Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together. \( y=2-x^{2},(1,1) \) An equation for the tangent line is (Type an equation.)

To find the equation of the tangent line to the curve \( y = 2 - x^2 \) at the point \( (1, 1) \), we first need to compute the derivative of the function to find the slope at that point. The derivative is \( \frac{dy}{dx} = -2x \). Evaluating this at \( x = 1 \), we get \( \frac{dy}{dx} = -2(1) = -2 \). Now, using the point-slope form of the equation of a line, we can write the equation of the tangent line as: \[ y - 1 = -2(x - 1) \] Simplifying this, we get: \[ y = -2x + 3 \] So, the equation for the tangent line is \( y = -2x + 3 \). On sketching, you'll see the curve opening downwards, intersecting the y-axis at (0,2), and the tangent line touching the curve at the point (1,1) while slanting downwards.