To evaluate the function \( f(x) = 3x^2 - 5x + 1 \) at \( x = -2 \), substitute \(-2\) for \( x \) in the function:

\[ 
f(-2) = 3(-2)^2 - 5(-2) + 1 
\]

Now, calculate step by step:

1. Calculate \((-2)^2\):
   \[
   (-2)^2 = 4
   \]

2. Multiply by 3:
   \[
   3 \times 4 = 12
   \]

3. Calculate \(-5 \times (-2)\):
   \[
   -5 \times (-2) = 10
   \]

4. Add the results together:
   \[
   12 + 10 + 1 = 23
   \]

Therefore, \( f(-2) = 23 \).

If you have any further questions or would like additional details, feel free to ask!

Here are some related questions for further exploration:

1. What is the value of \( f(x) \) when \( x = 0 \)?
2. How do you find the vertex of the quadratic function \( f(x) = 3x^2 - 5x + 1 \)?
3. How can you determine if this quadratic function opens upwards or downwards?
4. What are the roots of the function \( f(x) = 3x^2 - 5x + 1 \)?
5. How would you find the axis of symmetry for the function?
6. How do you complete the square to rewrite the function in vertex form?
7. What is the y-intercept of the function \( f(x) = 3x^2 - 5x + 1 \)?
8. How can you graph the function \( f(x) = 3x^2 - 5x + 1 \) using key features like intercepts and vertex?

**Tip:** When evaluating a function, carefully substitute values and perform arithmetic operations step by step to avoid errors.

Learn how to evaluate the function f(x) = 3x^2 - 5x + 1 at x = -2 step-by-step. Understand the arithmetic operations involved in substituting values into quadratic functions. Suitable for high school students.

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