The function $g(x) = (x - 2)^2 - 3$ is a quadratic function in vertex form, which is given as:

$g(x) = a(x - h)^2 + k$

Here, $h$ and $k$ represent the vertex of the parabola, and the axis of symmetry is the vertical line $x = h$.

For the function $g(x) = (x - 2)^2 - 3$:

• $h = 2$
• $k = -3$

So, the vertex is $(2, -3)$.

Axis of symmetry:

The axis of symmetry is the vertical line that passes through the vertex, so it is:

$x = 2$

Graph:

The parabola opens upwards since the coefficient of $(x - 2)^2$ is positive (the coefficient is 1), and the vertex is at $(2, -3)$. The general shape will resemble a U-shaped curve with its lowest point at the vertex.

Would you like me to graph it for you?

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Here are some additional questions to deepen understanding:

1. How does changing the value of $h$ affect the graph of the function?
2. What happens if the coefficient in front of $(x - 2)^2$ changes to a negative number?
3. How can you find the y-intercept of the function $g(x)$?
4. How would the graph look if the function were $g(x) = (x + 2)^2 - 3$?
5. How does the value of $k$ influence the graph's vertical position?

Tip: The vertex form of a quadratic function makes it easy to identify the vertex and axis of symmetry, simplifying graphing tasks!