To evaluate the function $f(x) = -3x^3 + 2x^2 + 7x - 8$ at $x = -2$, substitute $-2$ into the equation for $x$:

$f(-2) = -3(-2)^3 + 2(-2)^2 + 7(-2) - 8$

Step-by-Step Calculation:

1. Calculate $(-2)^3$: $(-2)^3 = -8$

   Substitute: $f(-2) = -3(-8) + 2(-2)^2 + 7(-2) - 8$

2. Calculate $(-2)^2$: $(-2)^2 = 4$

   Substitute: $f(-2) = -3(-8) + 2(4) + 7(-2) - 8$

3. Simplify each term:

   • $-3(-8) = 24$
   • $2(4) = 8$
   • $7(-2) = -14$
   • Constant term: $-8$

   Combine: $f(-2) = 24 + 8 - 14 - 8$

4. Add/Subtract the terms: $f(-2) = 24 + 8 = 32$ $32 - 14 = 18$ $18 - 8 = 10$

Final Answer:

$f(-2) = 10$

Let me know if you want further clarification or additional examples.

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Related Questions:

1. How does the degree of the polynomial affect its shape?
2. What are the roots of $f(x) = -3x^3 + 2x^2 + 7x - 8$?
3. How do you find the derivative of $f(x)$?
4. How can you determine where this function is increasing or decreasing?
5. How do you evaluate $f(x)$ for a fractional value of $x$?

Tip:

When substituting into polynomial functions, be careful with the signs, especially when dealing with negative exponents or odd powers!