P. 380. TThe 3 x-intercepts (-2,0), (1,0), and (3,0) will give you the 3 zeros of the cubic. They will also tell you 3 factors to use f(x)=a(x+2)(x-

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Presentation transcript:

p. 380

TThe 3 x-intercepts (-2,0), (1,0), and (3,0) will give you the 3 zeros of the cubic. They will also tell you 3 factors to use f(x)=a(x+2)(x- 1)(x-3). TThen use the 4 th point as x & f(x) values. 2=a(0+2)(0-1)(0-3) NNow solve for a! 2=6a so, a= 1 / 3 AAnswer: f(x)= 1 / 3 (x+2)(x-1)(x-3)

Ex: An eqn. for a polynomial function is f(n)=2n 3 +n 2 +2n+1. Show that this function has constant 3 rd order differences. (you check the 3 rd order diffs. because it’s a degree 3 polynomial)  First, write out the first several values; or find f(1), f(2), f(3),…, f(6).  f(1)=6 f(2)=25 f(3)=70 f(4)=153 f(5)=286 f(6)=481 Now subtract #s! (left from right) 1 st diffs Now subtract #s! (left from right) 2 nd diffs Now subtract #s! (left from right) 3 rd diffs ** This is called using finite differences.

 First, find finite differences. (Stop when the same number repeats all the way across!) The 2 nd differences are now a constant # across. This means the function will be a quadratic. (degree 2) So, use f(n)=an 2 +bn+c. Since you must find a, b, & c, you will need to make 3 eqns. with these 3 variables using the first 3 known values of the function.

a(1) 2 +b(1)+c= -2a+b+c= -2 a(2) 2 +b(2)+c=24a+2b+c=2 a(3) 2 +b(3)+c=129a+3b+c=12 ** Look familiar? It should! ** * Use inverse matrices to solve for a, b, &c! * This means the quadratic is f(n)=3n 2 -5n+0 or f(n)=3n 2 -5n

 f(1)= -2, f(2)=2, f(3)=12, f(4)=28, f(5)=50, and f(6)=78  1. [STAT] [1] this is the edit key  2. enter in all the x values in L1  3.enter in all the y values in L2  4.[STAT] arrow to CALC  5. Find the regression you need. ***Same as the finite order difference you found. ◦ linReg (ax +b) is linear degree 1 ◦ QuadReg is quadradic degree 2 ◦ CubicReg is cubic degree 3 ◦ QuartReg is quartic degree Write the equation in standard form

 When given special points ◦ – the x intercepts- are the special points  Writing the equation in factored form is the fastest. F(x) = a (x-p)(x – q) …  Substitute in x and y the xintercepts known as the zero’s and solve for a.  Last step leave in F(x) = a (x-p)(x – q) …

 Given many points not the x intercepts.  The answer may be in factored form in the book.  No Worries. They are equivalent!!!!  1.Find the finite difference.  2. Then use the calculator.  STAT CALC  It’s the fastest.  And it is in Standard form.

Don’t Forget you can use rref in the math for matrices as well.