The given problem shows the graph of a function $y = f(x)$ with:

• A maximum turning point at $(-2, 5)$
• A minimum turning point at $(3, -1)$

You are asked to sketch the graph of $y = f(x + 3) + 2$.

Explanation:

1. Horizontal Shift:

   • The expression $f(x + 3)$ indicates a horizontal shift of the graph by 3 units to the left. This means every point on the original graph moves 3 units to the left along the x-axis.

2. Vertical Shift:

   • The $+2$ outside the function indicates a vertical shift of 2 units upwards. This moves every point on the graph up by 2 units.

Transforming the key points:

• The maximum turning point $(-2, 5)$ on the original graph:

  • Horizontal shift: $(-2 - 3) = -5$
  • Vertical shift: $5 + 2 = 7$
  • New point: $(-5, 7)$

• The minimum turning point $(3, -1)$ on the original graph:

  • Horizontal shift: $(3 - 3) = 0$
  • Vertical shift: $-1 + 2 = 1$
  • New point: $(0, 1)$

Sketching the new graph:

The new graph will have:

• A maximum turning point at $(-5, 7)$
• A minimum turning point at $(0, 1)$

The overall shape remains the same as the original graph, but it's shifted to the left by 3 units and upwards by 2 units.

Would you like more details on sketching or have any questions?

Here are 5 relative questions to expand your understanding:

1. How do horizontal and vertical shifts affect the derivative of a function?
2. What happens to the symmetry of a graph after a transformation like $f(x + 3) + 2$?
3. Can you explain how the turning points change after applying horizontal and vertical stretches or compressions?
4. What is the effect of a negative coefficient on the function $f(x)$, like $f(-x)$?
5. How would the graph change if the transformation was $y = f(x-3) - 2$ instead?

Tip: When sketching transformations, always start by shifting key points like maximums, minimums, or intercepts to guide your drawing.