https://file.aiquickdraw.com/mathbot/file/17288565326609ast1mue.jpg

The problem in the image asks for calculating the values of \( f(x) \) for specific inputs, where the function is given as:

\[
f(x) = 4x^2 + 4x + 5
\]

You need to calculate \( f(x) \) for \( x = -2, -1, 0, 1, 2 \). Let's go through each one step by step:

1. **\( f(-2) \):**
   \[
   f(-2) = 4(-2)^2 + 4(-2) + 5 = 4(4) - 8 + 5 = 16 - 8 + 5 = 13
   \]
   So, \( f(-2) = 13 \).

2. **\( f(-1) \):**
   \[
   f(-1) = 4(-1)^2 + 4(-1) + 5 = 4(1) - 4 + 5 = 4 - 4 + 5 = 5
   \]
   So, \( f(-1) = 5 \).

3. **\( f(0) \):**
   \[
   f(0) = 4(0)^2 + 4(0) + 5 = 0 + 0 + 5 = 5
   \]
   So, \( f(0) = 5 \).

4. **\( f(1) \):**
   \[
   f(1) = 4(1)^2 + 4(1) + 5 = 4(1) + 4 + 5 = 4 + 4 + 5 = 13
   \]
   So, \( f(1) = 13 \).

5. **\( f(2) \):**
   \[
   f(2) = 4(2)^2 + 4(2) + 5 = 4(4) + 8 + 5 = 16 + 8 + 5 = 29
   \]
   So, \( f(2) = 29 \).

### Final Results:
- \( f(-2) = 13 \)
- \( f(-1) = 5 \)
- \( f(0) = 5 \)
- \( f(1) = 13 \)
- \( f(2) = 29 \)

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:

1. What is the vertex of the quadratic function \( f(x) = 4x^2 + 4x + 5 \)?
2. How does the value of \( f(x) \) change as \( x \) increases?
3. How would you solve \( f(x) = 0 \) for this quadratic function?
4. How can you find the axis of symmetry for the function \( f(x) = 4x^2 + 4x + 5 \)?
5. What is the domain and range of \( f(x) = 4x^2 + 4x + 5 \)?

**Tip:** Always substitute values carefully when evaluating a function, and make sure to handle negative numbers correctly.

Given the function f(x) = 4x^2 + 4x + 5, calculate the values of f(-2), f(-1), f(0), f(1), and f(2).

Learn how to calculate specific function values for the quadratic function f(x) = 4x^2 + 4x + 5. This step-by-step guide shows how to substitute values of x to find f(-2), f(-1), f(0), f(1), and f(2). Suitable for students in Grades 9-11.

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