math activity
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Quadratic Regressions STEM Project Week #4
In activity 1 we created a scatter plot on the calculator using a table of values that were given. Some of you were able to create a linear regression using your calculator’s linear regression application. In this activity, we will look at data that is best fit by a quadratic regression. Remember that a quadratic is in the shape of a parabola. The parabola can either open up or open down. We will create a scatterplot by hand, create a scatterplot on our calculator, and find a regression equation on our calculator. Remember, that a regression equation is an equation that “best fits” a given set of data. It best describes the relationship between the two variables. The regression equation follows the trend of the data.
Getting Started with a Data Set Projectiles When an object is thrown, we can measure the height above ground that the object is, versus the time that it take to reach that height. If we graph the time on the x-axis and height above ground on the y-axis, the path of the object resembles a parabola. The object that is thrown, launched or kicked is called a projectile. A projectile is any object that is subject to the force of gravitiy. A projectile travels with a parabolic trajectory due to the force of gravity. The force of gravity is 9.8 meters per second per second.
The following data was found when a person threw a rock off the side of a cliff. The table shows the vertical height (h) of the rock relative to the top of CLIFF as time (t) passed.
t (time in seconds)
0 1 3 5 6 8
h (height in meters)
2 19.2 23.8 -10.6 -42.4 -135.6
Plotting the Data by Hand
You will be plotting the data on the grid shown below, graphing time on the x-axis (time is the domain and the independent variable). You will be graphing height on the y-axis (height is the range and the dependent variable). Remember that the domain is the set of all possible values of an independent variable of a function and the range is the set of all possible values for a dependent variable of a function. These are the same steps that we followed in activity 1.
At this point we are stretching the mind’s capacity to use letters in conjunction with math. We have four letter sets, t, h, s and m. Two of the letters are variables and two of the letters are unit indicators: t represents time (replaces the traditional use of x in math). h represents height (replaces the traditional use of y in math). s indicates that the numerical value represents the number of seconds. m indicates that the numerical value represents the number of meters.
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Step 1: Label the t-axis (normally this is the x-axis but we are using t to represent time) and the h- axis (Normally this is the y-axis) of your grid. Let’s start with the t-axis.
• What is your smallest t-value? • What should be your smallest t-value on your t-axis? (Make sure that your smallest value is
on the x-axis. You need to have your t-axis go beyond the smallest value.) • What is your largest t-value? • What should be your largest t-value on your t-axis? (Make sure that your largest value is on
the x-axis. You need to have your t-axis go beyond the largest value.) • Label these values on your grid below. • What is your smallest h-value (height value)? Think about this, can your height be
negative? • What should be your smallest h-value on your graph? (Make sure that your smallest y-value
is on the y-axis. You need to have your h-axis go beyond the smallest value.) • What is your largest h-value (height value)? • What should be your largest h-value on your y-axis? (Make sure that your largest value is
on the h-axis. You need to have your h-axis go beyond the largest value.) • Label these values on your grid below.
Step 2: Now that your grid is labeled on the h and t-axis, plot the data points from the table onto the grid.
NOTE: It is very easy to confuse the graph showing the height of a projectile as a function of time with a graph that represents the object’s path, because they both are parabolas. However, even when a ball is tossed straight up and allowed to fall, its graph of height as a function of time is a parabola. Likewise if you drop something off a cliff, its graph of height as a function of time is also a parabola.
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Step 3: Look at your data. If you were to find an equation of a function that best fits this data, would this equation be a line? Why or why not? Would this equation be a parabola (quadratic)? Why or why not? **Make sure to remember that you are trying to find the equation that would “best fit” the graph. Not the equation that exactly fits the graph. If you have plotted the data correctly the equation that will best fit this data is a parabola (quadratic). Sketch a parabola that best fits this data. Explain why your sketch of a parabola best fits the data.
Finding the Equation for the Quadratic of Best Fit Using Your Calculator
Now, you will enter this data into the calculator and use the calculator to find the equation for your quadratic of best fit. Please use the instructions on the document labeled Regression Instructions on the Math 143 MLC webpage. First, find the set of instructions that says “Graphing a set of data points on the TI-83” (making a scatterplot). Follow these instructions to plot the points on your calculator. Once you have plotted the data on your calculator and you see a plot similar to the one that you sketched, observe the scatterplot. Does your graph on the calculator look like the graph that you sketched? Why or why not? If they do not look the same, you might want to explore reasons such as window size, domains and ranges, sketched points correctly or entered data correctly in the calculator.
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Once you have resolved any differences between the sketch and the calculator, look at the graph on your calculator. Do you think that a line best fits the data? Why or why not? Do you think that a quadratic best fits this data? Why or why not? If the data was entered and plotted correctly, it should most resemble a quadratic just like in your sketched graph. Now you want to find a quadratic that is a good "fit." We will find this equation using your calculator. In the document labeled Regressions, find the section labeled “Finding a Quadratic Regression”. Use those instructions to find the equation for the quadratic that best fits the data. What is the quadratic equation that best fits the data? After you have found a quadratic equation to best fit the data, try to find a cubic equation to fit the data. Follow the steps that you did for the quadratic regression but instead use the CubicReg key. What is the cubic equation that best fits the data? Graph this cubic equation in your calculator. Does this graph do a good job of approximating the data? Yes or No. If you answered the last question, “No, the cubic regression does not fit the data points” – STOP. Think about a cubic regression. A cubic regression should approximate data points better than a quadratic regression. Go back and determine what you did wrong. Explain what you did wrong and fix the issue. (Hint: double check the value of “a” in your cubic regression) Notice that the cubic regression and the quadratic regression both “fit” the data points, what are the similarities (differences) between the two?
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Pick either the cubic or the quadratic regression to model the data set. Explain why you chose that equation? (Hint: pick the quadratic – why?) Finally, we can use our model to make predictions or find other points on the curve that were not given. Using your model, after 4 seconds how high will the ball be? Locate this point on your graph. Both the sketched graph and the graph on your calculator. Label this point on your sketched graph. After 15 seconds how high will the ball be? Locate this point on your graph. Both the sketched graph and the graph on your calculator. Label this point on your sketched graph. If the cliff is 200 meters tall, when will the ball hit the ground? First you must determine what represents the ground. What represents the ground and why? Locate both point(s) where the sketched graph and the graph on your calculator indicate the ball hitting the ground. Label these point on your sketched graph. NOTE: Your graph will indicate that the ball hits the ground in two places. Locate both of these places on your graph. Why does only one of these make sense in this situation? What is the tallest height that the ball will reach relative to the cliff? Locate this point on your graph. Both the sketched graph and the graph on your calculator. Label this point on your sketched graph. In our situation, what values make sense for the height of the ball (what is the range)? Explain. In our situation, what values make sense for the time (what is the domain)? Explain.
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Path of a Projectile
The path of a projectile can be modeled by the equation h = h 0 +v
0 t +
1
2 at
2 where h is
the vertical height of the object at time t, h 0 is the initial height of the object, v
0 is
the initial velocity of the object, and a is the acceleration due to gravity. In our scenerio, since the height of the object is in meters, a should equal about -9.8 since this is the acceleration due to gravity. Check your equation and see if that is pretty
close. Notice that your coefficient of t 2 is
1
2 times a so your coefficient of t
2 should
be about -4.9. Also, in your sketch, does your parabola open up or down? What part of your quadratic equation tells whether the graph should open up or down?
In your equation, what is the initial height (the height when time, t = 0) of the ball relative to the cliff? Why do you think this is? In your equation, what is the initial velocity of the ball (v
0 in the equation or the coefficient of the t
term)? Exercises For the data sets below:
a. determine the domain and range, b. plot the data on graph paper making sure to label your “window” on your paper, c. sketch a quadratic of best fit, d. enter the data into your calculator, e. find the quadratic of best fit.
1. A person drops a rock off the top of a building. The table below shows the time in seconds after the rock has been dropped versus the height of the rock relative to the top of the building. (For appropriate completion of this problem, one piece of information is missing. What is the missing information? ________________________. Choose a number that would be appropriate for the missing peice of information__________________. Explain why this is a good choice and use this number to complete the problem.)
t (time in seconds)
0 1 3 5 8
h (height in meters)
1.5 -3.4 -42.5 -121.1 -312.1
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2. A football player kicks a football. The table below shows the time in seconds after the ball has been kicked versus the height of the ball off the ground.
t (time in seconds)
0 1 3 5 8
h (height in meters)
0 35.1 75.9 77.4 6.4