Transformations of Functions Worksheet Packet - 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 grid. Since I can't draw graphs here, I'll provide the completed tables, explain how to graph them, and describe the transformations.

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🔷 Part 1: Exponential Functions with Vertical Shifts

We are given three functions based on $ f(x) = 2^x $:

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

#### ✔ Step 1: Fill in the table for $ f(x) = 2^x $

| x | $ f(x) = 2^x $ |
|---|------------------|
| -2 | $ 2^{-2} = \frac{1}{4} = 0.25 $ |
| -1 | $ 2^{-1} = \frac{1}{2} = 0.5 $ |
| 0 | $ 2^0 = 1 $ |
| 1 | $ 2^1 = 2 $ |
| 2 | $ 2^2 = 4 $ |
| 3 | $ 2^3 = 8 $ |
| 4 | $ 2^4 = 16 $ |

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#### ✔ Step 2: $ f(x) = 2^x + 3 $ → Vertical shift up by 3 units

Add 3 to each value from above.

| x | $ f(x) = 2^x + 3 $ |
|---|----------------------|
| -2 | $ 0.25 + 3 = 3.25 $ |
| -1 | $ 0.5 + 3 = 3.5 $ |
| 0 | $ 1 + 3 = 4 $ |
| 1 | $ 2 + 3 = 5 $ |
| 2 | $ 4 + 3 = 7 $ |
| 3 | $ 8 + 3 = 11 $ |
| 4 | $ 16 + 3 = 19 $ |

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#### ✔ Step 3: $ f(x) = 2^x - 3 $ → Vertical shift down by 3 units

Subtract 3 from each value.

| x | $ f(x) = 2^x - 3 $ |
|---|----------------------|
| -2 | $ 0.25 - 3 = -2.75 $ |
| -1 | $ 0.5 - 3 = -2.5 $ |
| 0 | $ 1 - 3 = -2 $ |
| 1 | $ 2 - 3 = -1 $ |
| 2 | $ 4 - 3 = 1 $ |
| 3 | $ 8 - 3 = 5 $ |
| 4 | $ 16 - 3 = 13 $ |

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📈 Graphing Instructions (Part 1):

- Plot the points from each table.
- The original $ f(x) = 2^x $ starts at (0,1), increases rapidly.
- $ f(x) = 2^x + 3 $ is the same shape but shifted up 3 units → asymptote at $ y = 3 $.
- $ f(x) = 2^x - 3 $ is shifted down 3 units → asymptote at $ y = -3 $.

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🔷 Part 2: Transformations Involving Negatives and Base Changes

Now we have:

1. $ f(x) = -2^x $
2. $ f(x) = \left(\frac{1}{2}\right)^x $
3. $ f(x) = -\left(\frac{1}{2}\right)^x $

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#### ✔ Step 1: $ f(x) = -2^x $ → Reflect over x-axis

Take $ 2^x $ values and negate them.

| x | $ f(x) = -2^x $ |
|---|-------------------|
| -2 | $ -0.25 $ |
| -1 | $ -0.5 $ |
| 0 | $ -1 $ |
| 1 | $ -2 $ |
| 2 | $ -4 $ |
| 3 | $ -8 $ |
| 4 | $ -16 $ |

This is a reflection of $ 2^x $ across the x-axis.

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#### ✔ Step 2: $ f(x) = \left(\frac{1}{2}\right)^x $

Note: $ \left(\frac{1}{2}\right)^x = 2^{-x} $. This is a horizontal reflection or decay function.

| x | $ f(x) = \left(\frac{1}{2}\right)^x $ |
|---|------------------------------------------|
| -2 | $ \left(\frac{1}{2}\right)^{-2} = 2^2 = 4 $ |
| -1 | $ \left(\frac{1}{2}\right)^{-1} = 2^1 = 2 $ |
| 0 | $ 1 $ |
| 1 | $ \frac{1}{2} = 0.5 $ |
| 2 | $ \frac{1}{4} = 0.25 $ |
| 3 | $ \frac{1}{8} = 0.125 $ |
| 4 | $ \frac{1}{16} = 0.0625 $ |

This is an exponential decay curve, decreasing toward 0 as $ x \to \infty $.

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#### ✔ Step 3: $ f(x) = -\left(\frac{1}{2}\right)^x $ → Reflect decay over x-axis

Negate all values from above.

| x | $ f(x) = -\left(\frac{1}{2}\right)^x $ |
|---|-------------------------------------------|
| -2 | $ -4 $ |
| -1 | $ -2 $ |
| 0 | $ -1 $ |
| 1 | $ -0.5 $ |
| 2 | $ -0.25 $ |
| 3 | $ -0.125 $ |
| 4 | $ -0.0625 $ |

This is the decay curve reflected below the x-axis.

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📈 Graphing Instructions (Part 2):

- $ f(x) = -2^x $: Same shape as $ 2^x $, but flipped upside down.
- $ f(x) = \left(\frac{1}{2}\right)^x $: Starts high at left, decreases slowly → asymptote at $ y=0 $.
- $ f(x) = -\left(\frac{1}{2}\right)^x $: Same as above, but reflected below x-axis.

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✔ Summary of Transformations:

| Function | Transformation |
|--------|----------------|
| $ 2^x + 3 $ | Shift up 3 |
| $ 2^x - 3 $ | Shift down 3 |
| $ -2^x $ | Reflect over x-axis |
| $ \left(\frac{1}{2}\right)^x $ | Decay (base < 1) |
| $ -\left(\frac{1}{2}\right)^x $ | Decay + reflect over x-axis |

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🖼️ How to Graph:

1. Use the tables to plot points on the grids.
2. Connect the points smoothly with curves (not straight lines).
3. Draw asymptotes:
- For $ 2^x $: horizontal asymptote at $ y = 0 $
- For $ 2^x + 3 $: asymptote at $ y = 3 $
- For $ 2^x - 3 $: asymptote at $ y = -3 $
- For $ \left(\frac{1}{2}\right)^x $ and its negative: asymptote at $ y = 0 $

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✔ Final Answer (Tables Completed):

#### Part 1:

$ f(x) = 2^x $
| x | f(x) |
|---|------|
| -2 | 0.25 |
| -1 | 0.5 |
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |

$ f(x) = 2^x + 3 $
| x | f(x) |
|---|------|
| -2 | 3.25 |
| -1 | 3.5 |
| 0 | 4 |
| 1 | 5 |
| 2 | 7 |
| 3 | 11 |
| 4 | 19 |

$ f(x) = 2^x - 3 $
| x | f(x) |
|---|------|
| -2 | -2.75 |
| -1 | -2.5 |
| 0 | -2 |
| 1 | -1 |
| 2 | 1 |
| 3 | 5 |
| 4 | 13 |

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#### Part 2:

$ f(x) = -2^x $
| x | f(x) |
|---|------|
| -2 | -0.25 |
| -1 | -0.5 |
| 0 | -1 |
| 1 | -2 |
| 2 | -4 |
| 3 | -8 |
| 4 | -16 |

$ f(x) = \left(\frac{1}{2}\right)^x $
| x | f(x) |
|---|------|
| -2 | 4 |
| -1 | 2 |
| 0 | 1 |
| 1 | 0.5 |
| 2 | 0.25 |
| 3 | 0.125 |
| 4 | 0.0625 |

$ f(x) = -\left(\frac{1}{2}\right)^x $
| x | f(x) |
|---|------|
| -2 | -4 |
| -1 | -2 |
| 0 | -1 |
| 1 | -0.5 |
| 2 | -0.25 |
| 3 | -0.125 |
| 4 | -0.0625 |

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You can now use these values to plot each function on the provided grids. Each transformation affects the parent function $ 2^x $ in predictable ways: vertical shifts, reflections, and base changes affect growth vs. decay.

Let me know if you'd like a visual sketch description!