How can we find the points where the graph of the function f(x) = x² – 8 has a horizontal tangent line?

To determine the points at which the graph of the function f(x) = x² – 8 has a horizontal tangent line, we need to follow these steps:

1. Find the Derivative: The first step is to find the derivative of the function, which gives us the slope of the tangent line at any given point. For the function f(x) = x² – 8, we can use the power rule for differentiation.
2. Calculate the Derivative: The derivative f'(x) of the function is calculated as follows:

f'(x) = d(x²)/dx - d(8)/dx = 2x - 0 = 2x

3. Set the Derivative to Zero: A horizontal tangent line occurs where the slope (the derivative) is equal to zero:

2x = 0

4. Solve for x: To find the x-coordinate where the tangent is horizontal, solve the equation:

x = 0

5. Find the Corresponding y-coordinate: Now that we have the x-coordinate, we can find the corresponding y-coordinate by substituting x = 0 back into the original function:

f(0) = (0)² - 8 = -8

6. Conclusion: Therefore, the point at which the graph of the function f(x) = x² – 8 has a horizontal tangent line is:

(0, -8)

In summary, the function has a horizontal tangent line at the point (0, -8).