To solve the given problem, we need to transform the graph of the function $y = f(x)$ based on the given transformation $y = -3f(2x) + 4$.

Step-by-Step Explanation

1. Understanding the Original Graph: The original graph, $y = f(x)$, includes the following key points:

   • $(-2, 2)$
   • $(1, 2)$
   • $(2, 0)$
   • $(3, 0)$
   • $(4, 2)$ These points define the important features of the graph, including peaks, troughs, and intersections with the x-axis.

2. Transformation Breakdown: The transformation $y = -3f(2x) + 4$ includes several operations:

   • Horizontal Compression (by a factor of 2): The term $f(2x)$ means that all x-coordinates are compressed towards the y-axis by a factor of 2. This halves the x-coordinates of all points.

   • Vertical Stretch (by a factor of 3): The term $-3f(x)$ implies that the graph is stretched vertically by a factor of 3. Additionally, the negative sign reflects the graph about the x-axis.

   • Vertical Shift (upward by 4 units): The term $+4$ shifts the entire graph upward by 4 units.

Applying the Transformation to Key Points:

Let's now apply these transformations to each key point on the original graph.

• Point $(-2, 2)$:

  • Horizontal Compression: New x-coordinate = $-2/2 = -1$
  • Vertical Stretch and Reflection: New y-coordinate = $-3(2) = -6$
  • Vertical Shift: New y-coordinate = $-6 + 4 = -2$
  • Transformed Point: $(-1, -2)$

• Point $(1, 2)$:

  • Horizontal Compression: New x-coordinate = $1/2 = 0.5$
  • Vertical Stretch and Reflection: New y-coordinate = $-3(2) = -6$
  • Vertical Shift: New y-coordinate = $-6 + 4 = -2$
  • Transformed Point: $(0.5, -2)$

• Point $(2, 0)$:

  • Horizontal Compression: New x-coordinate = $2/2 = 1$
  • Vertical Stretch and Reflection: New y-coordinate = $-3(0) = 0$
  • Vertical Shift: New y-coordinate = $0 + 4 = 4$
  • Transformed Point: $(1, 4)$

• Point $(3, 0)$:

  • Horizontal Compression: New x-coordinate = $3/2 = 1.5$
  • Vertical Stretch and Reflection: New y-coordinate = $-3(0) = 0$
  • Vertical Shift: New y-coordinate = $0 + 4 = 4$
  • Transformed Point: $(1.5, 4)$

• Point $(4, 2)$:

  • Horizontal Compression: New x-coordinate = $4/2 = 2$
  • Vertical Stretch and Reflection: New y-coordinate = $-3(2) = -6$
  • Vertical Shift: New y-coordinate = $-6 + 4 = -2$
  • Transformed Point: $(2, -2)$

Final Result:

The transformed graph of $y = -3f(2x) + 4$ includes the following points:

• $(-1, -2)$
• $(0.5, -2)$
• $(1, 4)$
• $(1.5, 4)$
• $(2, -2)$

You can now plot these points and sketch the transformed graph. The graph will be a vertically stretched, reflected, and shifted version of the original, with the same basic shape but modified based on the transformation.

Would you like further details or assistance with this problem?

Related Questions:

1. How does horizontal compression affect the graph of a function?
2. Why does multiplying the function by a negative constant reflect the graph about the x-axis?
3. How does a vertical stretch differ from a vertical shift in terms of transformations?
4. What are other examples of function transformations that change both the x- and y-coordinates?
5. How would the graph change if the transformation was $y = 3f(2x) - 4$ instead?

Tip:

Always apply transformations in the correct order: handle horizontal changes (like compression/stretch) first, followed by vertical changes (like stretch/reflection), and finish with shifts.