weaks6_2

=**__ Mathematical Radical Group __**=

Follow the directions on the Vertical Motion page, to complete each of the sections below.
**__The Word Problem__:** How would you solve for time in the problem your group was assigned? Prepare the equation to solve for time, defining the variables and supporting your choice in equation.

3) A peregrine falcon dives toward a pigeon on the ground. When the falcon is at  a height of 100 feet, the pigeon sees the falcon, which is diving at 220 feet per second. Estimate to the nearest tenth of a second the time the pigeon has to escape.  Use the vertical motion equation: This is the correct equation because the peregrine falcon is diving at a certain speed, and is not simply dropped. The h is the height the peregrine falcon would be to reach the pigeon, or 0. The t is time to reach the height illustrated in h. The v is velocity, how fast the peregrine falcon is going, or -220f/s. The is the starting height of the peregrine falcon, or 100ft. The equation is

**__The Parabola:__** Use what you know about parabolas to predict the shape of the graph. What will it look like, and how do you know?

a=-16 b=-220 c=100 Because a is a negative whole number, the graph will open downwards and be narrow. Because c is 100, the parabola will have a y-intercept of 100. On the graph, this will illustrate when the peregrine falcon begins it's dive, and where the path is at a //t//(time) of 0. The peregrine falcon begins it's dive at 100ft. __Discriminant__ The discriminant is positive, and that means there will be two solutions, and two x-intercepts.

**__The Math__:** Pretend you are the teacher, and this section is your lesson. Post the calculations required to graph the equation and find the solution, and show all the steps to doing so. Teach your audience all the important parts of the graph and explain what they mean for the problem you were given.

Because the axis of symmetry is, you plug in -220 for //b// and -16 for //a//. Take the - of //b//, and multiply //a// by 2. When simplified, the result is 6.875. The Axis of Symmetry on the graph is simply the axis of symmetry. It is the line that is right in the middle of the parabola.

You plug in -6.875 for //t// in the equation because the AoS always gives the //x// value for the vertex and //t// is in the place of //x//. You are solving for //y//. After plugging in the AoS, you need to square it where you see, and multiply -220 by-6.875 where you see -220(-6.875). You get 47.265625 for, and you have to multiply that by -16. -220 multiplied by -6.875 is 1512.5. 47.265625 multiplied by -16 is -756.25. Next you have to combine -756.25 with 1512.5. -756.25+1512.5 is 756.25. Combine with 100. The //y// value for the vertex is 856.25. The vertex is (-6.875, 856.25) The best way to solve for roots is by using the quadratic formula because the equation is not factorable, the coefficient of the //x//^2 term is not 1, and there is an //x// term. Plug in -220 for //b//, and -16 for //a// outside the radical. Inside the radical, plug in our result for the discriminant that we solved earlier. Simplify. Separate the +/-. or

or

or

or (positive solution) Our answers/ solutions for //x// were the //x//- intercepts on the parabola and the roots on the graph. One is negative and one is positive, so one root is on the left of the //y//-axis, and one is on the right of the //y//-axis. The //y//-axis is the height, in feet, that the peregrine falcon is at. The //x//-axis is the time passed, in seconds.

**__The Solution and The Meaning__:** What do the different parts of the graph mean? What does the solution mean?

1) Our solution (0.44) means that the peregrine falcon will reach the pigeon after 0.44 seconds. 2) The vertex has no meaning because it is in negative time and can not conclude anything about the falcon. 3) The y-intercept is when peregrine falcon spots the pigeon and begins diving (100ft). Its when the path reaches a time of zero. 4) The positive root tells the time the object following a certain path will reach the hight of 0. The negative root tells when the object following the certain path would begin at a height of 0. 5) The peregrine falcon will be in the air for 0.44 seconds. We know this because when we solved for //t//(time) we got 4.44 and -14.19. We do not use the negative solution because we stay in real-time, not negative, to say that the peregrine falcon has already begun diving.

**__Reflection__** How did it feel to do the work and pretend to be the teacher? Do you think you understand the material more, now? Why or why not? Each member of your group needs to complete the reflection on their own. Post each person's reflection at the bottom of your wiki. Be sure to label each reflection with the name of the person who wrote it.

Sabrina's Reflection: The problem was kind of hard to figure out, but what I think was the hardest was the graph, the parabola, and gaph paper. We had to find a way to fit the whole thing on the graph paper using numbers (-6.875, 856.25) for the vertex. Also with those numbers we had to find out how we were going to scale the graph. The project kind of helped me understand quadratic equations and vertical motion better because it was sometimes confusing and overwhelming. Pretending to be a teacher was pretty interesting because we had to find out a way to make it as if we were teaching a student like a lesson plan. Since we are the ones always being taught, it was different to put ourselves in a teachers shoes.

Marshayla's Reflecton: One thing I found difficult was Part 4: What does the Vertex Represent?, because our vertex was in negative time. To make a project like this run smoother, I would probably find out if the vertex is in negative time, first. Also, I would have everyone have their own copy of the work, so we wont be missing anything important if someone is absent. This project helped me understand Vertical Motion better. It was very hard, and took a lot of time. I liked pretending to be a teacher.

Jenny's Reflection: One thing that was difficult to do was graphing the parabola because we kept having to continuously change the scale to fit the vertex on the graph. The vertex is (-6.875, 865.25) so we used a scale that goes by 20ft per line. To make the project go more smoothly next time, I would "be the teacher" and explain the work as I did the problem instead of after. I would also watch out for careless mistakes like forgetting to add a negative to a number. I have memorized the vertical motion equations for dropping or throwing objects because of this project. I also know and understand the graph better. Pretending to be the teacher was hard, long, took a lot of time, and made us think more about the problem.