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Alright here is a good example, it's pretty tough so I wanted to do an example how to find an equation of the graph
we have information like this.
What I have, is that I have the zeros.
And I have a point on the graph.
So this is how you find the equation of this graph with that given information.
Step one, state the zeros and their multiplicities.
-4, and what is the multiplicity at that zero, at least a three, right?
It could be five, seven, or any odd multiplicity but we are going to go with the smallest one.
The next zero is 0. The multiplicity at zero is straight through so it's 1.
The next zero is 2. The multiplicity is a 1 again.
So what is the degree of this polynomial? Well there are three zeros there, 1 zero there and 1 zero there so it is a fifth degree polynomial.
Does it look like a fifth.
Starts up and down it has only two turning points, but it has three zeros at this -4. So yes it looks like a fifth degree.
Write the factored form.
Start with the function notation F (X).
I don't know the leading coefficient so I put “a”.
But I do know the factors and their multiplicities. (X +4) is a factor because -4 is a zero. With a multiplicity of 3.
Then I know X = 0, so then I can put a X outfront. Multiplicity one.
Now for 2 being a zero and multiplicity one.
(X -2).
So here are my zeros. 0. -4. 2.
This is factored form. The only thing I don't know is this number a.
How do I find the number a?
I use the point that it's a given on the graph.
It has an X value and a Y value.
So X = 1 and Y = 5.
I have X's and I have a Y. So, here we go.
The Y is going to be a five.
And then every time I see a X I will replace it with a will 1.
So all of this in red will stay, and now here is where the one will be put in.
Every time there was a X there is a one. Every time there was a Y, there is a five.
There is only one variable to solve for. This is all multiplied together on the right.
See screen.
5 cubed is equal to 125.
Multiply that through, see screen.
Divide both sides by -125.
A = -1/25. That's my leading coefficient.
That's this guy right there.
So I erase the A, and write in -1/25.
That is the function notation of the graph.
If you put this in your calculator you will get this graph shown here.
It will go through the point (1, 5).
In the homework you probably have some examples like that.
Working through the same way. Start with the zeros, go to the multiplicity, and then write the factors. Solve for the missing value letter a.