## Polynomial functions: finding equation from graph, etc.

Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.
honest_denverco09
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### Polynomial functions: finding equation from graph, etc.

I have two questions :
1/ Please help me write a quadratic function whose graph has only one x-intercept, -4 and whose y-intercept, -8 in factored form. Thank a lot.
2/ If we have a graph of polynomial function, how can we write its equation ? Actually, if i have a graph of cubic or 4th-degree polynomial function, how can i know the sign of their leading coefficient, whether positive or negative ?
For EX : I have two following graphs :
Two is 4th-degree function :

How can i write these equations ?
Thanks a lot!

stapel_eliz
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I'm afraid I don't know what is meant by "the y-intercept 'in factored form'". The y-intercept is where the graph crosses the y-axis; "factoring" is not involved.

To learn how to find quadratics from their zeroes, try here. Thinking back to what you learned when you were factoring, solving, and graphing quadratics, you know that "one x-intercept" for a quadratic means one repeated root -- that is, there is one root that occurs twice -- and the graph touches the x-axis at the one zero (rather than passing through the axis).
2/ If we have a graph of polynomial function, how can we write its equation ? Actually, if i have a graph of cubic or 4th-degree polynomial function, how can i know the sign of their leading coefficient, whether positive or negative ?
To learn about polynomial behavior and the relationships between equations and graphs, try here.
For EX : I have two following graphs :
Two is 4th-degree function :

How can i write these equations ?
Once you've studied the lesson on polynomial behavior, you will understand why it is obvious that neither of these pictures displays a quadratic or a quartic polynomial!

The first graph is a positive odd-degree polynomial, probably a cubic. The second graph is a negative odd-degree polynomial, possibly of degree five.

honest_denverco09
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### Re: Polynomial functions: finding equation from graph, etc.

Thank a lot!

I can not open the links that you sent to me, but from your answer, i guess so :

If the graph of quadratic or 4th-degree functions that have the start-point below the x-axis, their leading coefficients will be positive, and if they have the start-point above the x-axis, their leading coefficients will be negative.
For EX in these pictures, the first graph has the positive "a" and the second one has the negative "a".
I just guess so, is that true a little bit ?

stapel_eliz
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I can not open the links that you sent to me....
I'm sorry to hear there was some difficulty. The links are working now.

Please review the material, keeping in mind what you already know about graphing linear functions (degree one, and thus odd) and graphing quadratic functions (degree two, and thus even), as they are strongly indicative of what you will encounter for end-behavior in all even and odd polynomial functions.

honest_denverco09
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### Re: Polynomial functions: finding equation from graph, etc.

Thank a lot!
Last edited by honest_denverco09 on Thu Apr 23, 2009 11:20 pm, edited 1 time in total.

honest_denverco09
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Joined: Tue Mar 31, 2009 2:06 am
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### Re: Polynomial functions: finding equation from graph, etc.

Here is the answer for my problem :

1/ We have x - intercept is only -4 , that means the vertex of this function's graph will be (-4, 0), and this x-intercept or this zero will be Double root.
As the theme, this polynomial function is a quadratic one, so it has two "square" (two zeros) or it is in form a(x - r1)(x - r2)
So, The equation will be y = a(x + 4)^2(x - r2)
We also have y-intercept is -8 , that means : The graph will be downward/the leading coefficient will be negative.
The point (0, -8) will be on the graph.
So far, we have : -8 = a(0 + 4)^2 and then solve for a, we'll get a = -2
Eventually, our equation is y = -2(x+4)^2

2/For this graph, i will do like this :

When we have a graph of any polynomial function, first, we need to identify the sign of "a", leading coefficient whether positive or negative by looking at its graph. In the second picture, the "a" is negative.
Then, we need to realize what the degree of this function. In this case, the degree is 5.
Set up the parenthesis based on the degree and the graph. In this case, we will have y = -()()^3()
Seek for the x-intercepts to fill out the set above. In this case, we have x-intercepts are -5, -2, 1 .So, y = -(x+5)(x+2)^3(x-1)