a) Determine an equation, in simplified form, for the family of cubic functions with zeros 2 +- sqrt5 and 0.
b) Determine an equation for the member of the family with graph passing through point (2,20)
If you mean roots of 0,2+√5 and 2-√5, then then
y = x^3 - 4x^2 - x
thank you again!
a) Oh boy, we're dealing with cubic functions! Let's start by writing the equation in factored form. Since the zeros are 2 + √5, 2 - √5, and 0, our factors will be (x - (2 + √5)), (x - (2 - √5)), and (x - 0). Simplifying this, our equation will be:
f(x) = (x - 2 - √5)(x - 2 + √5)(x - 0)
b) Now let's find the specific equation in the family of cubic functions that passes through the point (2, 20). We'll use this point to find the value of the constant k. Plugging in x = 2 and y = 20 into our equation, we get:
20 = (2 - 2 - √5)(2 - 2 + √5)(2 - 0)
Simplifying this, we have:
20 = (-√5)(√5)(2)
Multiply the square roots and 2 together:
20 = -5(2)
20 = -10? Uh oh, something went wrong. It seems our calculations took a Clowneruption. Let me double-check my steps and get back to you with the correct equations!
a) To determine an equation for the family of cubic functions with zeros 2 +- sqrt(5) and 0, we can use the fact that if a polynomial has a zero, (x-a), then the polynomial has a factor, (x-a), in its equation.
Let's start by considering the zeros given: 2 + sqrt(5), 2 - sqrt(5), and 0.
From the zeros 2 + sqrt(5) and 2 - sqrt(5), we can deduce two factors: (x - (2 + sqrt(5))) and (x - (2 - sqrt(5))).
To simplify, we can multiply these two factors to get the quadratic factor:
(x - (2 + sqrt(5)))(x - (2 - sqrt(5))) = (x - 2 - sqrt(5))(x - 2 + sqrt(5)).
Now, since 0 is also a zero, we know that there is another factor of x in the polynomial equation:
x*(x - 2 - sqrt(5))(x - 2 + sqrt(5)).
Combining the factors, we get the simplified form equation for the family of the cubic functions:
F(x) = x*(x - 2 - sqrt(5))(x - 2 + sqrt(5)).
This equation represents a family of cubic functions with zeros at 2 + sqrt(5), 2 - sqrt(5), and 0.
b) To determine the equation for the member of the family with the graph passing through the point (2,20), we need to substitute the x and y coordinates of the point into the equation we established for the family of cubic functions.
Using the given point (2, 20), we substitute x = 2 and y = 20 into the equation F(x) = x*(x - 2 - sqrt(5))(x - 2 + sqrt(5)):
20 = 2*(2 - 2 - sqrt(5))(2 - 2 + sqrt(5)).
After simplifying this equation, we get:
20 = 2*(-sqrt(5)*sqrt(5)).
20 = 2*(-5).
20 = -10.
From this evaluation, we can see that there is no member of the family of cubic functions that passes through the point (2, 20). The equation does not hold true for this particular point, indicating that there is no cubic function in the family with a graph that passes through (2, 20).
Since you have given me 4 distinct roots, the equation must be a quartic not a cubic
it would be
y = ax(x-2)(x+√5)(x-√5)
= ax(x-2)(x^2 - 5)
to find a, sub in your given point (2,20)