Questions: Find an equation for the tangent line to the graph of f(x)=x^2+4 at (-5,29). y=□

Solution Steps

To find the equation of the tangent line to the graph of the function at a given point, we need to determine the slope of the tangent line at that point. This is done by finding the derivative of the function and evaluating it at the given x-coordinate. Once we have the slope, we can use the point-slope form of a line to write the equation of the tangent line.

The function given is \( f(x) = x^2 + 4 \). To find the slope of the tangent line at the point \((-5, 29)\), we first compute the derivative of the function: \[ f'(x) = 2x \]

Next, we evaluate the derivative at \( x = -5 \): \[ f'(-5) = 2(-5) = -10 \] Thus, the slope of the tangent line at the point \((-5, 29)\) is \( m = -10 \).

Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), we substitute \( m = -10 \), \( x_1 = -5 \), and \( y_1 = 29 \): \[ y - 29 = -10(x + 5) \] Rearranging this gives: \[ y = -10x - 50 + 29 \] \[ y = -10x - 21 \]

Final Answer