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To find the equation of \( y \) given the derivative \( \frac{dy}{dx} = 2(x - 1) \) and the point \( (1, 2) \) on the curve, we can follow these steps:

1. **Integrate the Derivative**:
   The derivative given is \( \frac{dy}{dx} = 2(x - 1) \).
   To find \( y \), integrate \( 2(x - 1) \) with respect to \( x \).

   \[
   y = \int 2(x - 1) \, dx
   \]

2. **Perform the Integration**:
   Break it down:
   
   \[
   y = 2 \int (x - 1) \, dx
   \]

   Integrate \( x - 1 \) term-by-term:

   \[
   y = 2 \left( \frac{x^2}{2} - x \right) + C
   \]

   Simplify:

   \[
   y = x^2 - 2x + C
   \]

3. **Use the Point \( (1, 2) \) to Find \( C \)**:
   Substitute \( x = 1 \) and \( y = 2 \) into the equation to solve for \( C \).

   \[
   2 = (1)^2 - 2 \cdot 1 + C
   \]

   Simplify:

   \[
   2 = 1 - 2 + C
   \]
   \[
   2 = -1 + C
   \]
   \[
   C = 3
   \]

4. **Write the Final Equation**:
   Substitute \( C = 3 \) back into the equation:

   \[
   y = x^2 - 2x + 3
   \]

### Final Answer
The equation of \( y \) is:
\[
y = x^2 - 2x + 3
\]

Would you like further details on any of the steps? 

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Here are five related questions to deepen your understanding:

1. What would the solution be if the derivative were \( \frac{dy}{dx} = 3(x - 1) \) instead?
2. How would the integration change if \( \frac{dy}{dx} = 2(x + 1) \) instead?
3. Why do we add a constant \( C \) after integrating, and how is it determined?
4. How would you interpret the derivative \( \frac{dy}{dx} = 2(x - 1) \) graphically in terms of slope?
5. What does the point \( (1, 2) \) represent in terms of the original function \( y(x) \)?

**Tip:** Always use known points on the curve to solve for the constant \( C \) after integrating a derivative.

Find the equation of y, given the derivative \( \frac{dy}{dx} = 2(x - 1) \) and the indicated point (1, 2) on the curve.

Learn how to find the equation of y given the derivative \( \frac{dy}{dx} = 2(x - 1) \) and a point on the curve (1, 2). This solution involves calculus concepts such as integration and solving for constants. Suitable for high school students in Grades 11-12.

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