To find a linear relationship in a table, examine the rate of change between the input and output values; if the rate of change is constant, the relationship is linear.
Identifying Linear Relationships in Tables
When presented with a table of values, identifying a linear relationship involves checking if the rate of change between the input (independent variable, often 'x') and the output (dependent variable, often 'y') is constant. In simpler terms, for every consistent change in 'x', there should be a consistent change in 'y'.
Steps to Determine Linearity
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Calculate the Rate of Change:
- Choose any two points from the table: (x₁, y₁) and (x₂, y₂).
- Calculate the change in 'y' (Δy) and the change in 'x' (Δx):
- Δy = y₂ - y₁
- Δx = x₂ - x₁
- Divide Δy by Δx to find the rate of change (slope): m = Δy / Δx
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Repeat for All Points:
- Repeat step 1 using different pairs of points from the table. It's important to check multiple pairs to ensure consistency.
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Check for Consistency:
- If the rate of change (slope) is the same for every pair of points, the table represents a linear function.
- If the rate of change varies, the relationship is non-linear.
Example
Let's consider the following table:
| x | y |
|---|---|
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
-
First Pair (1, 3) and (2, 5):
- Δy = 5 - 3 = 2
- Δx = 2 - 1 = 1
- m = 2 / 1 = 2
-
Second Pair (2, 5) and (3, 7):
- Δy = 7 - 5 = 2
- Δx = 3 - 2 = 1
- m = 2 / 1 = 2
-
Third Pair (3, 7) and (4, 9):
- Δy = 9 - 7 = 2
- Δx = 4 - 3 = 1
- m = 2 / 1 = 2
Since the rate of change is consistently 2 for all pairs of points, this table represents a linear function.
Key Takeaway
According to the reference, if the rate of the change in the input to the change in the output is the same for every point in the table, then the table represents a linear function.
