https://file.aiquickdraw.com/mathbot/file/1726565078007ya3xfidi.jpg

Let's examine the three functions from the image you've uploaded. Each function seems to require filling out a table based on the given inputs \( x \). I'll go through each one step by step.

### Function 1: \( f(x) = x^2 \)
You need to square each value of \( x \) and fill in the table.

\[
f(x) = x^2
\]
- For \( x = -3 \), \( f(x) = (-3)^2 = 9 \)
- For \( x = -2 \), \( f(x) = (-2)^2 = 4 \)
- For \( x = -1 \), \( f(x) = (-1)^2 = 1 \)
- For \( x = 0 \), \( f(x) = (0)^2 = 0 \)
- For \( x = 1 \), \( f(x) = (1)^2 = 1 \)
- For \( x = 2 \), \( f(x) = (2)^2 = 4 \)
- For \( x = 3 \), \( f(x) = (3)^2 = 9 \)

| \( x \)  | \( f(x) = x^2 \) |
|:-------:|:-----------------:|
|   -3    |         9         |
|   -2    |         4         |
|   -1    |         1         |
|    0    |         0         |
|    1    |         1         |
|    2    |         4         |
|    3    |         9         |

### Function 2: \( f(x) = x^2 - 5 \)
Here, after squaring \( x \), subtract 5.

\[
f(x) = x^2 - 5
\]
- For \( x = -1 \), \( f(x) = (-1)^2 - 5 = 1 - 5 = -4 \)
- For \( x = 0 \), \( f(x) = (0)^2 - 5 = 0 - 5 = -5 \)
- For \( x = 1 \), \( f(x) = (1)^2 - 5 = 1 - 5 = -4 \)
- For \( x = 2 \), \( f(x) = (2)^2 - 5 = 4 - 5 = -1 \)
- For \( x = 3 \), \( f(x) = (3)^2 - 5 = 9 - 5 = 4 \)

| \( x \)  | \( f(x) = x^2 - 5 \) |
|:-------:|:---------------------:|
|   -1    |          -4           |
|    0    |          -5           |
|    1    |          -4           |
|    2    |          -1           |
|    3    |           4           |

### Function 3: \( f(x) = -3(x + 1)^2 + 2 \)
First, add 1 to \( x \), square the result, multiply by -3, and finally add 2.

\[
f(x) = -3(x + 1)^2 + 2
\]
- For \( x = -3 \), \( f(x) = -3((-3 + 1)^2) + 2 = -3((-2)^2) + 2 = -3(4) + 2 = -12 + 2 = -10 \)
- For \( x = -2 \), \( f(x) = -3((-2 + 1)^2) + 2 = -3((-1)^2) + 2 = -3(1) + 2 = -3 + 2 = -1 \)
- For \( x = -1 \), \( f(x) = -3((-1 + 1)^2) + 2 = -3(0^2) + 2 = 0 + 2 = 2 \)
- For \( x = 0 \), \( f(x) = -3((0 + 1)^2) + 2 = -3(1^2) + 2 = -3(1) + 2 = -3 + 2 = -1 \)
- For \( x = 1 \), \( f(x) = -3((1 + 1)^2) + 2 = -3(2^2) + 2 = -3(4) + 2 = -12 + 2 = -10 \)

| \( x \)  | \( f(x) = -3(x + 1)^2 + 2 \) |
|:-------:|:----------------------------:|
|   -3    |            -10               |
|   -2    |            -1                |
|   -1    |             2                |
|    0    |            -1                |
|    1    |            -10               |

Let me know if you'd like any clarifications or further explanations!

### Related Questions:
1. How do transformations like shifts and stretches affect the graph of a quadratic function?
2. Can you explain the differences between a vertex form and standard form of a quadratic equation?
3. What is the general process for solving quadratic equations by factoring?
4. How does completing the square work, and when is it useful in graphing quadratics?
5. What are the applications of quadratic functions in real-life scenarios?

### Tip:
When dealing with quadratic functions, understanding how the constants in each form (standard, factored, vertex) influence the graph will help in quickly sketching or predicting behavior.

Fill out the table of values for the following quadratic functions: f(x) = x^2, f(x) = x^2 - 5, and f(x) = -3(x + 1)^2 + 2.

Learn how to fill out tables for quadratic functions f(x) = x^2, f(x) = x^2 - 5, and f(x) = -3(x + 1)^2 + 2. Understand function transformations, including vertical shifts and reflections. Suitable for students in Grade 9-10.

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