Function table worksheet for graphing quadratic equations.

Since I can't view or access images directly, I can't see the specific details of your uploaded image. However, based on the description you provided — "Complete the function table and sketch the graph" with multiple functions like $ f(x) = x^2 - 6x + 11 $, $ f(x) = x + 3 $, etc. — I’ll walk you through how to solve one of these problems step-by-step, so you can apply it to all of them.

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🔧 General Steps to Solve Each Problem:

For each function:
1. Complete the function table by plugging in the given $ x $-values into the function to find $ f(x) $.
2. Plot the points $ (x, f(x)) $ on the coordinate grid.
3. Draw the graph — connect the points smoothly if it's a line or curve.

Let’s go through one example in detail:
> Problem 1: $ f(x) = x^2 - 6x + 11 $

#### Step 1: Complete the Function Table
Given $ x $-values: $ -2, -1, 0, 1, 2, 3 $

We compute $ f(x) = x^2 - 6x + 11 $ for each:

| $ x $ | $ f(x) = x^2 - 6x + 11 $ |
|--------|---------------------------|
| -2 | $ (-2)^2 - 6(-2) + 11 = 4 + 12 + 11 = 27 $ |
| -1 | $ (-1)^2 - 6(-1) + 11 = 1 + 6 + 11 = 18 $ |
| 0 | $ 0 - 0 + 11 = 11 $ |
| 1 | $ 1 - 6 + 11 = 6 $ |
| 2 | $ 4 - 12 + 11 = 3 $ |
| 3 | $ 9 - 18 + 11 = 2 $ |

So the completed table is:

| $ x $ | -2 | -1 | 0 | 1 | 2 | 3 |
|--------|----|----|---|---|---|---|
| $ f(x) $ | 27 | 18 | 11 | 6 | 3 | 2 |

#### Step 2: Sketch the Graph

Now plot the points:
- (-2, 27)
- (-1, 18)
- (0, 11)
- (1, 6)
- (2, 3)
- (3, 2)

This is a quadratic function (parabola), so it will form a U-shaped curve. Since the coefficient of $ x^2 $ is positive, it opens upward.

You can draw a smooth curve through the points.

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Let's Try Another One:

> Problem 2: $ f(x) = x + 3 $

This is a linear function.

#### Table:
| $ x $ | -2 | -1 | 0 | 1 | 2 | 3 |
|--------|----|----|---|---|---|---|
| $ f(x) $ | $ -2+3=1 $ | $ -1+3=2 $ | $ 0+3=3 $ | $ 1+3=4 $ | $ 2+3=5 $ | $ 3+3=6 $ |

So:
| $ x $ | -2 | -1 | 0 | 1 | 2 | 3 |
|--------|----|----|---|---|---|---|
| $ f(x) $ | 1 | 2 | 3 | 4 | 5 | 6 |

#### Graph:
Plot points: (-2,1), (-1,2), (0,3), (1,4), (2,5), (3,6).
Connect them with a straight line. This line has slope 1 and y-intercept 3.

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Tips for Other Problems:

- Linear functions (like $ f(x) = x + 3 $): Always make a straight line.
- Quadratic functions (like $ f(x) = x^2 $): Parabolas. Find vertex if needed.
- Absolute value (e.g., $ f(x) = |x| $): V-shape.
- Cubic (e.g., $ f(x) = x^3 $): S-shaped curve.

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Example: Problem 8: $ f(x) = x^3 $

| $ x $ | -2 | -1 | 0 | 1 | 2 |
|--------|----|----|---|---|---|
| $ f(x) $ | $ (-2)^3 = -8 $ | $ (-1)^3 = -1 $ | $ 0 $ | $ 1 $ | $ 8 $ |

Points: (-2,-8), (-1,-1), (0,0), (1,1), (2,8)
Graph: S-shaped curve passing through origin.

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Final Advice:

For each problem:
1. Plug in the $ x $ values.
2. Compute $ f(x) $ carefully.
3. Plot the points accurately.
4. Draw the correct shape (line, parabola, etc.).

If you want me to help with a specific function from the worksheet (say, #5 or #9), just tell me which one and I’ll solve it fully!

Would you like me to go through all nine problems? If so, list the functions or describe which ones you’re stuck on.