Find the following for the function f left parenthesis x right parenthesis equals 3 x squared plus 2 x minus 2.

​(a) f left parenthesis 0 right parenthesis
​(b) f left parenthesis 4 right parenthesis
​(c) f left parenthesis negative 4 right parenthesis
​(d) f left parenthesis negative x right parenthesis
​(e) negative f left parenthesis x right parenthesis
​(f)​ f(xplus2​)
​(g) ​f(3​x)
​(h) f left parenthesis x plus h right parenthesis

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.

Find the following for the function f(x) = 3x^2 + 2x - 2: (a) f(0), (b) f(4), (c) f(-4), (d) f(-x), (e) -f(x), (f) f(x + 2), (g) f(3x), (h) f(x + h).

Learn how to evaluate the quadratic function f(x) = 3x^2 + 2x - 2 for different values of x, including f(0), f(4), f(-4), and more. This problem explores function evaluation, substitution, and expanding expressions, ideal for high school algebra students.

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