Let’s look at the graphs of some basic functions.
The first function is y equals x. This is a linear function.
It is called the identity function.
It has its x - intercept and y - intercept both at zero, at the origin.
It is an odd function since it is symmetric with respect to the origin,
and its domain is all real numbers. x can be anything.
Its range is also all real numbers.
The second function is y equals x squared.
This is a basic quadratic function. It is called the square function.
Notice this one is symmetric with respect to the y - axis.
This one is said to be an even function.
The x - intercept is at 0. The y - intercept is at 0.
The domain here is the set of all real numbers.
Notice how the y - values are never lower than 0,
so the range will be the set of real numbers from 0 to infinity.
The third function is y equals x cubed.
Notice that this one has its x - intercept
and it y - intercept both at 0, at the origin.
The domain of this one is the set of all real numbers.
The range is also the set of all real numbers.
This one is symmetric with respect to the origin, so it is an odd function.
The fourth function we want to look at is the absolute value function,
y equals the absolute value of x.
This one also has its x and y - intercepts at the origin.
This one is symmetric with respect to the y - axis, so it is an even function.
Its domain, the set of allowed x - values, is all real numbers.
Again, here the y - values are never smaller than 0,
so the range is the set of real numbers from 0 to positive infinity.
The next function is y equals the square root of x.
First thing I notice on this one is that it has no symmetry.
The next thing you want to notice is that
the domain is all real numbers from 0 to infinity.
Alright, you know you cannot take the square root of a negative number,
not to get a real number anyway, so that is the domain.
The range would be the set of values from 0 to positive infinity again.
The sixth function we want to look at is y equals the cube root of x.
This one is also an odd function since it is
symmetric with respect the origin.
The domain is the set of all real numbers.
The range is the set of all real numbers,
and the x and y - intercepts are both at the origin.
The last function is called the reciprocal function.
Notice first of all this one has no x or y - intercepts.
x cannot equal 0, that would give us division by 0.
So the domain is the set of all real numbers except 0.
The range is also the set of all real numbers except 0,
because you will notice that the graph never touches the x - axis or crosses.
Finally, this is an odd function since it is
symmetric with respect to the origin.
These are very important basic functions.
You should be familiar with both the characteristics,
and the graphs of these functions.
Piecewise Functions are functions that are defined by
different rules in different parts of the domain.
Let’s examine a piecewise function closely.
The first thing you will notice is that it looks different from our usual functions.
This is a function f of x, and it is defined by the rule
y is equal to x plus 1 if x is greater than or equal to 0,
but for values of x less than 0, we have the same function but a different rule.
The first thing we would like to do is look at some specific function values.
This is asking us to find f of 0.
That means find the value of the function when x is equal to 0.
Well I need to decide which part of the rule for this function applies.
x equals 0 so that is here, so I want to use the first part of the rule.
Substitute 0 in place of the x, so we find that f of 0 is equal to 1.
The next one is f of 3. This is asking to find the value of the function when x is equal to 3.
Well again I need to decide which part of this rule applies.
3 is greater than or equal to 0, so we are going to use the first rule again.
Substitute 3 in place of the x, and you find that f of 3 is equal to 4.
One more. This is asking for f of negative 1.
We want to know the value of the function when x is equal to negative 1.
Well negative 1 is less than 0, so we want to use this rule.
Substitute negative 1 for the x, and we find that x of negative 1 is equal to negative 3.
The next thing we would like to find is the domain of this function.
The domain is the values that x is allowed to have.
So let’s look at these inequalities.
This tells me x can be any real number less than 0.
This tells me x could equal 0 or anything greater than 0.
That is the set of all real numbers.
The domain we could write in interval notation as negative infinity to infinity,
or we could just say it is just the set of all real numbers.
The next thing it asks us for is the range, but it will be easier
to answer this once we have drawn the graph so let’s stop and draw the graph.
Okay, well we need to examine the pieces of our piecewise function again.
Notice that both of them are linear pieces, so what we need to do is
graph this piece of a line for x greater than or equal to 0,
and graph the piece of this line for x less than 0.
Okay, well to graph a line we need just need to find two points,
maybe specific two points, but two points will do.
Alright, let’s look at the first one.
We want to start with x equal to 0 because that is the endpoint,
that is where something changes in this piecewise function.
I want to find the value of this piece of the function when x is equal to 0.
Okay, but we did that right? f of 0 is 1. Let’s plot this point.
Now we are still working on the first piece.
This rule is used for values of x greater than or equal to 0,
so I want to pick another value of x greater than 0
and use that to graph the rest of the line
or the portion of the line that I need. Alright, well look at our work.
We have used x equal 3, 3 is greater than 0.
We know that when x is 3, the value of the function is 4.
We can plot the point (3, 4).
Now we have already said that this piece of the graph is linear,
and we use this rule for values of x greater than or equal to 0.
Let’s draw that portion of our line. That takes care of the first piece.
Now let’s move to the second piece. It is also linear, and again I need to plot two points.
Alright, the first thing I am going to do is look at this endpoint.
This is where the graph changes, this is where strange things can happen if they are going to.
We want to substitute x equals 0 and
see what the value of the function would be there.
Okay well it would be 0 minus 2, that is negative 2.
But 0 is not included there, so to show this on my graph I want to use an open circle.
I am using the endpoint, and I want to make sure
my graph is complete but 0 is not included.
So this point cannot be included. Make sure you use an open circle there.
Now I need one more point. It needs to have an x - coordinate that is less than 0,
so let’s use this one we found earlier.
When x is negative 1, the value of the function is negative 3. So (negative 1, negative 3).
Again this portion of this piecewise function is a line, and we use this
rule for x less than 0. So we are going to draw the portion of the line for x less than 0.
Now that we have the graph for this piecewise function, let’s answer the last two questions.
This one asked us for the range. The range is the set of values y is allowed to have.
The first thing I want you to notice is that the graph has two pieces,
and the y - values are different in those two pieces.
For instance, here the y - values can be anything up to but not including negative 2.
So to write that in interval notation is going to be negative infinity to negative 2.
Use parentheses to signify the negative 2 is not included, by the open circle again.
Now, the other piece of the graph has y - values that start at 1
and they increase without bound.
So we would say this is the interval from 1 to infinity,
and we want to include the endpoint 1. Use a square bracket there.
The last question is for the intercepts, and we are lucky on this one.
We can sort of look at the graph and pick them out.
Never does this piecewise function touch, never does its graph touch the x - axis,
so there are no x - intercept. It touches the y - axis up here at positive 1,
so the y - intercept is 1, and as we said there are no x - intercepts.
Now be careful with the hole that is right here.
There is no point where that hole is, so there is no y - intercept there.
Be careful of that. Let’s look at one more example.
Let’s look at another piecewise function.
Notice that this one actually has three rules.
So we are going to use the square root of x as our rule for x greater than or equal to 0.
We are going to just use f of x equals four for the piece where x is just equal to 0.
The third piece is negative 2x. This is going to be a linear piece to the graph.
Okay, again let’s start by finding some specific function values.
When x is 0, we are going to use the middle piece of the graph.
It says the value of the function is 4 when x is 0.
The next one is when x is 1, find the value of the function.
1 is greater than 0, so I want to use this part of the function. This rule.
f of 1 is going to be the square root of 1, which is 1.
Finally let’s see what happens when x is negative 1.
We want to find the value of the function here.
Alright, negative 1 is between negative 3 and 0, so we want to use this rule.
So negative 2 times negative 1 gives me positive 2. Now let’s look at the domain.
Again to find the domain we always want to look at what values x is allowed to have.
x can be anything bigger than 0, any real number bigger than 0. It could be 0,
and it could be negative 3, and any real number between negative 3 and 0.
The important thing here is that x is never anything smaller than negative 3.
So our domain, our allowed x-values start at negative 3, include that using a bracket,
and then they increase without bound so to positive infinity. Now let’s look at the graph.
The first portion of the graph is the square root of x
but only for values of x greater than 0.
Well the square root of x is a basic function that you should know by now,
and let me just sketch what that might look like down here.
Okay, the square root of x has a point (0,0).
It has a point at (1,1) since that is the square root of 1.
If we go way out here at x equals 4, the positive square root of 4 is 2.
It looks like this, and its domain is all the real numbers from 0 to infinity
which is almost exactly what we needed. What is the difference?
Well, zero. This does not allow me to include the 0,
whereas our basic square root of x graph would.
How can I change that to work for my piecewise function?
I am just going to change this to a big open circle.
That point should not be included since x equal to 0 is not used in this rule.
Let me transfer that to the graph of my piecewise function.
Now let’s look at the middle portion.
This simply says the value of the function is 4, if x is equal to 0.
This is a single point that has coordinates (0,4).
The third portion of this function is defined by a linear rule.
Notice this one has two different endpoints,
so I want to make sure that I get the whole portion of the line
between those endpoints and including this one.
So let’s see. Let’s take x equal to negative 3.
That is going to be the leftmost point on our function on our graph.
If x is equal to negative 3, we are going to get negative 2 times negative 3.
That is positive 6. Okay, so let’s plot this point.
Alright, now I have decided that is the leftmost point on the graph.
I know that the line or the portion of the line is over here,
and I even know that it has a negative slope. But I need to know where it goes.
Exactly what does it do over there?
So what I am going to do is look at what happens at the endpoint.
Now I know that the endpoint is not included, so I am going to
take care of that on my graph by using an open circle again.
If x were equal to 0, this would be 0. That means I would have the point (0,0),
again it is not included so it is still going to stay this open circle,
and the portion of the line I have will be right here.
Now once we have the graph, we should be able to find the range and the intercepts.
Okay so let’s look at the range. This is values that y is allowed to have.
We are going to look for the smallest y - value. Well it is very close to 0
but not equal to 0, and then y can be anything bigger up to 6 on this side.
But look at this one. This one goes up forever and ever, so actually the range would be
the interval that goes from 0 to infinity, without including the 0.
Finally, the intercepts for this graph.
There is a y - intercept at 4 since there is a point that touches the y - axis there,
and the graph actually never touches or crosses the x - axis,
There is an open circle there, that point is not included so there is no x - intercept.
Continue to practice with piecewise functions until you can graph them comfortably
and answer all of these questions.
Now let's look some more at our basic functions.
We would like to be able to graph more complicated functions based on our basic functions.
We do this using transformations of the graphs. Let’s try a couple of examples.
Notice this one. Let’s read it first. y equal x plus 1 cubed, minus 2.
I do see one of our basic functions in there. It is a cubed function,
and the first thing I would like to do is draw y equals x cubed.
You remember what it looks like, right? Okay.
It has the point (0,0). It contains the point (1,1), the point (negative 1, negative 1).
Then I am going to go ahead and label this y equals x cubed
to remind us that that is our basic function,
and we are going to be working with this one
and doing some transformations. What kind of transformations?
Well let’s start right here. This is x plus 1 cubed, so you will need to think back.
Yes that moves this horizontally. In fact it is going to go to the left one unit.
Okay let’s remember that. We need to move this to the left one unit.
If you look at this other number, this minus 2 is sort of outside of that basic function.
This tells us whether to move the graph up or down.
Because it is negative 2, we need to move the graph down two units.
If you are very careful you can go ahead and do these at the same time in this problem.
So we are going to take this graph y equals x cubed,
take each point we have located and move it to the left one unit and down two.
Let's start with this one that is at the origin.
To the left one and down two will put that here.
Let’s take this one, to the left one unit and down two will put that one here.
Then finally the third point started out here.
If we go to the left one unit and down two.
Now we have to keep the shape of the same
because all we are doing is sort of sliding this one around.
This one is y equals x plus 1 cubed, minus 2.
Our second example is y equals 3x squared,
and the first thing we want to notice is that
it does involve one of our basic functions x squared.
Do you remember what x squared looks like? Right.
It is the parabola. It is going to have the points (0,0), (1,1),
and (negative 1,1), and remember it is this parabola shape.
Let’s label this y equals x squared. This is the basic function we are dealing with.
Now, what about this 3 that is multiplied in front of the x squared?
What does that do to the graph? Right, it stretches it.
It is going to make the y - coordinates grow faster for a given x - value.
So if I want to graph y equals 3x squared and I know this,
one way to approach that is to look at the y - coordinates of the points that I know
and multiply them by 3 and get our new y - coordinates. Let’s try that.
Remember, the x - coordinates are going to stay the same,
because for a given x - value we are just stretching it.
It is growing faster in this case.
Again I want to start with this point at the bottom of the parabola.
It has a y-coordinate of 0, and when I multiply that y - coordinate 0 by 3. It stays 0,
so this point does not move. I like to think of that as kind of the anchor in this case.
Now let’s look at this point. The coordinates of this point are (1,1).
The y-coordinate is 1, that is what is important,
and when I multiply that y-coordinate by 3
I get the new value for 3x squared. 1 times 3 is 3.
I have a new point, keeping the x - coordinate 1
but I have made the y - value three times bigger.
Okay, do that one more time. The y - coordinate for this point is at negative 1.
No it is not, the y - coordinate is 1. If I take that y - coordinate 1,
multiply it by the 3, I get 3 again. Keep the x the same. Plot the point (negative 1,3).
What I want you to notice is that it is still parabola shaped,
but it looks like someone stood at the top and pulled. It is stretched.
Let’s try a few more examples.
The next pair of examples deals with reflections of our basic graphs.
Let's look at this first one. Alright, notice how it's out in front, it is
like a negative 1 multiplied by our basic
function: the absolute value of x.
So, what effects should that have on the
graph of the absolute value of x?
Well, it's going to be negative 1 times the value of this basic function.
That's the y - value, so we're going to
multiply the y - values by negative 1 to get our new graph.
What does that look like when you do it?
Right, it just reflects it across the x - axis.
So, let's go through the steps and graph this one.
The basic function is the absolute value of x.
Do you remember what it looks like?
Good, it has the point (0,0), the point (1,1), and the point (negative 1,1).
Now the absolute value of x, that's the one
that's more V-shaped.
It's made of pieces of two straight lines.
So, there's your absolute value of x.
Okay now, we talked about how the negative
in the front is multiplying the
y - coordinates by negative 1.
It's going to reflect this across the x - axis.
What's going to happen to this point at the bottom of the V?
Right, nothing because the y - coordinate is 0,
and multiplying that by negative 1 is still 0.
Not only that, if you're reflecting
it across the x - axis, it's on the x - axis. It doesn't move.
What about this point here, it has the coordinates (1,1).
It's got to come down here, so the x doesn't change.
I want it to just be a reflection.
The y - coordinate needs to be the negative of what it was.
Okay and let's draw this piece.
Notice how it looks just like this reflected across the x - axis.
Same thing is going to happen on the other side.
This has a y - coordinate of 1.
We're going to make it negative because of this.
It's going to give you a new point at (negative 1, negative 1).
You can draw the other piece. This is y equals negative absolute value of x.
It's just a reflection of our basic function across the x - axis.
One more: I bet you're worried about the negative under the radical.
Well don't worry, we can take care of that.
It's not nearly as big a problem as you might think. What does it do?
If you think about the square root of x,
our basic function, we needed to use x - values that are positive.
This changes all of that because now, if I
put in a positive number for x, let's try 2,
I'd get the square root of negative 2. We can't do that.
It's not a real number, right.
But on the other hand, if I pick a
negative number now, like maybe I don't know
negative 2, you get the square root of a negative, negative 2.
That's the square root of positive 2, and this is good.
This is a real number, something we can handle okay.
If you think about the domain of this,
instead of 0 and positive numbers, I need to
use negative numbers and maybe 0. So what does all that mean?
It means that this negative in front of the
x takes the graph of the square root of x,
and it's going to reflect it across the y - axis.
Let's make this clear by actually drawing those graphs.
What does the square root of x look like?
Okay, it has a point at (0,0). It has a point at (1,1), and I'll put a third
point: the square root of 4 is 2.
Okay alright, so now we want to take care of this reflection.
As I said, it's a reflection across the
y - axis, so let's start with this point.
It doesn't move, it's already on the y - axis.
So when you reflect it, it stays where it is.
This point needs to come reflect over to here,
so where will its coordinates be?
Well, it's going to still have a
y - coordinate at 1, but it's x - coordinate
needs to be a negative 1, and then one more point.
This has an x - coordinate of 4, a y - coordinate of 2.
The y - coordinate doesn't change when it's reflected across the y - axis.
The x - coordinate does. It becomes negative, so this is now (negative 4,2).
Take a look at the difference in these two examples.
Here the negative is in front of, outside
of our basic function, and that shows you
it is a reflection across the x - axis.
Here, the x itself inside is negative, and
that is the one that reflects across the y - axis.
This next pair of examples asks us a totally
different question, though very much related to what we have just done.
Here you are given the graph, and the
instructions are to choose which of these is
the function that matches the graph.
Okay so we are going to do this by process of elimination.
The first thing I need to do is figure out which basic function is involved.
Okay so let's look at this one.
Which basic function looks like this, has this basic shape?
It is a tricky one because you actually have two possibilities.
Is it x cubed which looks roughly like this, or is it the cube root of x?
Remember that's the one that looks like it's
kind of turned on its side compared to that.
You're right, so this is the one we want to use.
This is definitely a cube root of x function.
Okay so we can eliminate some things right away.
Let's just cross these out. These have x cubed in them. They're not right.
Now, let's look for something else.
Alright it doesn't look like it's reflected, right?
It still has the same, it's still increasing,
still increasing, so it's not reflected.
That doesn't really help me any, so let's look for something else.
This one, right, has sort of this anchor point at (0,0), at the origin.
This one does not so. Let's see if we can figure out where
this point, how has it been moved?
If we started out at the origin, then we go to the right 2 units and up 1.
Okay the up 1 is the easier part to use, so let's look at these.
Up 1, I need to see the plus 1 out here, and I do.
Again it doesn't help me much, so I'm forced to think about is it moved left or
right, and how does that look in the equation or in the function?
Okay, so we said it's moved to the right 2.
Would I see a plus 2 in the function or a minus 2 in the function?
What do you think? Right, minus 2.
You want to be careful with that. It's kind of tricky.
Let's try another one. What basic function do you see here?
The overall shape looks to me like it's a parabola. It is y equals x squared
So, the basic function is y equals x squared.
All of these have an x squared in them, so that doesn't help me eliminate anything,
but it's a good start. One thing I notice right away is it's flipped over.
It looks like it's been reflected across the x - axis.
Well all of those also were reflected across the x - axis, so I need to keep looking.
This point is now at the top of the parabola since it's been reflected.
Usually it would be at (0, 0), so this one's been moved up 3 units.
Which of these looks like it might have been moved up 3 units?
This one? No, no, no. It's inside, so this is actually moved to the left 3.
That's wrong. Good, we can eliminate something.
This one has this plus 3. That's a possibility.
Let's skip C for just a minute.
Let's look at D: plus 3 out there, that's good.
That's been moved up 3. These are both good choices.
Now, what about C? Ah yes, you're paying attention.
This isn't really written in the right form, is it?
This is really negative 2 x squared minus 6.
That should go down 6 units, not up 3. C is out.
Okay, I've narrowed it down to B and D.
They look remarkably similar, except what? Except this 2.
There is no 2 over here, and what is it that 2 does?
Right, it stretches the graph.
Okay, so all we have to decide now is:
is this graph stretched, or is it a regular x squared graph? What do you think?
I agree. I think it's stretched too because if we look at y equals x squared,
look at how these other points are positioned relative to the bottom of the vertex.
It's 1 unit to the right and 1 unit up. Now I know it's reflected,
but here it's 1 unit to the right and 2 down.
That's stretched. So my answer has to be B.