Question

does the parabola $y=2x^{2}-17x+7$ have a tangent line whose slope is - 1? if so, find an equation for the line and the point of tangency. if not, why not?
select the correct choice below and, if necessary, fill in the answer boxes within your choice.
a. the parabola has a tangent line whose slope is - 1. the equation of the tangent line is $\boldsymbol{\square}$ and the point of tangency is $\boldsymbol{\square}$.
b. the parabola does not have a tangent line whose slope is - 1 because y does not have the value - 1 for any x-value in the domain of the curve.
c. the parabola does not have a tangent line whose slope is - 1 because $y$ does not have the value - 1 for any x-value in the domain of the curve.
d. the parabola does not have a tangent line whose slope is - 1 because the equation of the tangent line does not have the value - 1 for any x-value in the domain of the curve.

Explanation:

Step1: Find derivative of parabola

The derivative of $y=2x^2-17x+7$ is $y' = 4x - 17$.

Step2: Set slope equal to -1

Set $4x - 17 = -1$ and solve for $x$:
$4x = 16$
$x = 4$

Step3: Find y-coordinate of point

Substitute $x=4$ into original equation:
$y=2(4)^2-17(4)+7=2(16)-68+7=32-68+7=-29$

Step4: Find tangent line equation

Use point-slope form $y-y_1=m(x-x_1)$ with $(x_1,y_1)=(4,-29)$ and $m=-1$:
$y - (-29) = -1(x - 4)$
$y + 29 = -x + 4$
$y = -x -25$

Answer:

A. The parabola has a tangent line whose slope is -1. The equation of the tangent line is $y = -x -25$ and the point of tangency is $(4, -29)$.