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.