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To find the equation of the tangent line to the function \( f(x) = 2x^2 + 4 \) at \( x = 1 \), we can follow these steps:

### Step 1: Find the derivative of the function
The derivative \( f'(x) \) will give us the slope of the tangent line at any point \( x \).

\[
f'(x) = \frac{d}{dx}(2x^2 + 4) = 4x
\]

### Step 2: Evaluate the derivative at \( x = 1 \)
Substitute \( x = 1 \) into the derivative to find the slope of the tangent line at that point.

\[
f'(1) = 4(1) = 4
\]

Thus, the slope of the tangent line at \( x = 1 \) is 4.

### Step 3: Find the point on the curve at \( x = 1 \)
To find the y-coordinate at \( x = 1 \), substitute \( x = 1 \) into the original function \( f(x) \).

\[
f(1) = 2(1)^2 + 4 = 2 + 4 = 6
\]

Thus, the point of tangency is \( (1, 6) \).

### Step 4: Use the point-slope form to write the equation of the tangent line
The point-slope form of the equation of a line is:

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

Where:
- \( m \) is the slope (which is 4),
- \( (x_1, y_1) \) is the point of tangency \( (1, 6) \).

Substitute these values into the equation:

\[
y - 6 = 4(x - 1)
\]

Simplifying:

\[
y - 6 = 4x - 4
\]
\[
y = 4x - 4 + 6
\]
\[
y = 4x + 2
\]

### Final Answer:
The equation of the tangent line is \( y = 4x + 2 \).

Would you like any further details on how this was derived? Here are some related questions you can consider:

1. What are other methods to find the equation of a tangent line?
2. How can you verify that the tangent line and the curve intersect at only one point?
3. How does the derivative of a function geometrically represent the slope of the tangent line?
4. What happens to the tangent line when the function changes, for example, if \( f(x) \) is a cubic function?
5. Can you find the tangent line to this function at other points, like \( x = 2 \)?

### Tip:
Always remember to double-check both the derivative and the point of tangency when solving tangent line problems to avoid small errors.

Write an equation for the tangent line to f(x) = 2x^2 + 4 at x = 1.

Learn how to find the equation of the tangent line to the quadratic function f(x) = 2x^2 + 4 at x = 1 using derivatives and the point-slope form of a line. Suitable for high school calculus students (Grades 10-12).

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