Principles of Mathematics - MPM1DP

Unit 3: Linear Relations

Activity 6: Investigating Linear Relations

Content

Investigation: Patterns in Tables of Values

Complete all the components of this investigation as indicated in the instructions. 1. In a word processing file, set up and complete the table of values for the equation, y = 3x -1 as shown. The symbol, Δ, is the Greek letter Delta. In mathematical terms it reads, “the change in.”

y = 3x - 1		First Difference (Δy)
x	y
-2
-1
0
1
2

2. Notice that the x-values change by a consistent amount as we move down row by row. Look for any pattern in the y-values. You see that the x-values increment by 1 as we read down the chart. But, what about y?

   Calculate the first differences or change in y as you read down the chart. To do this, take the second value for y and subtract the first value for y.

   y = 3x - 1		First Difference (Δy)
   x	y
   -2	-7
   -1	-4	-4 - (-7) = 3
   0	-1
   1	2
   2	5

3. Continue to work down the chart. Take the third value for y and subtract the second and so on.
4. Describe the pattern formed by the first differences.
5. Repeat #1 - 4, for the equation y = -2x - 4.
6. Write a conclusion about first differences and linear relations

Enrichment

Try this out. Find the first difference in the following table.

Is this a table of values for a linear relation? Justify your answer.

Extend your thinking to second differences then. Complete another column, calculating second differences.

What did you discover about the second differences? What do you think this pattern tells us about the relation described in the table of values?

Check out the pattern by finding first and second differences for the following table of values.

After completing these tables check out the answers by clicking here.

Questions

1. Find the first differences for the table of values. What is the Δy? Does this table represent a linear relation? Justify your answer. Answer
2. Find the first differences for the table of values. What is the Δy? Does this table represent a linear relation? Justify your answer. Answer
3. Find the first differences for the table of values. What is the Δy? Does this table represent a linear relation? Justify your answer. Answer

Using Technology : First Differences

Open your QuattroPro file, tableValuesTemplate. Key in the title, First Differences, into cell C3.

Move the cursor to cell C5, and key in the formula, = A : B5 - A : B4. Copy and paste this formula into cells C6 : C8. The first difference column should now be completed.

Note: Quattro Pro is just one of several spreadsheet applications that you could use for this activity. If you do not have access to Quattro Pro, perhaps you an use Excel. Ask your teacher for help if you cannot find a suitable spreadsheet program.

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