# Math 1101 Project

**Background**

For this project, you will use the function

s(t) = at^{2} + bt + c .

s is the height (in feet) of an object propelled directly upward after t
seconds.

In physics, the coefficient b is usually written as v_{0} v and
represents the initial velocity (feet per

second) of the object. Initial velocity is the velocity at time t = 0 . For
instance, if a ball is

thrown up, it will have a positive initial velocity. If a ball is dropped, the
initial velocity is zero.

The constant c is normally written as s_{0} s and represents the initial
height or distance above the

ground of the object. Initial height is the height at time t = 0 . If a missile
is shot off from the

ground, then s_{0} = 0 . If however, it is shot off from the top of a
100-foot building, then s_{0} =100 .

**Collecting Data** (Monday, April 14 to Thursday, April 24 - no exceptions)

You will collect your CBL ball toss data in my office. You will toss a ball up
in the air and the

CBL motion detector will record the height of the ball. At least one member of
your group must

come to my office (during office hours or by appointment) to do the ball toss. I
prefer that both

members come to collect the data. You do not need to bring anything except your
TI-83/84

calculator. Group members will share the same data. I will keep a copy of your
data.

****************************************************************************

The CBL program records time data (in seconds) and the height of the ball above
the ground (in

feet). You will use this data to find a **quadratic** model for your ball
toss.

I will write a calculator program that will save your data and put it in L_{1}
and L_{2}. To execute this

program, hit the "PRGM" key. Select the program name. Hit "ENTER". Your time
data will be

in L_{1} and your height data will be in L_{2}

**Your Quadratic Model**

This assignment must be typed up and pictures of your calculator screen must be
inserted in

appropriate spots in your report. I should not have to “assemble” the report for
you. Your work

should be organized and easy to follow. Your papers must be stapled together.

1. Turn on one of the STATPLOTs. Check that L_{1} and L_{2} are
listed. Plot the data with dots,

not squares.

2. Since there are 80 data points, we will let the calculator set the window.
The

STATPLOT must be turned on. Hit the "ZOOM" key. Select "9: ZoomStat". Edit the

window so that you can see the x-axis. Include this STATPLOT in your report with
a

description.

3. Does the plot of your ball toss data open upward or downward? Is the vertex a
maximum

or minimum point? Will the coefficient a be positive or negative?

4. Find the vertex graphically (TRACE) or numerically (L_{1}
and L_{2}). In complete sentences,

explain what the x-coordinate of the vertex represents and what the y-coordinate
of the

vertex represents. Be sure to include units.

5. Write the function for the height in the form s(t) = a(t − h)^{2} + k
. Save this equation until

we find a.

6. Use Quadratic Regression on your calculator to find the best-fitting
quadratic for your

data. Graph this quadratic function with the STATPLOT of the data. Include this
picture

in your report with a description and the regression equation. Explain why this
quadratic

does not fit the ball toss data very well.

7. Now we will edit the data. We only want the data points that look like they
are on a

parabola.

- Hit **2nd , LIST** (on STAT key).

- Move the cursor to "OPS".

- Select "8: Select".

- You are now on the home screen. After the "Select ( " , type in "L_{1},
L_{2})".

- You are now on the graph screen. To answer the "Left Bound?", move the cursor
to the

first point that looks like it is on the parabola. Hit "ENTER".

- To answer the "Right Bound?", move the cursor to the last point that looks
like it is on

the parabola. Hit "ENTER".

Voila! Your data has been edited. If you made a mistake and want to start over,
run the

data program that I wrote for you. Include the coordinates of the first and last
data points

and a picture of the edited ball toss data (with a description) in your report.

8. Use Quadratic Regression on your calculator to find the best-fitting
quadratic for the

edited ball toss data. Graph this quadratic function with the STATPLOT of the
edited

data. Include this picture in your report with a description and the regression
equation..

9. Use the equation from part 8 to answer the following:

a) Does your initial velocity and initial height seem reasonable? Explain.

b) When does the ball hit the ground?

c) When is the ball 4 feet above the ground?

d) Find the interval of time when the ball is more than 4 feet above the ground.

Note: If your ball is never 4 feet above the ground, use 3 feet in 9c and 9d.

10. Now take the a from the equation in part 8, and plug it into the vertex form
equation in

part 5. Graph the vertex form equation with the STATPLOT of your edited ball
toss

data. How good is the fit? Include this picture in your report with a
description and the

vertex form equation.