Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together. \( y=-4 x^{3},(-1,4) \) An equation for the tangent line is (Type an equation.)

1. Differentiate the function with respect to \( x \). Since \( y = -4x^3 \), the derivative is \[ \frac{dy}{dx} = -4 \cdot 3x^2 = -12x^2. \] 2. Evaluate the derivative at the given point \( x = -1 \) to find the slope of the tangent line. \[ m = -12(-1)^2 = -12 \cdot 1 = -12. \] 3. Use the point-slope form of a line with the point \( (-1, 4) \) and slope \( m = -12 \). The point-slope form of a line is given by \[ y - y_1 = m(x - x_1). \] Substitute \( x_1 = -1 \), \( y_1 = 4 \), and \( m = -12 \) into the formula: \[ y - 4 = -12(x - (-1)) = -12(x + 1). \] 4. Solve for \( y \) to write the equation in slope-intercept form: \[ y - 4 = -12x - 12. \] Adding 4 to both sides: \[ y = -12x - 12 + 4, \] \[ y = -12x - 8. \] Thus, an equation for the tangent line is \[ y = -12x - 8. \]