The problem provided asks to analyze two functions based on the base function $f(x) = x^2$. Let's break down each part:

A) $g(x) = (x + 3)^2$

This function is a transformation of the base function $f(x) = x^2$. Specifically, it represents a horizontal shift of the graph of $f(x)$. The term $(x + 3)$ shifts the graph 3 units to the left.

• Horizontal transformations follow the rule $f(x - h)$, where $h$ determines the direction of the shift (left or right). Here, $h = -3$, so the shift is to the left.

B) $g(x) = x^2 + 3$

This function is a vertical shift of the base function $f(x) = x^2$. The addition of $+3$ outside the square moves the entire graph 3 units upward.

• Vertical transformations occur when a constant is added outside of the base function, affecting the vertical position of the graph.

Answers:

• A) $g(x) = (x + 3)^2$ is a horizontal shift to the left by 3 units.
• B) $g(x) = x^2 + 3$ is a vertical shift upward by 3 units.

Do you need further details on how transformations work?

Here are 5 related questions to deepen understanding:

1. What would happen if $g(x) = (x - 4)^2$?
2. How does adding a negative constant outside the function, like $g(x) = x^2 - 2$, affect the graph?
3. What is the effect of multiplying the function $f(x)$ by a constant?
4. How does the transformation change if we have $g(x) = (x + 3)^2 + 5$?
5. What is the difference between a horizontal stretch and a shift?

Tip: Horizontal transformations affect the input (inside the parentheses), while vertical transformations affect the output (outside the function).