https://file.aiquickdraw.com/mathbot/file/1726713103696c97q0xxw.jpg

what graph is it. is it the first, second, third, or fourth?

Let's break down the problem based on the information in the image. The function provided is:

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

You are asked to find the tangent line at the point \( P(3, 12) \).

### Steps to determine the correct graph:

1. **Find the slope of the tangent line using the derivative:**
   The derivative of \( f(x) = x^2 + 3 \) is:
   \[
   f'(x) = 2x
   \]
   At the point \( x = 3 \):
   \[
   f'(3) = 2(3) = 6
   \]
   The slope of the tangent line is 6.

2. **Equation of the tangent line:**
   The point-slope form of a line is:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \( m = 6 \) and \( (x_1, y_1) = (3, 12) \):
   \[
   y - 12 = 6(x - 3)
   \]
   Simplifying:
   \[
   y = 6x - 18 + 12 = 6x - 6
   \]
   So, the equation of the tangent line is \( y = 6x - 6 \).

### Now, based on the graph:

- The function \( f(x) = x^2 + 3 \) is a parabola shifted upward by 3 units.
- The tangent line at \( x = 3 \) has a slope of 6, which should appear quite steep.
- You need to identify which graph shows a parabola that is consistent with \( f(x) = x^2 + 3 \), and at \( x = 3 \), the tangent line should be steep and touch the curve exactly once.

Looking at the graphs in the image, the **second graph** seems most consistent with these features. It shows a parabola, and the tangent line appears to have a steep slope at \( x = 3 \).

So, the correct graph is **the second one**.

Would you like further explanation or have any other questions?

Here are 5 related questions:
1. How do you calculate the derivative of a general quadratic function?
2. Why is the derivative at a point considered the slope of the tangent line?
3. How does the point-slope form of a line work, and why is it useful?
4. What would the tangent line look like for a different function, like \( f(x) = x^2 - 5x + 6 \)?
5. How would you find the tangent line at a point if the function was more complex, like \( f(x) = x^3 + 2x \)?

**Tip:** When finding the tangent line, always first find the slope by taking the derivative and then apply the point-slope form of a line to make it easier.

What graph is it? Is it the first, second, third, or fourth?

Learn how to find the correct graph showing the tangent line for the quadratic function f(x) = x^2 + 3 at the point P(3, 12). The solution involves calculating the slope using the derivative, finding the equation of the tangent line, and matching it with the correct graph. Ideal for high school calculus students.

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