VARIATION WITH POWER REGRESSION ON TI NSPIRE CAS
TASK ONE: Triangular Numbers.
The triangular number Tn is a number that can be represented in the form of a
triangular grid of points where the first row contains a single element and each
subsequent row contains one more element than the previous one. This is illustrated
below for T1  1 , T2  3 , T3  6 . The triangular numbers are therefore 1, 1  2,
1  2  3, and so on. So the first few triangular numbers are 1, 3, 6, 10, 15.
1. Find the next three triangular numbers.
2. Complete the following table:
n
Tn
1
1
2
3
3
6
4
5
6
7
8
9
10
3. Enter the completed table above into the spreadsheet on your calculator.
4. Sketch the graph Tn versus n. When prompted, select n on the horizontal axis,
and Tn on the vertical axis
5. What sort of relationship between Tn and n can be deduced from the graph?
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6. Find the relationship between Tn and n using your graphics calculator. You
will need to use power regression. Write down your equation here in terms of
n and Tn .
7. What value of Tn do we have for n  0?
8. Add this data point to your spreadsheet and find the quadratic regression
equation.
7. Use your equation to find T20 .
8. Find n, if Tn  120.
Note that quadratic function models this data much better than the power
regression.
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5
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TASK TWO: Light Intensity.
In one description of how light travels, light particles spread out evenly in every
direction from the source of light. The intensity of the light, I particles per unit area ,
at a distance of r units from the globe can be described by the formulae:
k
(Equation 1)
I 2
r
A scientist measured the intensity of light at different distances with the following
results:
Distance 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 1.00 1.20 1.40
r (m)
Intensity, 37.50 24.00 16.67 12.24 9.38 7.41 6.00 4.17 3.06 2.34 1.50 1.04 0.77
I (units)
k
. Transpose the formula to make k the
r2
subject. Use the selection of the data in the table to calculate the approximate value of
k.
1. Equation 1 above indicates that I 
2. Plot the graph of intensity, I, against distance, r, by hand, on the set of axes below.
Describe the relationship between I and r.
I
40
30
20
10
0.25
0.5
0.75
1
1.25
1.5 r(m)
3. Enter the data points on your calculator and find the power regression equation
connecting intensity with distance. In a graph screen select Menu 4: Analyze
6: Regression 7: Show Power
4. Write down your equation with variables r and I.
5. Use your equation to find the intensity when r  1.30 metres.
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6. Use your equation to find the distance when the intensity is 10 units.
7. If the distance from the source doubles, how will the intensity be affected?
To save the equation as f1(x) follow
these steps in the spreadsheet view.
TASK THREE: Table of values.
Use power regression to fine the relationship between x and y as given in the
table:
x
2
3
4
5
6
7
8
y
.367
.441
.515
.576
.627
.679
.725
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