The task is to describe how to transform the graph of the function $f(x) = x^2$ into the given graph $g(x)$ and find a formula for $g(x)$. Based on the image, we have four transformations of the quadratic function $f(x) = x^2$ that need to be analyzed.

a) Transformation in the first graph:

• The vertex of the parabola has moved up by 2 units.
• Transformation: This is a vertical shift upwards by 2 units.
• Function term: $g(x) = x^2 + 2$.

b) Transformation in the second graph:

• The parabola has been shifted downward by 2 units.
• Transformation: This is a vertical shift downwards by 2 units.
• Function term: $g(x) = x^2 - 2$.

c) Transformation in the third graph:

• The parabola has been reflected across the x-axis, meaning it opens downward.
• Transformation: This is a vertical reflection and a shift downward by 2 units.
• Function term: $g(x) = -x^2 - 2$.

d) Transformation in the fourth graph:

• The vertex has been shifted to the right by 1 unit and upwards by 2 units.
• Transformation: This is a horizontal shift to the right by 1 unit and a vertical shift upwards by 2 units.
• Function term: $g(x) = (x - 1)^2 + 2$.

Would you like more details on any of these transformations, or do you have any specific questions?

Related Questions:

1. How do you perform a horizontal shift on a quadratic function?
2. What effect does a reflection over the x-axis have on the graph of a function?
3. How would you combine both horizontal and vertical shifts in a function transformation?
4. Can we reflect a parabola over the y-axis instead? What would the equation look like?
5. How does a vertical stretch or compression affect the graph of a quadratic function?

Tip:

To identify transformations easily, always start by looking at the vertex shift (horizontal or vertical) and whether the parabola has been reflected or stretched.