Graphing Square Root Functions

Objective

• Review over how to transform quadratic functions
• Go over how to graph the square root function
• Go over how to transform the square root function by stretching
• Go over how to transform the square root function by moving left/right
• Go over how to transform the square root function by moving up/down

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Graphing functions

So we know the basics of a graph right?

But what if you’re asked to skew the graph some?

For example:

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But what if we want to skew this graph to the left three slots?

What about to the right 3 slots?

What about up 3 slots?

Down 3 slots?

SO HOW DO WE DO IT?

MOVING THE GRAPH UP

Let’s look at a quick x-y chart to see what this means.

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X Y

0

3

1

2

3

4

7

12

So what’s the graph look like?

MOVING THE GRAPH DOWN

0

-3

1

2

3

-2

1

6

SO WHAT OTHER WAYS CAN WE MOVE THE GRAPH?

So moving the graph up and down was pretty simple, but what about moving the graph left or right?

This gets a little more complicated, so let’s experiment.

First off, we know that y goes up and down, so adding or subtracting something from y is pretty easy.

However, x is the variable that determines left to right, and we can’t just add something to the equation because that’s how we move up and down.

So, any ideas?

Well, let’s just try a few things and see what happens

MOVING THE GRAPH HORIZONTALLY

MOVING THE GRAPH LEFT

-3

0

-2

-1

0

1

4

9

That was different.

So then, what’s the graph look like?

So, why did we pick -3 as our first point and not 0?

MOVING THE GRAPH TO THE RIGHT

3

0

2

1

0

1

4

9

It makes sense, so what does the graph look like?

So as we can see, it did shift to the right 3 slots.

But that’s weird, why is it that subtracting from x moves the graph to the right,

but adding moves it to the left?

It almost seems like it’s an inverse operation.

Shrinking a parabola

0

0

1

2

3

2

8

18

So our graph changes from:

To:

STRETCHING A PARABOLA

0

0

2

4

6

2

8

18

Again, we went from this:

To this:

So how do we graph square root functions?

Well, to be honest, these types of graphs are basically the opposite of the quadratic functions.

Of course, just like the other graphs, one of the easiest ways to graph it is to look at the original square root function, and then try to find some ways to manipulate it so we don’t have to work so hard.

So, to start off, we need to original function.

So, to start off, let’s start like we would with any graph.

We pick a point.

0

So, when x = 0, then:

0

One of the easiest points to work with is 0, so:

Now, let’s try 1

So, when x = 1, then:

1

1

Now, let’s try 4

4

So, when x = 4, then:

2

Now, let’s try 9

So, when x = 9, then:

9

3

Alright, great. So, what now?

Well, now there are a few things we can do with this graph.

We can: stretch this graph

We can move the graph left or right

And finally we can move the graph up or down.

So, let’s look at some examples of how to do all of these things.

Stretching the graph:

So, stretching the graph is very similar to how we stretch a quadratic.

So, to start, let’s look at a quick example:

Now, let’s look at the graph:

So, obviously this graph looks like it’s been stretched up some.

But how do we know by how much?

Well, it’s because, each point on the x axis is being pushed twice as far as the point on the y-axis.

I know this may sound confusing, but looking at the original graph we can see it:

As you can see, we can tell that this graph is skinnier, or in other words, it’s stretched.

So what if we’re just given the graph, how can we tell what the stretch is?

Well, we look at the two closest points to the center of the graph.

However far away they are from the center, that’s the stretch.

For example:

2

1

So the stretch is 2/1, or 2.

But:

1

1

So the stretch is 1/1, or 1.

MOVING THE GRAPH TO THE LEFT OR RIGHT

So, to move the graph left or right, we do the exact same as if it’s a quadratic.

For example:

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The graph of this looks like:

As we can see, the graph is the exact same, but moved 2 units to the right.

Now let’s look at another example:

As we can see, the graph is moved to the left 2 units now.

So, just like with quadratics, the graph will move the opposite way that the number is.

So, if the number is negative, the graph moves to the right.

If the number is positive, the graph moves to the left.

MOVING THE GRAPH TO UP OR DOWN

Lastly, to move the graph up or down, we do the exact same as if it’s a quadratic.

For example:

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The graph of this looks like:

As we can see, the graph is the exact same, but moved 1 unit up.

Now let’s look at another example:

As we can see, the graph is moved down 1 unit.

So, just like with quadratics, the graph will move in whatever direction the number’s sign is.

So, if the number is negative, the graph moves down.

If the number is positive, the graph moves up.

SO THAT’S HOW SQUARE ROOT FUNCTIONS WORK

So now let’s look at some examples of all of these different types of transformations mixed together.

Example 1

Find the equation from the following graph:

So, as we can see, the graph seems like it was moved to the right 3 units.

Which means our equation starts with:

We can also see, the graph looks like it went down 2 units.

Adding to our equation, we can see that:

Finally, we can see that from the first point:

To the middle point:

The graph seems to go up 2, and over 1:

2

1

So, our equation looks like:

Example 2

Find the equation from the following graph:

So, as we can see, the graph seems like it was moved to the left 2 units.

Which means our equation starts with:

We can also see, the graph looks like it went down 1 unit.

Adding to our equation, we can see that:

Finally, we can see that from the first point:

To the middle point:

The graph seems to go up 4, and over 1:

So, our equation looks like:

4

1