One relationship in tables and points
Turn each table row into a coordinate point and check whether intermediate inputs are allowed.
Begin with the activity ↓One relationship in tables and points
Put input two and output five in one table row. With input on the horizontal axis and output on the vertical axis, that row becomes (2, 5). A table and graph show the same input-output pair in different forms.
Put input two and output five in one table row. With input on the horizontal axis and output on the vertical axis, that row becomes (2, 5). A table and graph show the same input-output pair in different forms.
The rule f(x) = 2x + 1 gives outputs 1,3,5,7,9 at integers 0,1,2,3,4. Select a row and read horizontal then vertical. The reversed point (5, 2) does not represent the same input-output pair.
If only integer inputs are allowed, as with notebook counts, these five points form the entire graph on this domain. Do not join them in a way that implies 1.5 is allowed. The input conditions matter more than the alignment of the points.
Now allow every number on the number line from zero through four, including both endpoints. This is a real interval. The same formula gives four at input 1.5. Including all intermediate inputs makes the graph a line segment. The screen lets you sample intermediate inputs in steps of 0.5.
The five rows alone do not uniquely determine a formula. The line segment is determined because we separately specify that the formula applies at every allowed input. A few observed points do not automatically establish the relationship between them.
Switch between integer inputs and a real interval. The formula stays the same while the included inputs change. Next we compare increasing rules with a function whose output stays fixed.
Listen one paragraph at a time
Hold on to this distinction
Match the table’s (input, output) order to the axes. Joining points depends on the domain and the specified intermediate values, not just the picture’s shape.
Change it and check
Choose the domain and select a table row to locate its point. In interval mode choose 1.5 to check a value between integers.
f(x) = 2x + 1 · Domain: integers 0,1,2,3,4
| x | f(x) |
|---|---|
| 0 | 1 |
| 1 | 3 |
| → 2 | 5 |
| 3 | 7 |
| 4 | 9 |
f(2) = 5 · (2, 5)
Only the five integer points belong. Do not join them.
The axes are input x and output f(x), each with unit intervals. The line segment covers a real interval; the selectable inputs are just samples of that interval.
Try with paper and objects
First draw the integer table and five points. In a second graph allowing all intermediate inputs, add the line segment given by the formula and explain the difference.
Check and explain another example
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6 · output value
2 × 2.5 + 1 = 6, and 2.5 belongs to the domain.
Does your explanation include these ideas?
Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.
Continue, or pause here
Compare proportional, offset, constant and squaring rules using the same criterion.