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.
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)
ty!
that you!
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how the hell do you do this.
To find an equation for the cubic function f(x), we need to know that the zeros give us the factors of the function. In this case, the zeros are -5, 2, and 4, so the factors of the cubic function are (x + 5), (x - 2), and (x - 4).
The general form of a cubic function is f(x) = a(x - r)(x - s)(x - t), where "a" is the leading coefficient and "r", "s", and "t" are the zeros.
We already determined the factors, so we can set up the equation as follows:
f(x) = a(x + 5)(x - 2)(x - 4)
Now, to find the value of "a", we can use the point (-10, -4) which gives us the equation:
-4 = a(-10 + 5)(-10 - 2)(-10 - 4)
Simplifying this equation:
-4 = a(-5)(-12)(-14)
-4 = 840a
Solving for "a":
a = -4/840 = -1/210
Therefore, the equation for f(x) is:
f(x) = (-1/210)(x + 5)(x - 2)(x - 4)
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 )