For the following, find the equation, roots, axis of symmetry, vertex, and graph that go together. i (1,2) F X = -2, -2. ii (3,13) iii (1, -5)

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1 Name: Date: Hour: Practice with Quadratics and Parabolas (40 Formative Points) For the following, find the equation, roots, axis of symmetry, vertex, and graph that go together. Equation A y = x 2 + 4x + 4 Axis of Symmetry a X=3 Vertex Roots Graph i (1,2) E X = 0.184, e B y = 3x 2 + 6x 1 b X=1 ii (3,13) F X = -2, -2 f C c X=1 iii (1, -5) G X = , g y = 2x 2 4x 3 D d X= -2 iv (-2, 0) H X = 2.581, h y = x 2 + 6x + 4 Final Matches Equation Axis of Symmetry Vertex Roots Graph A B C D

2 A new movie: Skyscraper starring Dwayne The Rock Johnson is coming out this summer; and the poster is totally absurd (see right): On Twitter, writer James Smythe kicked off a long discussion thread with some different parabolic arcs Johnson might follow depending on the momentum he starts the jump with: Stating, I ve mocked up some parabolas for The Rock s SKYSCRAPER jump. [One] assuming he jumped up for a bit first; [one] assuming he ran forward and somehow didn t lose momentum; [and a third] for a sort of squat-thrust thing. Whichever you choose, rest in peace The Rock, as you are dead now. Today, we are going to take some real world applications of parabolas and with a little help from physics, determine what velocity The Rock would need to be running at when he jumps off the scaffolding in order to make it to the window; if it is even possible! Our equation that we will be using today is a few of the kinematics equations used in physics to describe the motion of objects, lucky for us there s a quadratic and a linear equation!!! d = 1 2 at2 + v o t d = v 0 t d = Distance Traveled a = Acceleration of Gravity (9.81m/s 2 ) t = Time v0 = Initial Velocity

3 Vertical Distance (in Meters) Algebra 2 in the Workplace The coordinate plane below was obtained by using Dwayne The Rock Johnson s known height. The distance in meters represented by the coordinate plane is proportionate to The Rock s height as portrayed in the poster. You will be using this image to determine if The Rock could actually make this jump Horizontal Distance (in Meters) Step 1: On the coordinate plane, do your best to SKETCH the graph of a parabola (a smooth curve) where the scaffolding The Rock jumped off is the vertex. Be sure your graph goes through The Rock (after all it is his path) and that it makes it to the base of the window (so The Rock does not fall to his death!) In order to determine the initial velocity (v0) The Rock will need to have jumping off the scaffolding, we will need to determine the additional variables in our kinematics equation. Step 2: We need to determine the VERTICAL and HORIZONTAL distances that The Rock needs to travel from the scaffolding he jumped off of to the open window in the side of the building. Use the given coordinate play to determine the: Vertical Distance: Horizontal Distance:

4 Step 3: We now need to determine HOW LONG it would take The Rock to vertically fall the vertical distance you identified in Step 2; assuming that The Rock did not jump up when he jumped from the scaffolding. THIS ONE NOT This one To find this time we can use one of our kinematics equations! d = 1 2 at2 + v o t We are trying to find t in this equation, so that is our unknown variable. That means that we need to find values of d, a, and v0. Since we are trying to find the time it takes to fall VERTICALLY all of our other variables have to be in terms of vertical distance/velocity/acceleration: d = The total vertical distance The Rock needs to travel a = The vertical acceleration of The Rock (Gravity) v0 = Initial vertical velocity You have found the value of d already in Step 2; the acceleration of gravity (a) is a known constant of 9.81m/s 2. For the initial vertical velocity, we need to consider how The Rock is leaving the scaffolding. Is he jumping up first to gain additional vertical velocity? Is he starting from a resting position and just jumping off (not up)? Based on your answer to those questions, what do you believe the value of v0 will be? Record the following values you will use for the vertical fall situation: d = a = v0 = t =??? Step 4: Plug in the values that you know into the kinematics equation: d = 1 2 at2 + v o t and SOLVE for the unknown value of t.

5 Step 5: We are now going to determine the initial HORIZONTAL velocity (v0) that The Rock must have in order to make the necessary HORIZONTAL distance from the scaffolding to the window. This makes v0 the unknown variable. To do this, we will use our second kinematics linear equation: d = v 0 t Since we are talking about HORIZONTAL distance, all of our variables will be in terms of horizontal distance/initial velocity. d = Horizontal distance v0 = Initial horizontal velocity t = Time to travel horizontal distance You have already calculated the horizontal distance The Rock needs to travel in Step 2 and you have already calculated the time The Rock has to travel this horizontal distance in Step 4 (This is the same as the time to drop the vertical distance, because when The Rock is done dropping vertically, he better have reached the window or he s a goner!) Record the following values you will use for the horizontal situation: d = t = v0 =??? Step 6: Plug in the values that you know into the kinematics equation: d = v 0 t and SOLVE for the unknown value of v0. Step 7: Does The Rock live?!?!?! In Step 6, you have calculated the necessary initial velocity (in meters per second) convert meters per second to miles per hour. There are meters in 1 mile and 3600 seconds in 1 hour. The MINIMUM initial velocity The Rock MUST have to make the jump (in mph): For comparison, Usain Bolt s (the fastest man alive) top speed is 27.4 mph. Based on this, do you believe that The Rock would survive the jump? Why or why not?