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To find the equation of the tangent line to the curve \( y = 3x^2 - 4 \) at \( x = 2 \), we need to follow these steps:

1. **Find the derivative** of the function to get the slope of the tangent line.
   
   The given equation is \( y = 3x^2 - 4 \). The derivative \( \frac{dy}{dx} \) represents the slope of the tangent line at any point \( x \).

   \[
   \frac{dy}{dx} = \frac{d}{dx}(3x^2 - 4) = 6x
   \]

2. **Evaluate the slope at \( x = 2 \)**.
   
   Substituting \( x = 2 \) into the derivative:

   \[
   \text{slope} = 6(2) = 12
   \]

   So, the slope of the tangent line at \( x = 2 \) is 12.

3. **Find the y-coordinate at \( x = 2 \)**.
   
   To get the point where the tangent touches the curve, substitute \( x = 2 \) into the original equation:

   \[
   y = 3(2)^2 - 4 = 12 - 4 = 8
   \]

   Thus, the point of tangency is \( (2, 8) \).

4. **Use the point-slope form of the equation of a line**.
   
   The point-slope form is given by:

   \[
   y - y_1 = m(x - x_1)
   \]

   where \( m \) is the slope, and \( (x_1, y_1) \) is the point on the line. Substituting \( m = 12 \), \( x_1 = 2 \), and \( y_1 = 8 \):

   \[
   y - 8 = 12(x - 2)
   \]

   Simplifying this equation:

   \[
   y - 8 = 12x - 24
   \]
   \[
   y = 12x - 16
   \]

So, the equation of the tangent line is \( y = 12x - 16 \).

Would you like any further details or explanations?

Here are some related questions you might find useful:

1. How do you find the derivative of more complex functions like trigonometric or exponential functions?
2. What is the geometric interpretation of the derivative in terms of slopes and tangents?
3. Can the tangent line ever be vertical? Under what conditions?
4. How does one find the normal line to a curve at a given point?
5. What happens to the tangent line as \( x \to \infty \) for polynomials of higher degrees?

**Tip:** When finding the tangent line, always ensure you evaluate both the slope and the point of tangency correctly for accurate results.

Find the equation of the tangent to the curve y = 3x^2 - 4 at x = 2.

Learn how to find the equation of the tangent line to the quadratic curve y = 3x^2 - 4 at x = 2. This problem involves calculating the derivative to get the slope and applying the point-slope form of the line. Suitable for Grades 10-12.

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