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Let's solve the problem step by step. The given function is:

\[
f(x) = -2 + 3x^2
\]

You are asked to calculate three things:

1. \( f(a) \)
2. \( f(a + h) \)
3. \(\frac{f(a + h) - f(a)}{h}\)

### Step 1: Calculate \( f(a) \)
Substitute \( a \) for \( x \) in the function \( f(x) \):

\[
f(a) = -2 + 3a^2
\]

### Step 2: Calculate \( f(a + h) \)
Substitute \( a + h \) for \( x \) in the function \( f(x) \):

\[
f(a + h) = -2 + 3(a + h)^2
\]
Now, expand \( (a + h)^2 \):

\[
(a + h)^2 = a^2 + 2ah + h^2
\]

Thus:

\[
f(a + h) = -2 + 3(a^2 + 2ah + h^2) = -2 + 3a^2 + 6ah + 3h^2
\]

### Step 3: Calculate \( \frac{f(a + h) - f(a)}{h} \)
Now, substitute the expressions for \( f(a + h) \) and \( f(a) \):

\[
f(a + h) - f(a) = \left(-2 + 3a^2 + 6ah + 3h^2\right) - \left(-2 + 3a^2\right)
\]

Simplifying:

\[
f(a + h) - f(a) = 6ah + 3h^2
\]

Now, divide by \( h \):

\[
\frac{f(a + h) - f(a)}{h} = \frac{6ah + 3h^2}{h}
\]

Simplify by dividing each term by \( h \):

\[
\frac{f(a + h) - f(a)}{h} = 6a + 3h
\]

### Final Results:
1. \( f(a) = -2 + 3a^2 \)
2. \( f(a + h) = -2 + 3a^2 + 6ah + 3h^2 \)
3. \( \frac{f(a + h) - f(a)}{h} = 6a + 3h \)

Would you like more details or have any questions? Here are five related questions you might find useful:

1. How do you compute the derivative of \( f(x) \) using the limit definition?
2. Can you find the derivative of \( f(x) = -2 + 3x^2 \) using standard differentiation rules?
3. What happens to the expression for \( \frac{f(a+h)-f(a)}{h} \) as \( h \to 0 \)?
4. How do you use the result to find the slope of the tangent line at a specific point?
5. How do the terms involving \( h \) contribute to higher-order derivatives?

**Tip:** Always simplify expressions before plugging in values for easier calculations, especially when dealing with limits and derivatives!

Given the function f(x) = -2 + 3x^2, calculate the following values: f(a), f(a + h), and (f(a + h) - f(a)) / h

Learn how to evaluate a function and compute the difference quotient for f(x) = -2 + 3x^2. This problem involves key algebraic steps and the application of the limit definition of derivatives. Suitable for high school students in Grades 10-12.

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