From the provided image, the task involves selecting the graph of the function $f(x) = x^2 - 1$ alongside its parent function $y = x^2$, and describing the transformation.

Key Observations:

1. Parent Function: The parent function is $y = x^2$, which is a parabola opening upwards with its vertex at the origin $(0, 0)$.
2. Given Function: $f(x) = x^2 - 1$ represents a vertical translation of the parent function downwards by 1 unit.

Transformation:

• The function $f(x) = x^2 - 1$ moves the graph of $y = x^2$ 1 unit downward.
• The vertex shifts from $(0, 0)$ to $(0, -1)$.

Correct Graph:

The correct graph is the one where the vertex of $f(x)$ is located at $(0, -1)$, and the shape of the parabola remains the same as the parent function.

Completing the Sentence:

"The graph of $f$ is a translation 1 unit downward of the parent function."

Would you like me to verify specific details about the transformations or explain further?

Here are 5 related questions to expand your understanding:

1. What other transformations, such as horizontal shifts, would modify the graph of $y = x^2$?
2. How would the graph change if the equation was $f(x) = x^2 + 1$?
3. What effect does a coefficient $a$ in $f(x) = a(x^2 - 1)$ have on the graph?
4. How can we determine transformations algebraically versus visually?
5. What happens to the graph if the equation changes to $f(x) = -(x^2 - 1)$?

Tip: Always identify the vertex and any scaling factors to quickly understand transformations!