Question 1 ( 20 points) Given the parabola y = -(x-2) and the line: y = x-8. a. Graph them on the same set of axes and find the parabola's negativity interval. b. Find the interval where the graph of the parabola is above the graph of the line. c. Find the equation of a tangent line to the graph of the parabola, which has a slope of 4. Question 2 ( 15 points) Given the following equation: ax + ax = x-1+2a where "a" is a constant. a. Find its two solutions and simplify the results. b. Can you find values of the constant "a", for which the equation would have only one solution? If not, explain why and if so, find them. Question 3 (20 points) Given the function: f (x) = 2. extl - e. e2x-1 +16 a. Does the function intersect the x- axis? If so, find the intersection point (or points) and if not, explain why. b. Find the equation of the tangent line to the graph of the function at its extreme point. Question 4 (20 points) Given the function: y = xv4x-x2 . Find its domain and all of its extreme points Question 5 (25 points) Given the function: y = (x-a)? x2 -4 a. The slope of the tangent line to the graph of the function at x =0 is 2. Find "a". Plug a = 2 (even if you got a different value in part a) and answer: b. Find vertical asymptotes and discontinuity points (If they exist) c. Find increase and decrease intervals (if they exist). d. Find the function's positivity interval.d. Find the function's positivity interval. Formula sheet -Special product identities: (a+b)(a-b) = a -b? (atb) =a+2ab+b - Quadratic formula: ax + bx + c=0 the solutions: x, , _-b# vb -4ac 2a -Trinomial factorization: a(x-x,)(x-x2) -The slope of a line passing through the points (x,, y, ) and (x,, y, ) is: a = VI - yz X1 - X2 -The equation of a line with slope "a", passing thorough (x, y,) : y- y, = a(x- x,) -The x coordinate of the vertex of the parabola: y = ax +bx+c is: x= - b 2a Derivatives: Power rule: (x" )' = nx" product rule: (f . g)' = f'8 + f .8' Quotient rule: f'8 85 Chain rule: [(f(x))" ] =(f(x))". f'(x) Exponential: (e() ) =er. f '(x) Logarithmic: (In(f(x))) = f (x) f ( x ) Exponents: a" . a' = qty a' a (a' )' = ary (a . b)* = a* . b' Na' = a Logarithms: alos. b = b a" =b->n=log b log, (x. y) = log, x +log, y = log, x-log, y y log, (x." ) = m. log, x log x log, *= log b