The problem asks you to write an equation for the function in blue, which is a transformation of the basic function $y = x^2$, represented by the green graph.

Let's analyze the transformation:

1. Vertical shift:

   • The blue parabola's vertex is higher than that of the green parabola. The green parabola has a vertex at $(0,0)$, and the blue one appears to have a vertex at $(0, 5)$. So, the blue graph is shifted up by 5 units.

2. Horizontal shift:

   • Both graphs share the same x-coordinate for their vertex, at $x = 0$. There is no horizontal shift.

3. Vertical stretch:

   • The blue graph is narrower than the green graph, indicating a vertical stretch. To determine the factor, we can compare the y-values at a specific x-coordinate.
   • For example, at $x = 1$, the green parabola has $y = 1^2 = 1$, while the blue parabola has $y = 10$. Therefore, the blue parabola has been stretched vertically by a factor of 10.

Equation:

The general form of the transformed function is:

$y = a(x - h)^2 + k$

Where $(h, k)$ is the vertex, and $a$ is the vertical stretch factor.

• $h = 0$ (no horizontal shift)
• $k = 5$ (shift up by 5)
• $a = 10$ (vertical stretch by a factor of 10)

Thus, the equation of the blue graph is:

$y = 10x^2 + 5$

Let me know if you'd like further details or have questions!

Here are 5 related questions:

1. What is the effect of a negative value for $a$ in $y = a(x - h)^2 + k$?
2. How does a horizontal shift affect the graph of a quadratic function?
3. What happens to the graph if $k$ is negative instead of positive?
4. Can you describe the effect of fractional values of $a$ on the graph of $y = x^2$?
5. How would you reflect the graph across the x-axis?

Tip: For a transformation, always examine the vertex and the points on the graph to identify shifts and stretches.