A. positive and negative square roots of 7
b. finding the derivative and plugging in the positive and negative square roots of 7 yields a slope of -2 for both x values, meaning that the points must be parallel.
c. this is where i'm stuck again. i think this question is confusing to me because instead of giving the equation, i'm given the slope and the point and i don't know how to work backwards to give a and b values for this "family of curves." correct me if i'm wrong, but part c seems independent of the answers to part a and part b.

C.

yes, part c is completely independent of parts a and b.

they give you information about a point and the slope of a tangent line at that particular point. then they tell you to find the values of a and b.

since the derivative gives the tangent line slope, we can differentiate  with respect to , with   and   as constants.

my reasoning is this: notice how the formula for the derivative from part b was in terms of x and y. we do know that   and   and the value of the derivative (the slope) being , so if we differentiate, we might be able to solve for one of our constants.

note that  because   is a constant, not a function of . so if we differentiated, we could end up solving for .



our derivative has an unknown constant of a. but we can find this constant with our information.

the tangent line at point   has to have a slope of . substitute 1 for , 3 for , and   for .



to get the value of , use this value of a along with the given coordinates on the equation of the curve.



i confirmed my answer using a graphing calculator like desmos, attached. it seems like a reasonable tangent line.