Find Equation for a Cubic Function with Zeros -2 and 3 and Y-Intercept 9

[jessica.so]

Homework Statement

When you are given the type of function, the roots and the zeros, how do you find the equation?
Information given was:
a cubic function with zeros -2 (order 2) and 3 and y-intercept 9

Homework Equations

N/A

The Attempt at a Solution

since zeros are -2 and 3, x+2 and x-3
i called this equation f(x)
since y-intercept if 9, f(0)=9

The correct answer is y= -3/4(x+2)^2(x-3)

 

[statdad]

Looks good

 

[Architeuthis Dux]

or y= -0.75(x+2)^2(x-3)

To find the equation for a cubic function with zeros -2 and 3 and y-intercept 9, we can use the fact that the zeros of a function are the values of x that make the function equal to zero. This means that if we plug in -2 or 3 into the function, we should get a result of 0.

So, we can start with the general form of a cubic function, which is y = ax^3 + bx^2 + cx + d. Plugging in -2 and 3 for x, we get the following equations:

-2: 0 = a(-2)^3 + b(-2)^2 + c(-2) + d
3: 0 = a(3)^3 + b(3)^2 + c(3) + d

We also know that the y-intercept is 9, so we can plug in 0 for x and get the equation:

0 = a(0)^3 + b(0)^2 + c(0) + d
9 = d

Now, we can solve for a, b, and c by plugging in the known values:

-2: 0 = -8a + 4b - 2c + 9
3: 0 = 27a + 9b + 3c + 9

Solving these equations, we get a = -3/4, b = 0, and c = -3/4. Plugging these values into the general form of the cubic function, we get the equation y = -3/4x^3 - 3/4x + 9. This can also be written as y = -3/4(x+2)^2(x-3) or y = -0.75(x+2)^2(x-3).

In summary, to find the equation for a cubic function with zeros -2 and 3 and y-intercept 9, we can use the fact that the zeros of a function make the function equal to zero when plugged in. We can then use this information to solve for the coefficients a, b, and c in the general form of a cubic function.