Given the function $f(x) = 3x^2 + 2x - 2$, let's calculate the required values:

(a) $f(0)$

Substitute $x = 0$ into the function: $f(0) = 3(0)^2 + 2(0) - 2 = -2$

(b) $f(4)$

Substitute $x = 4$ into the function: $f(4) = 3(4)^2 + 2(4) - 2 = 3(16) + 8 - 2 = 48 + 8 - 2 = 54$

(c) $f(-4)$

Substitute $x = -4$ into the function: $f(-4) = 3(-4)^2 + 2(-4) - 2 = 3(16) - 8 - 2 = 48 - 8 - 2 = 38$

(d) $f(-x)$

Substitute $x = -x$ into the function: $f(-x) = 3(-x)^2 + 2(-x) - 2 = 3x^2 - 2x - 2$

(e) $-f(x)$

Negate the function: $-f(x) = -(3x^2 + 2x - 2) = -3x^2 - 2x + 2$

(f) $f(x + 2)$

Substitute $x + 2$ into the function: $f(x + 2) = 3(x + 2)^2 + 2(x + 2) - 2$ First expand $(x + 2)^2$: $(x + 2)^2 = x^2 + 4x + 4$ Now substitute and expand: $f(x + 2) = 3(x^2 + 4x + 4) + 2(x + 2) - 2 = 3x^2 + 12x + 12 + 2x + 4 - 2 = 3x^2 + 14x + 14$

(g) $f(3x)$

Substitute $3x$ into the function: $f(3x) = 3(3x)^2 + 2(3x) - 2 = 3(9x^2) + 6x - 2 = 27x^2 + 6x - 2$

(h) $f(x + h)$

Substitute $x + h$ into the function: $f(x + h) = 3(x + h)^2 + 2(x + h) - 2$ First expand $(x + h)^2$: $(x + h)^2 = x^2 + 2xh + h^2$ Now substitute and expand: $f(x + h) = 3(x^2 + 2xh + h^2) + 2(x + h) - 2 = 3x^2 + 6xh + 3h^2 + 2x + 2h - 2$

Let me know if you'd like further details on any part! Here are some related questions:

1. How do you find the derivative of $f(x)$?
2. What is the vertex of the parabola $f(x) = 3x^2 + 2x - 2$?
3. How do you find the zeros of the function $f(x) = 3x^2 + 2x - 2$?
4. What is the axis of symmetry for the parabola $f(x)$?
5. How do you graph $f(x) = 3x^2 + 2x - 2$?

Tip: When finding $f(x + h)$, carefully expand the squares and simplify terms to avoid mistakes.