1. at the state fair, erin and her cousin ride the ultra drop roller coaster. when the ride plummets down the first hill, it dips below the loading platform. at the bottom, a camera snaps the riders' picture before hurtling them back toward the sky. the equation y = x2 – 9x + 20 models the roller coaster's path over time. the variable y represents height (in feet) above or below the platform. at y = 0, the roller coaster is even with the platform. the variable x represents the amount of time (in seconds) since the ride began. factor and graph the equation to better understand erin's ride. part i: write the equation in factored form. (1 point) part ii: find the vertex of the parabola. hint: to find the x-value of the vertex, take the average of the x-values of the x-intercepts or use the first part of the quadratic formula (2 points: 1 point for each coordinate value) part iii: identify the intercepts. a. what are the x-intercepts? hint: use the equation from part i. (2 points) b. what is the y-intercept? hint: use the equation y = x2 – 9x + 20. (1 point) part iv: sketch the graph of y = x2 – 9x + 20. identify the vertex and x- and y-intercepts on your sketch. (3 points) part v: use the graph to answer the questions. a. between what times does the roller coaster dip below the platform? (1 point) b. what is the height and time at which erin's picture is taken during the roller coaster ride? hint: erin's picture is taken at the lowest point of the roller coaster. (1 point) completing the square 2. solve the equation by completing the square. show your work. x2 – 30x = –125 step 1: add to both sides of the equation. (2 points) step 2: factor the left side of the equation. show your work. (2 points) hint: it is a perfect square trinomial. step 3: take the square root of both sides of the equation from step 2. (1 point) step 4: simplify the radical and solve for x. show your work. (1 point) 3. a community is developing plans for a pool and hot tub. the community plans to form a swim team, so the pool must be built to certain dimensions. answer the questions to identify possible dimensions of the deck around the pool and hot tub. part i: the dimensions of the pool are to be 25 yards by 9 yards. the deck will be the same width on all sides of the pool. including the deck, the total pool area has a length of (x + 25) yards, and a width of (x + 9) yards. a. write an equation representing the total area of the pool and the pool deck. use y to represent the total area. hint: the area of a rectangle is length times width. (1 point) b. rewrite the area equation in standard form. hint: use the foil method. (1 point) c. rewrite the equation from part b in vertex form by completing the square. hint: move the constant to the other side, add to each side, rewrite the right side as a perfect square trinomial, and finally, isolate y. (4 points: 1 point for each step in the hint) d. what is the vertex of the parabola? what are the x- and y-intercepts? hint: use your answer from part a to identify the x-intercepts. use your answer from part b to identify the y-intercept. use your answer from part c to identify the vertex. (4 points: 1 point for each coordinate point) e. graph the parabola. use the key features of the graph that you identified in part d. (3 points) in this problem, only positive values of x make sense. why? (1 point) g. what point on your graph shows a total area that includes the pool but not the pool deck? (1 point) h. the community decided on a pool area that adds 6 yards of pool deck to both the length and the width of the pool. what is the total area of the pool and deck when x = 6 yards? (2 points) part ii: a square hot tub will be placed in the center of an enclosed area near the pool. each side of the hot tub measures 6 feet. it will be surrounded by x feet of deck on each side. the enclosed space is also square and has an area of 169 square feet. find the width of the deck space around the hot tub, x. step 1: write an equation for the area of the enclosed space in the form y = (side length)2. hint: don't forget to add x to each side of the hot tub. (1 point) step 2: substitute the area of the enclosed space for y in your equation. (1 point) step 3: solve your equation from part b for x. (3 points) step 4: what is the width of the deck around the hot tub? hint: one of the answers from part c is not reasonable. (1 point) comparing quadratic functions 4. two different quadratic functions have graphs with the same vertex. which function's graph increases faster between x = 2 and x = 3? function a function b x y –1 12.5 0 5 1 0.5 2 –1 3 0.5 4 5 5 12.5 a. what is the vertex of each function's graph? (1 point) b. what is the average rate of change in function a between x = 2 and x = 3? (2 points) c. what is the average rate of change in function b between x = 2 and x = 3? (2 points) d. which function's graph increases faster between x = 2 and x = 3? (1 point)