Given the function $g(x) = f(x+3) + 5$, let's analyze how the vertex of the original quadratic function $f(x)$ changes.

Step 1: Identify the effect of $f(x+3)$

The term $f(x+3)$ represents a horizontal shift of the original function $f(x)$. Specifically, it shifts the graph 3 units to the left.

• The vertex of the original function $f(x)$ is at $(5, -3)$.
• Shifting it 3 units to the left gives the new x-coordinate of the vertex: $5 - 3 = 2$

Step 2: Identify the effect of adding 5

The term $+5$ represents a vertical shift of the graph 5 units upward.

• The original y-coordinate of the vertex is $-3$.
• Shifting it 5 units upward gives the new y-coordinate of the vertex: $-3 + 5 = 2$

Step 3: Combine the shifts

After applying the shifts, the new vertex of the function $g(x) = f(x+3) + 5$ is: $(2, 2)$

Thus, the vertex of the function $g(x)$ is $(2, 2)$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How does changing $f(x+k)$ affect the graph of a function?
2. What happens if a constant is subtracted from the function $f(x)$?
3. How do vertical stretches or compressions affect the vertex of a parabola?
4. How can we determine the axis of symmetry from the vertex form of a quadratic function?
5. What is the standard form of a quadratic function?

Tip: The vertex form of a quadratic function $f(x) = a(x-h)^2 + k$ makes it easy to identify the vertex $(h, k)$ directly.