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The task requires determining the equation of the tangent \( t(x) \) to the graph of the function \( f(x) = 2x^2 - 7 \) at the point \( P(-2 | 1) \).

### Step-by-Step Solution:

1. **Find the derivative of \( f(x) \):**
   The derivative \( f'(x) \) gives us the slope of the tangent line at any point \( x \). For the given function:
   \[
   f(x) = 2x^2 - 7
   \]
   The derivative is:
   \[
   f'(x) = \frac{d}{dx}(2x^2 - 7) = 4x
   \]

2. **Determine the slope of the tangent line at \( x = -2 \):**
   The slope of the tangent line at the point \( P(-2, 1) \) is given by:
   \[
   f'(-2) = 4(-2) = -8
   \]
   So, the slope of the tangent line is \( -8 \).

3. **Find the equation of the tangent line:**
   The equation of the tangent line has the form:
   \[
   t(x) = f'(x_0)(x - x_0) + f(x_0)
   \]
   where \( x_0 = -2 \) and \( f(x_0) = 1 \).

   Substituting the values, we get:
   \[
   t(x) = -8(x + 2) + 1
   \]

4. **Simplify the equation:**
   Expanding and simplifying the equation of the tangent:
   \[
   t(x) = -8(x + 2) + 1 = -8x - 16 + 1 = -8x - 15
   \]

Thus, the equation of the tangent line is:
\[
t(x) = -8x - 15
\]

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Do you want further details on any part of this solution, or do you have any other questions?

Here are five related questions you might find helpful:
1. How do you interpret the geometric meaning of the derivative in relation to tangents?
2. What happens to the slope if the function \( f(x) \) changes?
3. How would you find the tangent at another point on the same function?
4. Can you solve the same problem for a cubic function, e.g., \( f(x) = x^3 \)?
5. What is the significance of the point of tangency in practical problems?

**Tip:** When finding the tangent line at a specific point, always check if the point lies on the curve by substituting the point’s coordinates into the original function.

Determine the equation of the tangent to the graph of f(x) = 2x^2 - 7 at the point P(-2 | 1).

Learn how to find the equation of the tangent line to the quadratic function f(x) = 2x^2 - 7 at the point P(-2 | 1). This problem involves calculating the derivative and using the tangent line formula. Suitable for high school students in Grades 10-12.

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