f(x) = a(x+5)(x-2)(x-4)
f(-10) = a(-5)(-12)(-14) = -840a = -4
a = 1/210
f(x) = 1/210 (x+5)(x-2)(x-4)
Given that f(x) is a cubic function with zeros at −5, 2, and 4, find an equation for f(x) given that f(−10)=−4.
6 answers
ty!
A cubic function is a function of the form:
f ( x ) = a ^ 3 + b x ^ 2 + c x + d
You must find 4 coefficients:
a , b , c , and d
You now zero pionts:
x = - 5 , y = 0
x = 2 , y = 0
and
x = 4 , y = 0
You also know :
f ( - 10 ) = - 4
Now put this values in equation:
f ( x ) = a x ^ 3 + b x ^ 2 + c x + d
For x = - 5
a x ^ 3 + b x ^ 2 + c x + d = a * ( - 5 ) ^ 3 + b * ( - 5 ) ^ 2 + c * ( - 5 ) + d = 0
a * ( - 125 ) + b * 25 + c * ( - 5 ) + d = 0
- 125 a + 25 b - 5 c + d = 0
For x = 2
a x ^ 3 + b x ^ 2 + c x + d = a * 2 ^ 3 + b * 2 ^ 2 + c * 2 + d = 0
8 a + 4 b + 2 c + d = 0
For x = 4
a x ^ 3 + b x ^ 2 + c x + d = a * 4 ^ 3 + b * 4 ^ 2 + c * 4 + d = 0
64 a + 16 b + 4 c + d = 0
For x = - 10
a x ^ 3 + b x ^ 2 + c x + d = a * ( - 10 ) ^ 3 + b * ( - 10 ) ^ 2 + c * ( - 10 ) + d = - 4
a * ( - 1000 ) + b * 100 + c * ( - 10 ) + d = - 4
- 1000 a + 100 b - 10 c + d = - 4
Now you have system of 4 equaions with 4 unknow:
- 125 a + 25 b - 5 c + d = 0
8 a + 4 b + 2 c + d = 0
64 a + 16 b + 4 c + d = 0
- 1000 a + 100 b - 10 c + d = - 4
The soluitions are :
a = 1 / 210 , b = - 1 / 210 , c = - 11 / 105 , d = 4 / 21
Your equation:
f ( x ) = ( 1 / 210 ) x ^ 3 - ( 1 / 210 ) x ^ 2 - ( 11 / 105 ) x + 4 / 21
Or:
f ( x ) = ( 1 / 210 ) x ^ 3 - ( 1 / 210 ) x ^ 2 - 2 * 11 / 210 x + 40 / 210
f ( x ) = ( 1 / 210 ) ( x ^ 3 - x ^ 2 - 22 x + 40 )
f ( x ) = a ^ 3 + b x ^ 2 + c x + d
You must find 4 coefficients:
a , b , c , and d
You now zero pionts:
x = - 5 , y = 0
x = 2 , y = 0
and
x = 4 , y = 0
You also know :
f ( - 10 ) = - 4
Now put this values in equation:
f ( x ) = a x ^ 3 + b x ^ 2 + c x + d
For x = - 5
a x ^ 3 + b x ^ 2 + c x + d = a * ( - 5 ) ^ 3 + b * ( - 5 ) ^ 2 + c * ( - 5 ) + d = 0
a * ( - 125 ) + b * 25 + c * ( - 5 ) + d = 0
- 125 a + 25 b - 5 c + d = 0
For x = 2
a x ^ 3 + b x ^ 2 + c x + d = a * 2 ^ 3 + b * 2 ^ 2 + c * 2 + d = 0
8 a + 4 b + 2 c + d = 0
For x = 4
a x ^ 3 + b x ^ 2 + c x + d = a * 4 ^ 3 + b * 4 ^ 2 + c * 4 + d = 0
64 a + 16 b + 4 c + d = 0
For x = - 10
a x ^ 3 + b x ^ 2 + c x + d = a * ( - 10 ) ^ 3 + b * ( - 10 ) ^ 2 + c * ( - 10 ) + d = - 4
a * ( - 1000 ) + b * 100 + c * ( - 10 ) + d = - 4
- 1000 a + 100 b - 10 c + d = - 4
Now you have system of 4 equaions with 4 unknow:
- 125 a + 25 b - 5 c + d = 0
8 a + 4 b + 2 c + d = 0
64 a + 16 b + 4 c + d = 0
- 1000 a + 100 b - 10 c + d = - 4
The soluitions are :
a = 1 / 210 , b = - 1 / 210 , c = - 11 / 105 , d = 4 / 21
Your equation:
f ( x ) = ( 1 / 210 ) x ^ 3 - ( 1 / 210 ) x ^ 2 - ( 11 / 105 ) x + 4 / 21
Or:
f ( x ) = ( 1 / 210 ) x ^ 3 - ( 1 / 210 ) x ^ 2 - 2 * 11 / 210 x + 40 / 210
f ( x ) = ( 1 / 210 ) ( x ^ 3 - x ^ 2 - 22 x + 40 )
that you!
thank*
how the hell do you do this.
2 answers