Let's go through the questions based on the functions $f(x) = x^2$, $g(x) = x^2 + 2$, and $h(x) = x^2 - 3$.

1. Effect of $g$ and $h$ on $f$:

   • The functions $g(x)$ and $h(x)$ are transformations of the parent function $f(x) = x^2$. Specifically:
     • $g(x) = x^2 + 2$ is a vertical shift upwards by 2 units.
     • $h(x) = x^2 - 3$ is a vertical shift downwards by 3 units.
   • These transformations move the graph of $f(x)$ vertically without affecting its shape.

2. Coordinates of the vertices of $f$, $g$, and $h$:

   • The vertex of $f(x) = x^2$ is at $(0, 0)$.
   • For $g(x) = x^2 + 2$, the vertex is shifted up by 2, so the vertex is $(0, 2)$.
   • For $h(x) = x^2 - 3$, the vertex is shifted down by 3, so the vertex is $(0, -3)$.

3. Predicting the vertical shift from the function's equation:

   • Yes, the vertical shift can be identified by looking at the constant term in the equation.
     • For $f(x) = x^2$, there is no constant term, so it remains at $y = 0$.
     • For $g(x) = x^2 + 2$, the constant $+2$ indicates a shift up by 2.
     • For $h(x) = x^2 - 3$, the constant $-3$ indicates a shift down by 3.

4. Effect on the domain and range with a vertical shift:

   • The domain of all three functions remains the same, $(-\infty, \infty)$, as the vertical shift does not affect the input values.
   • The range changes based on the shift:
     • For $f(x)$, the range is $[0, \infty)$.
     • For $g(x)$, after shifting up by 2, the range becomes $[2, \infty)$.
     • For $h(x)$, after shifting down by 3, the range becomes $[-3, \infty)$.

5. Finding the equation from a graph with a vertical shift:

   • To determine the equation from a vertically shifted graph:
     • Identify the vertex of the parabola.
     • Compare the vertex’s $y$-coordinate to that of the parent function $f(x) = x^2$ at $(0, 0)$.
     • The difference in the $y$-coordinate gives the vertical shift, allowing you to write the equation as $x^2$ plus or minus that shift.

Would you like further details on any part of this explanation?

Here are some related questions you might consider:

1. What would happen if we applied horizontal shifts instead of vertical shifts to these functions?
2. How would the graphs of $f(x) = x^2$, $g(x) = x^2 + 2$, and $h(x) = x^2 - 3$ compare in terms of symmetry?
3. If we changed the functions to include a coefficient on $x^2$, how would that affect the graphs?
4. What are the transformations if we add a term inside the square, such as $(x - 1)^2$?
5. How does changing the sign of $x^2$ (e.g., $f(x) = -x^2$) affect the vertex and range?

Tip: Recognizing vertical shifts by the constant term can be very helpful for quickly sketching transformations without plotting every point!