AP Precalculus Practice Test: Unit 1 Question #33 Transforming a Function Using a Table of Values

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To transform a function using a table of values, we will apply specific transformations based on changes made to the function’s formula. These transformations typically include:

1. **Vertical Shifts**: Adding or subtracting a constant to the function.
2. **Horizontal Shifts**: Adding or subtracting a constant inside the function’s argument.
3. **Vertical Stretch/Compression**: Multiplying the function by a constant.
4. **Reflection**: Reflecting the function over the x-axis or y-axis.

The table of values helps visualize how these transformations affect the function by showing the input-output pairs before and after the transformations.

### Example Problem:

Let's say the original function is \( f(x) = x^2 \), and we want to apply the following transformations:
1. **Shift up by 3 units**.
2. **Stretch vertically by a factor of 2**.
3. **Shift left by 1 unit**.

### Step 1: Start with the Original Function \( f(x) = x^2 \)

We will use the following table of values for \( f(x) = x^2 \):

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

### Step 2: Apply the Transformation "Shift Up by 3 Units"

To shift the graph up by 3 units, we add 3 to each \( f(x) \) value.

| \( x \) | \( f(x) = x^2 \) | \( f(x) + 3 \) |
|--------|-------------------|-----------------|
| -3 | 9 | 12 |
| -2 | 4 | 7 |
| -1 | 1 | 4 |
| 0 | 0 | 3 |
| 1 | 1 | 4 |
| 2 | 4 | 7 |
| 3 | 9 | 12 |

### Step 3: Apply the Transformation "Stretch Vertically by a Factor of 2"

To stretch the graph vertically by a factor of 2, we multiply each \( f(x) \) value by 2.

| \( x \) | \( f(x) + 3 \) | \( 2(f(x) + 3) \) |
|--------|-----------------|-------------------|
| -3 | 12 | 24 |
| -2 | 7 | 14 |
| -1 | 4 | 8 |
| 0 | 3 | 6 |
| 1 | 4 | 8 |
| 2 | 7 | 14 |
| 3 | 12 | 24 |

### Step 4: Apply the Transformation "Shift Left by 1 Unit"

To shift the graph left by 1 unit, we replace \( x \) with \( x + 1 \) in the function. This will change each \( x \)-value to \( x + 1 \) in the table.

| \( x \) | \( 2(f(x) + 3) \) | Shift Left: \( x + 1 \) |
|--------|-------------------|-------------------------|
| -3 | 24 | -4 |
| -2 | 14 | -3 |
| -1 | 8 | -2 |
| 0 | 6 | -1 |
| 1 | 8 | 0 |
| 2 | 14 | 1 |
| 3 | 24 | 2 |

### Final Table of Transformed Values:

| \( x \) | Transformed Function \( 2(f(x) + 3) \) |
|--------|----------------------------------------|
| -4 | 24 |
| -3 | 14 |
| -2 | 8 |
| -1 | 6 |
| 0 | 8 |
| 1 | 14 |
| 2 | 24 |

### Conclusion:

This table represents the function \( f(x) = x^2 \) after applying a vertical shift of 3 units up, a vertical stretch by a factor of 2, and a horizontal shift to the left by 1 unit. The transformed function can now be written as:

\[
f(x) = 2(x + 1)^2 + 3
\]

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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