Find an expression for a cubic function f if f(1)=8 and f(-1)=f(0)=f(2)=0
the roots indicate that
f(x) = a(x+1)(x)(x-2)
Since f(1) = a(2)(1)(-1) = -2a = 8, a = -4
I figured it out already. :)
To find the expression for a cubic function given some conditions, we can use the fact that a cubic function has the general form: f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
Given that f(1) = 8, we can substitute these values into the equation to get an equation with one unknown:
f(1) = a(1)^3 + b(1)^2 + c(1) + d = 8
a + b + c + d = 8 ---(Equation 1)
Given that f(-1) = f(0) = f(2) = 0, we can substitute these values into the equation and get three more equations:
f(-1) = a(-1)^3 + b(-1)^2 + c(-1) + d = 0
-a + b - c + d = 0 ---(Equation 2)
f(0) = a(0)^3 + b(0)^2 + c(0) + d = 0
d = 0 ---(Equation 3)
f(2) = a(2)^3 + b(2)^2 + c(2) + d = 0
8a + 4b + 2c + d = 0 ---(Equation 4)
Now, we have a system of four equations (Equations 1, 2, 3, and 4) with four unknowns (a, b, c, and d). We can solve this system to find the values of a, b, c, and d.
First, substitute Equation 3 into Equation 1:
a + b + c + 0 = 8
a + b + c = 8 ---(Equation 5)
Next, substitute Equation 3 into Equation 2:
-a + b - c + 0 = 0
-a + b - c = 0 ---(Equation 6)
Now, we have a system of equations (Equations 5 and 6) with three unknowns (a, b, and c).
Subtract Equation 6 from Equation 5:
a + b + c - (-a + b - c) = 8 - 0
2a = 8
a = 4
Substitute the value of a = 4 into either Equation 5 or Equation 6 to solve for b and c.
Let's use Equation 5:
4 + b + c = 8
b + c = 8 - 4 = 4
Now, we have:
a = 4
b + c = 4
We can choose any value for b and solve for c:
Let's choose b = 2:
2 + c = 4
c = 4 - 2 = 2
Therefore, we have:
a = 4
b = 2
c = 2
Finally, we substitute these values of a, b, c, and d = 0 into the general form of a cubic function:
f(x) = ax^3 + bx^2 + cx + d
f(x) = 4x^3 + 2x^2 + 2x + 0
f(x) = 4x^3 + 2x^2 + 2x
So, the expression for the cubic function f is f(x) = 4x^3 + 2x^2 + 2x.