In the image you provided, the task involves graphing transformations of the parent function $f(x) = x^2$ and identifying the type of transformation. Let me walk through each of the parts visible in the image:

Part a. $g(x) = x^2 + 2$

• Graph: The graph of $g(x) = x^2 + 2$ is a vertical shift of the parent function $f(x) = x^2$ upward by 2 units.
• Transformation: Vertical translation up by 2 units.

Part b. $h(x) = (x + 2)^2$

• Graph: The graph of $h(x) = (x + 2)^2$ is a horizontal shift of the parent function $f(x) = x^2$ to the left by 2 units.
• Transformation: Horizontal translation to the left by 2 units.

Part c. $k(x) = 2x^2$

• Graph: The graph of $k(x) = 2x^2$ stretches the graph of $f(x) = x^2$ vertically by a factor of 2, making the parabola narrower.
• Transformation: Vertical stretch by a factor of 2.

Part d. $p(x) = -x^2$

• Graph: The graph of $p(x) = -x^2$ reflects the parent function $f(x) = x^2$ across the x-axis.
• Transformation: Reflection over the x-axis.

If you need a more detailed breakdown of any part or further explanations, feel free to ask!

Follow-up Questions:

1. What would happen if $g(x) = x^2 - 3$? How would the graph shift?
2. How do horizontal shifts differ from vertical shifts in transformation?
3. What would be the transformation if $k(x) = \frac{1}{2}x^2$?
4. What kind of transformation occurs if the function is $h(x) = (x - 4)^2$?
5. Can you explain why the graph reflects over the x-axis in $p(x) = -x^2$?

Tip:

When dealing with horizontal shifts, the transformations inside the parentheses (like $(x + 2)$) move the graph in the opposite direction to the sign.