Let's examine the two graphs provided to identify their corresponding equations based on the standard form of a quadratic function, $f(x) = ax^2 + bx + c$. Since the equation you're asked to match is based on $f(x) = x^2$, we'll focus on the transformations that adjust the graph of $f(x) = x^2$.

(a) Graph Analysis:

The graph is an upright parabola (opening upwards), with the vertex at the origin $(0, 0)$. This matches the basic graph of $y = x^2$, meaning there are no transformations (like shifts, stretches, or compressions).

Equation for (a): $y = x^2$

(b) Graph Analysis:

The second graph is an inverted parabola (opening downwards), with the vertex at $(0, 2)$. The vertex form of a quadratic function is: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex. In this case, the vertex is $(0, 2)$, so we substitute: $y = a(x - 0)^2 + 2 \quad \Rightarrow \quad y = a x^2 + 2$ Since the parabola opens downward, the coefficient $a$ must be negative. Observing that the parabola has a similar "width" to $y = x^2$, we can assume $a = -1$.

Equation for (b): $y = -x^2 + 2$

Would you like more details on how these transformations work or have any other questions? Here are 5 related questions for practice:

1. What would be the equation of a parabola that is stretched vertically by a factor of 2 and opens upwards?
2. How does the graph of $y = (x - 3)^2$ differ from $y = x^2$?
3. What effect does a negative coefficient in front of $x^2$ have on the graph of a quadratic function?
4. How would the equation change if the vertex of the parabola in graph (b) was at $(2, -3)$?
5. What is the vertex form of the equation for a parabola with a vertex at $(h, k)$?

Tip: When identifying transformations of quadratic functions, always start by identifying the vertex and whether the parabola opens upward or downward.