Questions: Find an equation for the tangent line to the graph of the given function at (-5,16). f(x)=x^2-9

Solution Steps

To find the equation of the tangent line, we first need to determine the slope of the tangent line at the given point. The slope of the tangent line is the value of the derivative of the function at that point.

Given the function: \[ f(x) = x^2 - 9 \]

The derivative of the function is: \[ f'(x) = \frac{d}{dx}(x^2 - 9) = 2x \]

Next, we evaluate the derivative at \( x = -5 \) to find the slope of the tangent line at the point \((-5, 16)\).

\[ f'(-5) = 2(-5) = -10 \]

Thus, the slope of the tangent line at the point \((-5, 16)\) is \(-10\).

The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \]

where \( m \) is the slope, and \((x_1, y_1)\) is the point on the line. Substituting the values we have:

• \( m = -10 \)
• \( (x_1, y_1) = (-5, 16) \)

The equation becomes: \[ y - 16 = -10(x + 5) \]

Final Answer