Transformations Of Functions Worksheet Packet, 51% OFF - Free Printable

Let's solve this step by step. The task is to create a table of values for each function and then graph them on the coordinate grid (which we'll describe since we can't draw here).

We'll go through each function, compute the values of $ f(x) $ for the given $ x $-values, and explain how the transformations affect the graph.

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🔷 Problem 3: Transformations of $ f(x) = x^3 $

We have three functions:
1. $ f(x) = x^3 $
2. $ f(x) = x^3 + 5 $
3. $ f(x) = x^3 - 5 $

These are vertical shifts of the basic cubic function.

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#### ✔ Table for $ f(x) = x^3 $

| $ x $ | $ f(x) = x^3 $ |
|--------|------------------|
| -2 | $ (-2)^3 = -8 $ |
| -1 | $ (-1)^3 = -1 $ |
| 0 | $ 0^3 = 0 $ |
| 1 | $ 1^3 = 1 $ |
| 2 | $ 2^3 = 8 $ |
| 3 | $ 3^3 = 27 $ |
| 4 | $ 4^3 = 64 $ |

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#### ✔ Table for $ f(x) = x^3 + 5 $

This is the original function shifted up by 5 units.

So add 5 to each value of $ x^3 $:

| $ x $ | $ f(x) = x^3 + 5 $ |
|--------|----------------------|
| -2 | $ -8 + 5 = -3 $ |
| -1 | $ -1 + 5 = 4 $ |
| 0 | $ 0 + 5 = 5 $ |
| 1 | $ 1 + 5 = 6 $ |
| 2 | $ 8 + 5 = 13 $ |
| 3 | $ 27 + 5 = 32 $ |
| 4 | $ 64 + 5 = 69 $ |

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#### ✔ Table for $ f(x) = x^3 - 5 $

Shifted down by 5 units.

Subtract 5 from each $ x^3 $:

| $ x $ | $ f(x) = x^3 - 5 $ |
|--------|----------------------|
| -2 | $ -8 - 5 = -13 $ |
| -1 | $ -1 - 5 = -6 $ |
| 0 | $ 0 - 5 = -5 $ |
| 1 | $ 1 - 5 = -4 $ |
| 2 | $ 8 - 5 = 3 $ |
| 3 | $ 27 - 5 = 22 $ |
| 4 | $ 64 - 5 = 59 $ |

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📈 Graphing Instructions (Problem 3)

- Plot the points from each table.
- The original $ f(x) = x^3 $ passes through $ (-2,-8), (-1,-1), (0,0), (1,1), (2,8), \dots $
- $ f(x) = x^3 + 5 $ is the same shape but every point moved up 5 units.
- $ f(x) = x^3 - 5 $ is the same shape but every point moved down 5 units.

✔ All three graphs are cubic curves with the same "S" shape, just vertically shifted.

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🔷 Problem 4: More Transformations

Functions:
1. $ f(x) = 2x^3 $
2. $ f(x) = (x+2)^3 $
3. $ f(x) = (x-2)^3 $

These involve vertical stretch and horizontal shifts.

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#### ✔ Table for $ f(x) = 2x^3 $

This is a vertical stretch by factor of 2 → outputs are doubled.

| $ x $ | $ f(x) = 2x^3 $ |
|--------|-------------------|
| -2 | $ 2(-8) = -16 $ |
| -1 | $ 2(-1) = -2 $ |
| 0 | $ 2(0) = 0 $ |
| 1 | $ 2(1) = 2 $ |
| 2 | $ 2(8) = 16 $ |
| 3 | $ 2(27) = 54 $ |
| 4 | $ 2(64) = 128 $ |

→ The graph is steeper than $ x^3 $ because it grows faster.

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#### ✔ Table for $ f(x) = (x+2)^3 $

This is a horizontal shift left by 2 units.

Think: $ (x + 2)^3 $ means the graph is shifted left 2.

So plug in $ x $, compute $ (x+2)^3 $:

| $ x $ | $ f(x) = (x+2)^3 $ |
|--------|----------------------|
| -2 | $ (0)^3 = 0 $ |
| -1 | $ (1)^3 = 1 $ |
| 0 | $ (2)^3 = 8 $ |
| 1 | $ (3)^3 = 27 $ |
| 2 | $ (4)^3 = 64 $ |
| 3 | $ (5)^3 = 125 $ |
| 4 | $ (6)^3 = 216 $ |

Note: This graph has the same shape as $ x^3 $, but starts at $ x = -2 $ instead of $ x = 0 $. It's shifted left 2.

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#### ✔ Table for $ f(x) = (x-2)^3 $

This is a horizontal shift right by 2 units.

| $ x $ | $ f(x) = (x-2)^3 $ |
|--------|----------------------|
| -2 | $ (-4)^3 = -64 $ |
| -1 | $ (-3)^3 = -27 $ |
| 0 | $ (-2)^3 = -8 $ |
| 1 | $ (-1)^3 = -1 $ |
| 2 | $ (0)^3 = 0 $ |
| 3 | $ (1)^3 = 1 $ |
| 4 | $ (2)^3 = 8 $ |

→ Shifted right 2 units.

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📈 Graphing Instructions (Problem 4)

- $ f(x) = 2x^3 $: Same shape as $ x^3 $, but stretched vertically — it’s steeper.
- $ f(x) = (x+2)^3 $: Same shape, shifted left 2 units.
- $ f(x) = (x-2)^3 $: Same shape, shifted right 2 units.

All are still cubic functions with the characteristic S-shape.

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✔ Summary of Transformations

| Function | Transformation |
|---------------------|------------------------------------|
| $ x^3 + 5 $ | Up 5 units |
| $ x^3 - 5 $ | Down 5 units |
| $ 2x^3 $ | Vertical stretch by 2 |
| $ (x+2)^3 $ | Left 2 units |
| $ (x-2)^3 $ | Right 2 units |

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✔ Final Step: Graphing

On the provided grid:
- Use the tables above to plot points.
- Connect the points smoothly with a curve that resembles an "S" (cubic shape).
- For vertical shifts: move all points up/down.
- For horizontal shifts: move all points left/right.
- For vertical stretch: make the curve steeper.

> 💡 Tip: Start with the basic $ y = x^3 $ graph, then apply transformations one at a time.

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Let me know if you'd like a visual sketch or explanation of how to plot these on the grid!