An equation of the tangent line to the graph $f ( x ) = x ^ { 2 } - x$ at the point (-2, 6) is

A) $y = - 5 x - 4$
B) $y = - 5 x - 2$
C) $y = - 5 x + 2$
D) $y = - 5 x + 3$
E) $y = - 5 x + 4$

Question

An equation of the tangent line to the graph $f ( x ) = x ^ { 2 } - x$ at the point (-2, 6) is

A) $y = - 5 x - 4$
B) $y = - 5 x - 2$
C) $y = - 5 x + 2$
D) $y = - 5 x + 3$
E) $y = - 5 x + 4$

Quizplus · Accepted Answer

To find the equation of the tangent line to the graph at a specific point, we first need to find the derivative of the function, which gives us the slope of the tangent line at any point $x$. The derivative of $f(x) = x^2 - x$ is $f'(x) = 2x - 1$. To find the slope of the tangent line at $x = -2$, we substitute $x = -2$ into the derivative: $f'(-2) = 2(-2) - 1 = -4 - 1 = -5$. So, the slope of the tangent line at the point (-2, 6) is -5.Next, we use the point-slope form of a line, $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is the point on the line. Substituting the slope (-5) and the point (-2, 6) gives us $y - 6 = -5(x + 2)$. Simplifying this equation gives $y = -5x - 10 + 6$, which simplifies further to $y = -5x - 4$, matching option A.

An Equation of the Tangent Line to the Graph f(x)=x2−xf ( x ) = x ^ { 2 } - xf(x)=x2−x

An Equation of the Tangent Line to the Graph $f ( x ) = x ^ { 2 } - x$