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We are given that the slope of the tangent line to a curve is represented by the derivative function:

\[
f'(x) = 6x^2 - 4x + 3.
\]

We need to find the equation of the curve, which requires integrating the derivative \( f'(x) \) to obtain \( f(x) \), the original function.

### Step 1: Integrating \( f'(x) \)

To find \( f(x) \), we integrate \( f'(x) \) term by term:

\[
f'(x) = 6x^2 - 4x + 3.
\]

Integrating each term:

- \(\int 6x^2 \, dx = 2x^3\),
- \(\int -4x \, dx = -2x^2\),
- \(\int 3 \, dx = 3x\).

Thus, the general form of \( f(x) \) is:

\[
f(x) = 2x^3 - 2x^2 + 3x + C,
\]

where \( C \) is the constant of integration.

### Step 2: Using the Point (0, 1)

We are given that the point \( (0, 1) \) is on the curve, meaning that when \( x = 0 \), \( f(x) = 1 \). Substituting this into the equation:

\[
f(0) = 2(0)^3 - 2(0)^2 + 3(0) + C = 1,
\]

which simplifies to:

\[
C = 1.
\]

### Step 3: Final Equation of the Curve

Now that we have the value of \( C \), the equation of the curve is:

\[
f(x) = 2x^3 - 2x^2 + 3x + 1.
\]

This is the required equation of the curve.

---

Would you like further clarification on any of these steps?

Here are 5 related questions to expand your understanding:
1. How do you interpret the meaning of the derivative in terms of the curve?
2. Why is it important to include the constant of integration when finding the antiderivative?
3. How does the point given in the problem help determine the unique curve?
4. How would you find the equation of the tangent line at a specific point on the curve?
5. How could you verify if your equation satisfies the slope function \( f'(x) \)?

**Tip:** When integrating to find a function, always remember to include the constant of integration unless additional conditions are given to determine it.

If the point (0, 1) is on the curve, find an equation of the curve, given that the slope of the tangent line to a curve is f'(x) = 6x^2 - 4x + 3.

Learn how to find the equation of a curve whose slope of the tangent line is given as f'(x) = 6x^2 - 4x + 3, with the point (0, 1) on the curve. This step-by-step solution includes the use of derivatives and integration, crucial concepts for calculus students.

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