A cubic polynomial function f(x) has leading coefficient -2 and intercepts the y-axis at 2. If f(1)=1, and f(-2)=-2, find f(-1) and write the complete function.

y = -2 x^3 + b x^2 + c x + d

y(0) = 2 so d = 2

y = -2 x^3 + b x^2 + c x + 2

1 = -2 + b + c + 2 or b+c=1
-2 = 16 + 4 b - 2 c + 2 or -20 = 4 b - 2 c
or -10 = 2 b - c

2 b - 1 c = -10
1 b + 1 c = 1
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3 b = - 9
b = -3 etc

I need a cubic trinomial in standard form

To find the value of f(-1), we can use the given information that f(1) = 1 and f(-2) = -2.

Let's start with determining the x-intercepts of the cubic polynomial function. Since the y-intercept is at (0, 2), this means that the function crosses the x-axis at x = 0, which is one x-intercept.

Since the function is a cubic polynomial, it will have a total of three x-intercepts. We already know one of them is at x = 0, so we need to find the other two.

Using the fact that f(1) = 1, we know that when x = 1, the function has a y-value of 1. This means that (1, 1) is a point on the graph of the function. Since the function is cubic, it can potentially intersect the x-axis at this point. Therefore, one of the x-intercepts is at x = 1.

Similarly, using the fact that f(-2) = -2, we know that when x = -2, the function has a y-value of -2. This means that (-2, -2) is another point on the graph of the function. Again, since the function is cubic, it can potentially intersect the x-axis at this point as well. Therefore, one more x-intercept is at x = -2.

Now we know all three x-intercepts: x = 0, x = 1, and x = -2.

To find the complete function, let's write it in factored form using the x-intercepts:

f(x) = (x - 0)(x - 1)(x + 2)

Since the leading coefficient is -2, we multiply the expression by -2:

f(x) = -2(x - 0)(x - 1)(x + 2)

Expanding this, we get:

f(x) = -2(x² - x)(x + 2)
= -2(x³ + x² - 2x - 1)

Therefore, the complete function is f(x) = -2x³ - 2x² + 4x + 2.

Now, to find f(-1), we substitute x = -1 into the function:

f(-1) = -2(-1)³ - 2(-1)² + 4(-1) + 2
= -2(-1) - 2(1) - 4 + 2
= 2 - 2 - 4 + 2
= -2.

Therefore, f(-1) = -2.