Application of Quadratics in space exploration Part II

Question

You are going to come up with an equation to map out John Glenn’s descent back to Earth in Friendship 7. You are supplied with 2 graphs that have relevant values on both the x and y-axis. You only need to use 1 of them, but there is an extra one in case you wish to use it. You are going to plot the following points on this graph: (0,160), (10,150), (17.5,130), (22,110), (27,70), (30,0);

These coordinates represent approximate targeted points that we are trying to get Friendship 7 to go through in its parabolic (parabola) path as John Glenn comes back to Earth. Come up with a function, a graph of “best fit”, that represents the line that Friendship 7 will travel. Sketch your equation on the same graph as your given plotted points.

Answer 1

You can either use you graphing calculator (like a TI-84 Plus CE) or an online computer tool such as desmos.

To use your TI-84 Plus CE:

1) Push the "stat" button (4 buttons above the 8)

2) Hit "enter" when "EDIT" and "1:Edit" are highlighted

3) You will get a table to enter your data in. If there any numbers in the table, clear them out as follows: Scroll up to the header marked "L1, L2, L3, etc. Highlight the top of the column that has data in it (for instance if L1 has data in it, highlight "L1"). Push the "clear" button then the "enter" button. Repeat until the table is empty.

4) Enter your data in the table. Put the x-values in the L1 column and the y-values in the L2 column. It will look like this when you are done:

L1 L2 L3 L4 L5

0 160

10 150

17.5 130

22 110

27 70

30 0

5) Push the "stat" button again. Scroll right to highlight "CALC". Scroll down to "5:QuadReg" and hit "enter"

6) You will see the following:

Xlist:L1

Ylist:L2

FreqList:

Store RegEQ:

Calculate

Scroll down to "Calculate" and hit "enter"

7) You will get the following reult:

y = ax²+bx+c

a = -0.2667501861

b= 3.40486531

c = 154.8987085

This is the answer. The quadratic equation that best fits is y = -0.2667501861x² + 3.40486531x + 154.8987085

To use desmos:

1) Google "desmos" and select the results that says "graphing calculator"

2) In the upper left hand corner, click the "+" button and select table:

3) Enter your data in the table. When you are done, it will look like this:

x₁ y₁

0 160

10 150

17.5 130

22 110

27 70

30 0

4) Click the "+" again in the upper left hand corner but this time select "f(x) expression"

5) In the box that pops up, type in the equation as follows:

y1 ~ ax12 + bx1 + c

To do this, type the letter "y" then the underscore button (shift "dash") the the number "1". This will give you the "y₁". Then use the "cursor right" button to get out of the "subscript". Then type the squiggle button "~" )usually just to the left of the "1"). Then type "a", the "x" and then the underscore again and the "1". Use the "cursor right" to get out of the "subscript" then type the "^" (shift 6) and then "2". Then use the cursor righ again to get out of the "superscript". This will give you "y₁~ax₁²". Then type "+bx" and then use the underscore again and then "1". Now you will have "y₁~ax₁² + bx₁". Make sure to use the "cursor right" key to get out of the "subscript". Type "+c" and you will see the following pop up:

y₁~ax₁² + bx₁+c

STATISTICS RESIDUALS

R2 = 0.9515 e₁ plot

PARAMETERS

a = -0.26675

b = 3.40487

c = 154.899

Again, this is the answer for the best quadratic that fits this data. On the right side of the screen, if you scroll out, will be the graph of the points and the graph of the best-fit equation.