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We are given the graph of \( y = x^2 \) (solid black curve) and asked to identify which function corresponds to the dotted orange curve from the provided options.

### Observing the Dotted Graph:
- The dotted graph represents a parabola that is similar to \( y = x^2 \), but it has been **shifted horizontally** and **vertically**.
- The vertex of the dotted graph appears to be at \( (1, 5) \), whereas the vertex of \( y = x^2 \) is at the origin \( (0, 0) \).

### Applying the Vertex Form of a Parabola:
The general form for a parabola is:

\[
y = a(x - h)^2 + k
\]

where:
- \( (h, k) \) is the vertex of the parabola,
- \( a \) determines the direction and width of the parabola.

In this case, the dotted parabola seems to be shifted horizontally by \( +1 \) (i.e., \( h = 1 \)) and vertically by \( +5 \) (i.e., \( k = 5 \)).

Thus, the equation of the dotted parabola is:

\[
y = (x - 1)^2 + 5
\]

### Answer:
The correct function representing the dotted graph is \( y = (x - 1)^2 + 5 \).

Let me know if you'd like further details! Here are five related questions for practice:

1. How would the equation change if the vertex were at \( (-1, -5) \)?
2. What is the effect of the coefficient \( a \) on the width of the parabola?
3. How would you graph \( y = (x + 3)^2 - 2 \)?
4. What is the vertex of the parabola \( y = (x - 4)^2 + 1 \)?
5. How does shifting a parabola horizontally and vertically affect its equation?

**Tip:** The signs in \( (x - h)^2 \) and \( k \) directly tell you the horizontal and vertical shifts.

The graph of y = x^2 is the solid black graph below. Which function represents the dotted graph?

Learn how to identify the transformation of the quadratic function y = x^2 into the form y = (x - 1)^2 + 5 by analyzing its horizontal and vertical shifts. This problem covers key algebraic concepts, including quadratic transformations and the vertex form of a parabola.

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