We are given two functions:

• $f(x) = x^2$, which is a basic parabola.
• $g(x) = (x + 5)^2 + 1$, which is a transformation of $f(x)$.

Describing the Transformation:

• The term $(x + 5)$ indicates a horizontal translation. The graph is shifted 5 units to the left.
• The $+1$ indicates a vertical translation. The graph is shifted 1 unit up.

So, the graph of $g(x)$ is a translation 5 units left and 1 unit up from the graph of $f(x)$.

Identifying the Graph:

Looking at the options in the image:

• We need to find the graph where $g(x)$ (the red parabola) is 5 units to the left and 1 unit higher than $f(x)$ (the blue parabola).

By analyzing the provided graphs:

• The third graph (bottom left) shows the red graph $g(x)$ shifted 5 units to the left and 1 unit up from the blue graph $f(x)$, which matches the transformation.

Thus, the correct graph is the third one (bottom left).

Would you like more details on transformations or help with another problem?

Related Questions:

1. How do you find the vertex of a transformed quadratic function?
2. What is the effect of adding a constant outside the parentheses on the graph of a function?
3. What other types of transformations can be applied to $f(x) = x^2$?
4. How do horizontal and vertical translations affect the domain and range of the function?
5. Can you describe the reflection of $f(x) = x^2$ across the x-axis or y-axis?

Tip:

Always remember that a transformation inside the parentheses (e.g., $(x + 5)$) affects the graph horizontally, while a transformation outside (e.g., $+1$) affects the graph vertically.