Determine the equation of a cubic function with zeros 2 √3 and -1 whose graph passes through the point (2, -18)
y = a(x-2)(x-√3)(x+1)
since y(2) = -18, this is impossible. So maybe you meant the zeroes are
2±√3 and -1. In that case,
y = a(x-(2-√3))(x-(2+√3))(x+1) = a(x^3 - 3x^2 - 3x + 1)
since y(2) = -18, that means that
a(-9) = -18
so a = 2 and y = 2(x^3 - 3x^2 - 3x + 1)
To find the equation of a cubic function with given zeros and a point it passes through, we can use the factored form of the function.
Given zeros: 2 √3 and -1
Since the zeros are given, we know that the factors of the cubic function are (x - 2 √3) and (x + 1).
To find the third factor, we need to use the given point (2, -18).
Let's start by finding the third factor:
Substitute the given zero, 2 √3, into the cubic function to get:
(2 √3 - 2 √3) = 0
So, the third factor is (x - 2 √3).
Now, let's find the equation of the cubic function:
Multiply the three factors together:
(x - 2 √3)(x + 1)(x - 2 √3)
Expanding the expression:
(x - 2 √3)(x^2 - 2 √3x + x - 2 √3)
Simplifying:
(x - 2 √3)(x^2 - √3x - √3x - 6)
Expanding and combining like terms:
x^3 - √3x^2 - √3x^2 + 6x - 2 √3x + 3x + 2 √3 - 6
Simplifying:
x^3 - 2 √3x^2 + 9x - 4 √3
Therefore, the equation of the cubic function with zeros 2 √3 and -1, and passing through the point (2, -18) is:
f(x) = x^3 - 2 √3x^2 + 9x - 4 √3
To determine the equation of a cubic function with given zeros and one additional point, we can use the fact that the zeros of a function correspond to the factors of its equation.
Step 1: Determine the factors
Since the zeros are 2√3 and -1, the factors of the cubic function are (x - 2√3), (x + 1), and an unknown factor, let's call it (x - a), where 'a' is yet to be determined.
Step 2: Determine the complete equation
Multiplying the factors together, we get:
(x - 2√3)(x + 1)(x - a)
Step 3: Use the given point to determine 'a'
Since the graph passes through the point (2, -18), we can substitute these values into the equation:
(-18) = (2 - 2√3 + 1)(2 + 1)(2 - a)
Simplify this equation:
(-18) = (3 - 2√3)(2 - a)
Step 4: Solve for 'a'
Expand the equation:
-18 = 6 - 3√3 - 2a + 2√3a
Combine like terms:
-24 = 6√3 - 2a + 2√3a
Bring all 'a' terms to one side:
2a - 2√3a = 6√3 + 24
Factor 'a' on the left side:
a(2 - 2√3) = 6√3 + 24
Divide both sides by (2 - 2√3):
a = (6√3 + 24) / (2 - 2√3)
Step 5: Simplify 'a'
To simplify 'a', we rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:
a = [(6√3 + 24) / (2 - 2√3)] * [(2 + 2√3) / (2 + 2√3)]
Multiply the numerators and denominators:
a = [(12√3 + 48 + 12√3√3 + 48√3) / (4 - 12)]
Simplify:
a = [60 + 20√3] / (-8)
Simplify further:
a = -7.5 - 2.5√3
Step 6: Determine the complete equation
Now that we have the value of 'a', substitute it back into the equation:
(x - 2√3)(x + 1)(x - (-7.5 - 2.5√3))
Simplify:
(x - 2√3)(x + 1)(x + 7.5 + 2.5√3)
Expand and multiply the factors:
(x^2 - 2√3x + x - 2√3)(x + 7.5 + 2.5√3)
Simplify:
(x^2 - √3x - 2√3)(x + 7.5 + 2.5√3)
Multiply using the distributive property:
x^3 + (7.5 + 2.5√3)x^2 - √3x^2 + (21.25 + 7.5√3 + 5√3 + 2√3^2)x - 14.14√3 - 4.71√3^2
Simplify:
x^3 + (7.5 + 2.5√3 - √3)x^2 + (28.75 + 12.5√3)x - 14.14√3 - 14.14
Therefore, the equation of the cubic function is:
f(x) = x^3 + (7.5 + 2.5√3 - √3)x^2 + (28.75 + 12.5√3)x - 14.14√3 - 14.14