve for y' from our graph equations, but you can't say to solve y = 3x + 5 at a certain point for x, without actually writing that instruction out in words. With function notation, you can say to solve the expression when x=10, 50, or 25,000, just by writing f(10), f(50), or f(25,000).

There are a few other terms you will likely see when dealing with functions. Domain refers to all the values of x that are valid for the expression. Range refers to all the values of y in the expression. Domains and ranges could be infinite, as in the case of a straight line that goes forever in either direction, or they could have defined numbers as in line segments. The notation for domain and range sets is like [x₁, x₂] or [y₁, y₂], where those numbers represent the two extremes of the domain (furthest left and right) or range (highest and lowest), and the square brackets indicate that those numbers are included in the range. (If you had an 'open' point on a line, this indicates that that exact point is excluded from the line, and if this occurred as an endpoint, you would write (x1, x2) to denote that the number is not included in the set. Also, you could have on the line a closed and an open point, in which case you'd have something like (x1, x2] or vice versa.)

The domain (the x-values) for line segment A is [-8, 1] and its range (y-values) is [5, 8]. In this case the domain and range extremes are the actual end points of the line, but pay attention to the particular question as this is not always the case. Remember, this notation is defining the domain and range (all the values that x or y could take), not the end points of the line.

Similarly for line segment B, the domain is [2, 7] and the range is [-3, 6].

Line C demonstrates a very important concept about functions. Line C is not a function. We should expand our definition of a function to say that it is a way of relating two variables, where each number in the domain corresponds to ONLY ONE number in the range. That means, that for every x, there can only be one y. To test this on a graph, you can use the vertical line test to prove a line is a function or not. If you draw a vertical line through your function at any value for x, the vertical line will pass through the function line ONLY ONCE. If it passes through it more than once, then you are not looking at a graph of a function. Therefore, looking back to line c, you can tell that a vertical line would pass through the line twice at some positions, and so this proves it is not a function of x. Similarly, the circle in D does not pass the vertical line test, and so is not a function either.

These concepts may seem different, but you will become accustomed to them. Just remember our definition of what a function actually is, and what the notation is saying, and you will be fine. When you begin to grasp these concepts and advance to higher levels, you will get to see some really cool stuff with functions. (sneak peek.... f(x) = x⁶ + x⁵ + x⁴ + x³ + x² + x + 1.... it's still a function!)

If any of that isn't clear and you'd like something more explained or clarified, please don't hesitate to leave a comment for me! I'm always glad to help!

Graphing - Standard Form of the Equation

Just a short explanation for what is meant by "standard form" of the equation of the line. We have been looking at line equations in the form of y=mx+b. However, you may be asked to express this in standard form, or as a standard form equation.  Graphing standard form equations will give you the exact same line as graphing something expressed as y=mx+b... standard form is just a different way of displaying the equation.  (Please hit the Like button and/or the +1 button if this post is helpful for you!)

The general notation for a standard form equation is Ax + By = C, where A, B, and C are coefficients, and the x and y are the same variables we've been looking at but in a different position from what we recognize.

To express in standard form, you simply just rearrange the y = mx + b form such that you have x and y on the same side, equal to a number. Let's look at some examples:

Given that y = 3 x+ 5, standard form of this is 3x - y = (-5).

Given y = (1/2)x -15, standard form of this is (1/2)x - y = 15... also, if you don't want to have any fractions in your answer, you can multiply everything by the number in the denominator, such that we now get x - 2y = 30. Both expressions mean the same thing and will produce the same line. (In fact, convince yourself that no matter what you do to the equation, so long as you do it to both sides, the line is the same. eg. Multiply it all by 100, you get 100x-200y=30000... looks different, but it's not! Reduce it down and see for yourself!)

For graphing standard form equations, you still might want to go from standard form to the mx+b form, for which you may need to do a bit more math, but it's still quite straight forward.

Given 5x - 15y = 10, you just have to rearrange things to get y by itself on one side:
(-15y) = (-5x) + 10
y = (1/3)x - (2/3)...
and then you can see it is a line with slope 1/3 and y-intercept (-2/3).

Both types of equations mean the same thing. They are just expressed differently, and y=mx+b gives immediate information about the line without having to do a lot of work. However, you should be able to use both forms interchangeably.  Convince yourself that graphing standard form equations will give you the same line as graphing y=mx+b equations.  They just look different because the numbers are rearranged.  This should be obvious because if you start with a standard form equation, and convert it to y=mx+b and graph it, you have only rearranged things not added or removed anything.  You do not have a new line.

Also, from these equations, you should be able to tell that whenever you have an equation with 2 variables (x and y), and there aren't any exponents on either term, then you are dealing with a straight line. So while an equation in standard form may not immediately look like a straight line equation to you until it looks like y = mx + b, because it has an x and a y in it (without an exponent... exponents make the graph do cool things later), it is automatically a straight line.

(check this post for some additional pointers)

If you liked my post, you can help me spread the word by tweeting about it!  Also, please remember to click the Like button or +1 if it helped!

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.

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!

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!)

b=y-mx
= 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)
y=(6x-7)/5

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)
m=(-7/2)

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!

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.