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The given function is:

\[
y = x^4 - 20x^2 + 64
\]

We are asked to find the slope of the tangent line at \( x = 2 \).

### Step 1: Find the derivative (slope of the tangent)
The slope of the tangent line is given by the derivative of the function. Let's differentiate the function \( y = f(x) \) with respect to \( x \):

\[
y' = \frac{d}{dx} \left( x^4 - 20x^2 + 64 \right)
\]

Using the power rule for derivatives:

\[
y' = 4x^3 - 40x
\]

### Step 2: Evaluate the derivative at \( x = 2 \)
To find the slope of the tangent line at \( x = 2 \), substitute \( x = 2 \) into the derivative:

\[
y'(2) = 4(2)^3 - 40(2)
\]
\[
y'(2) = 4(8) - 40(2) = 32 - 80 = -48
\]

Thus, the slope of the tangent line at \( x = 2 \) is \( -48 \).

### Step 3: Find the equation of the tangent line
Now, we use the point-slope form of the equation of a line:

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

Where:
- \( m = -48 \) (the slope),
- \( x_1 = 2 \),
- \( y_1 = f(2) \) (the value of \( y \) at \( x = 2 \)).

First, calculate \( y_1 \) by substituting \( x = 2 \) into the original function:

\[
y(2) = (2)^4 - 20(2)^2 + 64
\]
\[
y(2) = 16 - 80 + 64 = 0
\]

So, \( y_1 = 0 \). Now, using the point \( (2, 0) \) and the slope \( m = -48 \), the equation of the tangent line is:

\[
y - 0 = -48(x - 2)
\]
\[
y = -48x + 96
\]

Thus, the equation of the tangent line is:

\[
y = -48x + 96
\]

Would you like further clarification or additional details on any part of the solution?

Here are five related questions you could explore:

1. What is the second derivative of the function, and what does it tell us about the concavity of the graph at \( x = 2 \)?
2. How would the tangent line change if \( x \) was a different value, such as \( x = 1 \)?
3. What is the y-intercept of the tangent line found here?
4. How can you use this tangent line approximation to estimate values of the function near \( x = 2 \)?
5. What would happen if the function had an additional term, such as \( x^3 \)?

**Tip**: When differentiating powers of \( x \), remember that the power rule \( \frac{d}{dx} x^n = n x^{n-1} \) is essential for finding the slope of tangent lines.

Find the slope and the equation of the tangent line to the graph of the function at x = 2. y = x^4 - 20x^2 + 64; x = 2

Learn how to find the slope and equation of the tangent line to the function y = x^4 - 20x^2 + 64 at x = 2. This solution explains key concepts such as differentiation using the power rule and finding tangent lines using the point-slope form of a line equation.

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