*y* = 2*f*(−*x* + 4) − 7

reflect about the *x*-axis, shift right 4 units, stretch vertically by a factor of 2, shift down 7 unitsreflect about the *y*-axis, shift right 4 units, stretch vertically by a factor of 2, shift down 7 units     reflect about the *y*-axis, shift left 4 units, stretch vertically by a factor of 2, shift up 7 unitsreflect about the *y*-axis, shift left 4 units, stretch horizontally by a factor of 2, shift up 7 unitsreflect about the *x*-axis, shift left 4 units, stretch horizontally by a factor of 2, shift up 7 units

It seems you are describing a series of transformations for a function \( f(x) \) under different conditions. Let’s go over each transformation step by step to clarify the meaning:

### Transformation 1 (Original Transformation):
The equation you provided is:
\[ y = 2f(-x + 4) - 7 \]

This transformation involves the following steps:
1. **Reflection about the y-axis**: The term \( -x \) inside the function reflects the graph about the y-axis.
2. **Shift right 4 units**: The term \( -x + 4 \) represents a shift to the right by 4 units.
3. **Stretch vertically by a factor of 2**: The factor 2 multiplying \( f \) outside indicates a vertical stretch by a factor of 2.
4. **Shift down 7 units**: The subtraction of 7 at the end shifts the graph down by 7 units.

### Analysis of Other Transformations

Now, let’s compare the transformation you’ve provided to the other options:

#### Option 1: **Reflect about the x-axis, shift right 4 units, stretch vertically by a factor of 2, shift down 7 units**
- **Reflection about the x-axis** would be represented by multiplying the entire equation by -1, like:
  \[
  y = -2f(-x + 4) - 7
  \]
  This reflects the graph over the x-axis. The other parts (shift right by 4, vertical stretch by 2, and shift down by 7) remain the same.

#### Option 2: **Reflect about the y-axis, shift right 4 units, stretch vertically by a factor of 2, shift down 7 units**
This is exactly the transformation you provided:
\[
y = 2f(-x + 4) - 7
\]
This matches your original equation.

#### Option 3: **Reflect about the y-axis, shift left 4 units, stretch vertically by a factor of 2, shift up 7 units**
To reflect about the y-axis, shift left 4 units, and stretch vertically, we would expect:
\[
y = 2f(-x - 4) + 7
\]
Here:
- \( -x - 4 \) shifts the graph left by 4 units,
- The factor of 2 stretches it vertically,
- Adding 7 shifts the graph up by 7 units.

#### Option 4: **Reflect about the y-axis, shift left 4 units, stretch horizontally by a factor of 2, shift up 7 units**
To reflect about the y-axis, shift left 4 units, and stretch horizontally by a factor of 2, you need to modify the argument of the function as follows:
\[
y = f\left(\frac{-x - 4}{2}\right) + 7
\]
Here:
- \( \frac{-x - 4}{2} \) shifts left by 4 and stretches horizontally by a factor of 2,
- Adding 7 shifts the graph up by 7 units.

#### Option 5: **Reflect about the x-axis, shift left 4 units, stretch horizontally by a factor of 2, shift up 7 units**
For a reflection about the x-axis, shift left by 4 units, and stretch horizontally by a factor of 2:
\[
y = -f\left(\frac{x - 4}{2}\right) + 7
\]
Here:
- The negative outside reflects about the x-axis,
- The horizontal stretch comes from dividing the \( x \)-term by 2,
- Shifting left by 4 is handled by \( x - 4 \),
- Adding 7 shifts the graph up by 7 units.

### Summary
- The equation you originally provided represents **a reflection about the y-axis, a shift right by 4 units, a vertical stretch by a factor of 2, and a shift down by 7 units**.
- Other options involve different reflections, shifts, or stretches.

Would you like further details on any of these transformations?

### Related Questions:
1. What is the general formula for reflecting a function over the y-axis?
2. How does a vertical stretch affect the appearance of a function’s graph?
3. How can you differentiate between horizontal and vertical transformations in a function?
4. What does it mean when a function is shifted vertically by a certain number of units?
5. How would the graph of \( y = f(-x) \) compare to \( y = f(x) \)?

### Tip:
Always apply horizontal transformations inside the function and vertical transformations outside the function to maintain clarity in transformations.

y = 2*f*(-x + 4) − 7 reflect about the x-axis, shift right 4 units, stretch vertically by a factor of 2, shift down 7 units; reflect about the y-axis, shift right 4 units, stretch vertically by a factor of 2, shift down 7 units; reflect about the y-axis, shift left 4 units, stretch vertically by a factor of 2, shift up 7 units; reflect about the y-axis, shift left 4 units, stretch horizontally by a factor of 2, shift up 7 units; reflect about the x-axis, shift left 4 units, stretch horizontally by a factor of 2, shift up 7 units

Learn how to apply function transformations including reflections, horizontal and vertical shifts, and stretches. This problem breaks down key transformations such as reflection over the x- and y-axes, vertical and horizontal stretches, and both upward and downward shifts. Suitable for high school students in Grades 10-12.

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