Problem Statement
Jefferson High Schools Soccer team has just won the championships. To celebrate their victory, the school will be putting on a firework display. The director needed our help to figure out out how to launch celebratory fireworks safetly and to figure out the timing mechanism when being launched from the top of a school tower.
Process and Solution
Getting started on this problem we asked questions and listed out things that we needed to know to get us started...
 Height of the tower?  Angle of the launcher?  Staring launch point and end point in yards  Highest Peak We were then given all the things we needed to get started..  65 degree launch angle  92 ft/ Sec inital velocity  The height of the tower 160ft h(t) = 160 + 92t  16t^2 d(t) = 92t/ tan 65 
Solution
Where the fireworks will land:
My group and I first approached the problem, and knew if we needed to figure out the landing we need to find the x intercpets (the x intercepts are points only on the x axis) which will tell us whrere it will launch and where specifically it will land. So we took our function for height and time: h(t) = 160 + 92t  16t^2 rearaged it to something more familar h(t) = 16t^2 + 92t + 160 and plugged it into the quadtratic formula to find our launch point and end point. This would find our horizontal distance. 

After we got our x intercepts: (1.40, 0) and (7.15) we realized that this wouldnt actually give us the horizontal distance it gave us the specific times of where it would launch and land. So with our specific time we plugged it into our function d(t) = 92t/ tan 65
which will give us distance when we plug in a certain time.
92(7.15) / tan 65 = 306.7
In conclusion the end point of the fire work is at 7.15 seconds the firework will land 306.7ft anywhere after that, the crowd will be safe.
Horizontal distance: (7.15, 306.7)
Peak of the firework:
Now that we found our x intercepts I knew that if we were looking for the peak of the firework so we had to find the vertex. (The vertex is the highest point in the middle of the x intercepts. I immedietly thought to put our function into vertex form because that's the first step you do to solve for vertex. When we did solve it we realized that it was way too confusing.
We knew that the vertex is right in the center of the x intercepts so we chose to find the vertex a different way...
which will give us distance when we plug in a certain time.
92(7.15) / tan 65 = 306.7
In conclusion the end point of the fire work is at 7.15 seconds the firework will land 306.7ft anywhere after that, the crowd will be safe.
Horizontal distance: (7.15, 306.7)
Peak of the firework:
Now that we found our x intercepts I knew that if we were looking for the peak of the firework so we had to find the vertex. (The vertex is the highest point in the middle of the x intercepts. I immedietly thought to put our function into vertex form because that's the first step you do to solve for vertex. When we did solve it we realized that it was way too confusing.
We knew that the vertex is right in the center of the x intercepts so we chose to find the vertex a different way...
We added the x intercepts to find the total distance then, divide by two to find the middle value after that you add that value to one of the x intercepts to find the middle point which would be our vertex. We got 2.88 seconds
That gave us the specific time it would reach its highest point so we plugged it into to our function that gives us the height in relation to time h(t) = 160 + 92t  16t^2 160 + 92(2.88)  16(2.88) ^2 = 292.2 Highest Peak: (2.88, 292.2) 
Problem Evaluation
Its actually crazy looking at this problem because at the beginning of the semster when Mr. Carter showed us I had no idea what I was looking at. We have progressed a lot and grew so much in this qudratics unit. I liked this problem because it was a good way to end our unit. Everything we spent learning for a semster was put into this problem. I feel that this problem made my group and I work hard and made us work together by sharing our ideas.
Self Evaluation
I feel like I desrve an A because I constantly challenged myself through this unit. I was able to share my ideas, and work with group mates who challenged my thinking. I was able to challenge their thinking as well. Whenever I came across something challenging I was able to stay patient and keep a good mindset, which played a role during this unit. For example when I had to dervive the quadrtic formula, I stuck by it no matter how challenging it got. It eventually took 3 days to solve it.