What is the equation of the normal line to the graph of $f ( x ) = x ^ { 2 } - 2 x$ at $x = 1$ ?

A)y = x
B)y = -x
C)y + 1 = x - 1
D)x = 1
E)y = -1

Question

What is the equation of the normal line to the graph of $f ( x ) = x ^ { 2 } - 2 x$ at $x = 1$ ?

A)y = x
B)y = -x
C)y + 1 = x - 1
D)x = 1
E)y = -1

Quizplus · Accepted Answer

The normal line to a curve at a given point is perpendicular to the tangent line at that point. To find the equation of the normal line to the graph of $f(x) = x^2 - 2x$ at $x = 1$, first find the derivative $f'(x) = 2x - 2$. At $x = 1$, $f'(1) = 2(1) - 2 = 0$, indicating the slope of the tangent line is 0. Therefore, the normal line, being perpendicular, will have an undefined slope, indicating it is a vertical line passing through $x = 1$, which is described by the equation $x = 1$.

What Is the Equation of the Normal Line to the Graph