If f(x) = 3x ^ 2 - x + 2 find f(2) f(- 2) f(a) f(- a) f(a + 1) 2f(a) f(2a) f(a ^ 2) [f(a)] ^ 2 and f(a + h)

Given the function \( f(x) = 3x^2 - x + 2 \), let's calculate the values of each requested expression.

1. **Find \( f(2) \):**
   \[
   f(2) = 3(2)^2 - (2) + 2 = 3(4) - 2 + 2 = 12 - 2 + 2 = 12
   \]

2. **Find \( f(-2) \):**
   \[
   f(-2) = 3(-2)^2 - (-2) + 2 = 3(4) + 2 + 2 = 12 + 2 + 2 = 16
   \]

3. **Find \( f(a) \):**
   \[
   f(a) = 3a^2 - a + 2
   \]

4. **Find \( f(-a) \):**
   \[
   f(-a) = 3(-a)^2 - (-a) + 2 = 3a^2 + a + 2
   \]

5. **Find \( f(a + 1) \):**
   \[
   f(a + 1) = 3(a + 1)^2 - (a + 1) + 2
   \]
   First expand \( (a + 1)^2 = a^2 + 2a + 1 \):
   \[
   f(a + 1) = 3(a^2 + 2a + 1) - (a + 1) + 2
   \]
   Now simplify:
   \[
   f(a + 1) = 3a^2 + 6a + 3 - a - 1 + 2 = 3a^2 + 5a + 4
   \]

6. **Find \( 2f(a) \):**
   \[
   2f(a) = 2(3a^2 - a + 2) = 6a^2 - 2a + 4
   \]

7. **Find \( f(2a) \):**
   \[
   f(2a) = 3(2a)^2 - (2a) + 2 = 3(4a^2) - 2a + 2 = 12a^2 - 2a + 2
   \]

8. **Find \( f(a^2) \):**
   \[
   f(a^2) = 3(a^2)^2 - a^2 + 2 = 3a^4 - a^2 + 2
   \]

9. **Find \( [f(a)]^2 \):**
   \[
   [f(a)]^2 = (3a^2 - a + 2)^2
   \]
   Expand:
   \[
   [f(a)]^2 = (3a^2)^2 - 2(3a^2)(a) + 2(3a^2)(2) + (-a)^2 - 2(-a)(2) + 2^2
   \]
   Simplify:
   \[
   [f(a)]^2 = 9a^4 - 6a^3 + 12a^2 + a^2 - 4a + 4 = 9a^4 - 6a^3 + 13a^2 - 4a + 4
   \]

10. **Find \( f(a + h) \):**
    \[
    f(a + h) = 3(a + h)^2 - (a + h) + 2
    \]
    Expand \( (a + h)^2 = a^2 + 2ah + h^2 \):
    \[
    f(a + h) = 3(a^2 + 2ah + h^2) - (a + h) + 2
    \]
    Now simplify:
    \[
    f(a + h) = 3a^2 + 6ah + 3h^2 - a - h + 2
    \]

### Summary of Results:
1. \( f(2) = 12 \)
2. \( f(-2) = 16 \)
3. \( f(a) = 3a^2 - a + 2 \)
4. \( f(-a) = 3a^2 + a + 2 \)
5. \( f(a + 1) = 3a^2 + 5a + 4 \)
6. \( 2f(a) = 6a^2 - 2a + 4 \)
7. \( f(2a) = 12a^2 - 2a + 2 \)
8. \( f(a^2) = 3a^4 - a^2 + 2 \)
9. \( [f(a)]^2 = 9a^4 - 6a^3 + 13a^2 - 4a + 4 \)
10. \( f(a + h) = 3a^2 + 6ah + 3h^2 - a - h + 2 \)

Would you like any further details or clarifications on these calculations?

---

### Follow-up Questions:
1. How do you derive the formula for the square of a binomial, like in the case of \( f(a + 1) \)?
2. Can you apply these techniques to find the derivative of \( f(x) \)?
3. What happens if you plug specific values like \( a = 1 \) or \( h = 0 \) into these expressions?
4. How would you solve for \( a \) given that \( f(a) = 0 \)?
5. What are some other functions of the form \( f(x) = ax^2 + bx + c \) and how do they behave differently?

### Tip:
When expanding expressions like \( (a + h)^2 \), it's often helpful to first expand the square term and then simplify step by step. This method avoids mistakes and keeps calculations clear.

This page provides step-by-step solutions to the problem of evaluating the function f(x) = 3x^2 - x + 2 for various inputs like f(2), f(-2), f(a), and more. Learn about function substitution, binomial expansion, and simplification techniques. Suitable for high school algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation