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Help me sove this and graph it x^4-4x^2+3

How do I solve so I can graph this? Should i set up an x_Y TABLE AND use -3... on for x? Sorry I missed the x^4th in the equation. How would I solve this with a graphing calculator?


Lori -

Missing the exponent on the first term. x^? - 4x^2 + 3 ???

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3 Answers

Lori, I have one more way to "tackle" your problem. Has your class been introduced to Synthetic Division yet? If so, then you can line up the coefficients while trying the possible solution of x=1 (Factor of ( X-1)) 1/ 1 0 -4 0 3 You begin by bringing down the first column and 1 1 -3 -3 "synthesize" with multiplications & additions. 1 1 -3 -3 0 When you reach zero, you have a root! Now, you have the factors (X-1)and (X3 + X2 -4X -3) and you can try another root...say X=-1 (Footnote: I used the exponents behind the variable) -1/ 1 1 -3 -3 Continue the process until you can manage the -1 0 +3 remaining roots. Each bottom number comes 1 0 -3 0 from a multiplication & addition step. Now, the remaining factor is X2 - 3 which leads to X = sqrt(3).plus or minus Your factors would be (X-1)(X+1)(X2 - 3)= 0 This leads to X=1, -1, +- sqrt(3). I just wish I had seen your problem sooner...sorry! Charles S. 3/26/2013

Sorry Kevin, but the function " y = x^4 - 4x^2 + 3 " is not always positive! Another way to solve this kind of problem is: let's assume that z=x^2, then original function become a quadratic one y =  z^2 - 4z + 3. Next step is to find the zeros of function, the values of x where graph touch or intercept the x-axis, so the value of y will be zero. So, we have to solve equation z^2 - 4z + 3 = 0 . {The easier way to find the roots of ax^2+bx+c=0, I believe, is using the formula x = (-b"+"or"-" square root from (b^2-4ac))/2a} . In our equation variable is "z", a=1, b=-4, c=3 . Substitute all letters by given numbers and we have z=1 and z=3 . But z=x^2 , then x^2=1 and x^2=3. So we have max possible number of roots - four, which are negative square root from 3, -1, 1, positive square root from 3. Let's plot all those coordinates onto x-axis, they divide the x-axis onto 5 parts: 1. (negative infinity , 1.73] and line of graph of original function is approaching the x-axis, crossing it in -1.73 , going down, making "U-turn" at the midpoint of a segment [-1.73 , -1], going up till point (0,3), intercept the y-axis at 3, at the same way making "U-turn" there and going down, intercept point 1, still going down till midpoint of segment [1,1.73], one more "U-turn" and going up to Universe :-) through the 1.73 ...........


Assuming it is x^4 - 4x^2 + 3, factor the expression:

(x^2 - 1) (x^2 - 3) = (x + 1)(x - 1)(x^2 - 3)

You then have vertices at x = -1, x = 1, and x^2 = 3

Since the equation is positive, you will have the graph opening upwards, with 3 vertices. Plot your points and graph If necessary, you may plot additional points to assist with defining the graph