Given y=−x2−2x+8, the equation of the tangent line at the point with abscissa −2 is...
Explanation
Given the function y=−x2−2x+8. We want to find the equation of the tangent line at the point with abscissa x=−2.
Finding the Ordinate
Substitute x=−2 into the curve equation:
y=−(−2)2−2(−2)+8
y=−4+4+8
y=8
So, the point of tangency is (−2,8).
Finding the Gradient of the Tangent Line
The gradient of the tangent line m is the value of the first derivative y′ at the abscissa of the point of tangency.
y′=−2x−2
Substitute x=−2:
m=−2(−2)−2
m=4−2
m=2
Determining the Equation of the Tangent Line
Use the point-slope form for the equation of a line passing through (x1,y1) with gradient m:
y−y1=m(x−x1)
Substitute x1=−2, y1=8, and m=2:
y−8=2(x−(−2))
y−8=2(x+2)
y−8=2x+4
y=2x+4+8
y=2x+12
Thus, the equation of the tangent line is y=2x+12.