We are given that the vertex of the quadratic function $f(x)$ is $(6, 4)$, and the new function is defined as $g(x) = f(-x) + 2$.

Let's break down the transformation:

1. Reflection across the y-axis: The term $f(-x)$ reflects the graph of $f(x)$ over the y-axis. This will change the x-coordinate of the vertex from 6 to -6, because reflecting over the y-axis inverts the sign of the x-coordinate.

2. Vertical translation: The $+2$ outside of $f(-x)$ shifts the entire graph upwards by 2 units. This means the y-coordinate of the vertex will increase by 2 from 4 to 6.

Vertex of $g(x)$

• The x-coordinate changes from 6 to -6.
• The y-coordinate changes from 4 to $4 + 2 = 6$.

Thus, the new vertex of the function $g(x)$ is $(-6, 6)$.

Do you need any further clarifications or details?

Related questions:

1. What is the effect of reflecting a function across the x-axis instead of the y-axis?
2. How does a vertical shift affect the graph of a quadratic function?
3. What happens to the vertex if we apply a horizontal shift instead of a vertical shift?
4. How do horizontal and vertical reflections affect the symmetry of a parabola?
5. Can you derive the general form of a quadratic function based on its vertex?

Tip:

When transforming functions, always handle horizontal and vertical changes separately, as their effects on the graph are distinct.