g(x) = x^4-8x^3-16x^2+128x
the graph goes through the x through
(-4,0) , (0,0) (4,0), (8,0)
horizontal axis has a scale of 2,4,6,8,10 (each tick increases by 2)
Y-axis value that is the highest is (-4,30) and (8,30)
Assume
g(x) = k(x-p)(x-q)(x-r)(x-s) p<q<r<s
I need help finding K and P please.
k = ?
p = ?
q = 0
r = 4
s = 8
IF you need more information let me know please.
you have the zeroes.
the only thinking is to put them in order.
equation will be
g(x)=k(x+4)(x)((x-4)(x-8)
changing order to make s>r>q>p
g(x)=k(x+4)(x)(x-4)(x-8)
so that give you pqrs Now on k, I don't understand. You say it has zeros at 8, and -4, but then you staed it is Highest at those points...I dunno at this point on k.
never mind forget how i put (-4,30) and and the (8,30) bit... Forget i said that. I just meant to say the highest endpoints you can see before the graph cuts off is those points
To find the values of k and p, we can use the given information about the graph of the function g(x) = x^4 - 8x^3 - 16x^2 + 128x.
1. The graph goes through the x-intercepts (-4, 0), (0, 0), (4, 0), and (8, 0):
The given x-intercepts help us find the values of p, q, r, and s. Since the roots of the function correspond to the x-intercepts, we know that p, q, r, and s are the values of x when g(x) = 0. In this case, the values are p = -4, q = 0, r = 4, and s = 8.
2. The y-axis value that is highest is (x, 30):
To find the value of k, we can use the point (x, 30) where x = -4 or x = 8, as mentioned. We substitute the x-values and the corresponding y-value into the equation g(x) = k(x-p)(x-q)(x-r)(x-s):
For x = -4:
0 = k(-4 - p)(-4 - 0)(-4 - 4)(-4 - 8)
Since the expression on the right side is multiplied by k, we can divide both sides of the equation by the product of the four differences:
0 = k(-4 - p)(-4)(-8)(-12)
For x = 8:
0 = k(8 - p)(8 - 0)(8 - 4)(8 - 8)
Similarly, we divide both sides by the product of the four differences:
0 = k(8 - p)(8)(4)(0)
From these equations, we can see that the values of k(-4 - p) and k(8 - p) must be zero for g(x) to be zero at x = -4 and x = 8, respectively.
To find the value of k, we need to recognize that for g(x) = 0, the function must equal zero for any value of x. Therefore, for g(x) to be true, k must be 0 as well.
So, we have:
k = 0.
For p, since k(-4 - p) = 0, we can have any value for p, including p = -4. However, since g(x) = x^4 - 8x^3 - 16x^2 + 128x, it is common to express g(x) with the roots in ascending order. Therefore, the simplified form would have p = -4.
So, the values are:
k = 0
p = -4
q = 0
r = 4
s = 8.