To determine which of the options represents a linear function with the same slope as the table, we first need to find the slope from the given data points.
The table provides the following points:
- (−4, 14)
- (−2, 8)
- (3, -7)
- (5, -13)
To find the slope (m) between any two points (x₁, y₁) and (x₂, y₂), we can use the formula:
\[ m = \frac{y₂ - y₁}{x₂ - x₁} \]
Let's calculate the slope between the first two points: (−4, 14) and (−2, 8).
Using the points:
- (x₁, y₁) = (−4, 14)
- (x₂, y₂) = (−2, 8)
Calculating the slope:
\[ m = \frac{8 - 14}{-2 - (-4)} = \frac{-6}{2} = -3 \]
Now, we'll calculate the slope using the second pair of points (3, -7) and (5, -13):
Using the points:
- (x₁, y₁) = (3, -7)
- (x₂, y₂) = (5, -13)
Calculating the slope:
\[ m = \frac{-13 - (-7)}{5 - 3} = \frac{-13 + 7}{2} = \frac{-6}{2} = -3 \]
Now that we have established the consistent slope of -3, we need to evaluate the options for linear functions that can represent this slope.
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6x - 2y = 12: Rearranging this into slope-intercept form (y = mx + b):
\[ -2y = -6x + 12 \] \[ y = 3x - 6 \]
The slope here is 3, not -3. Therefore, this option does not match.
-
James increases his muscle mass by 3 kg per day: This can be expressed as a linear function where the slope (rate of change of muscle mass per day) is 3. Thus, the slope is 3, which does not match -3. Therefore, this option does not match.
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Cynthia has $200 in her savings account and spends $3 per day on coffee: The linear function here represents the amount of money she has over time. It can be expressed as:
\[ y = 200 - 3x \]
The slope here is -3 (since for every day (x), she spends $3 reducing her total savings). Thus, this option matches the slope we calculated.
The correct options that represent a linear function with the same slope as the table are:
- Cynthia has $200 in her savings account and spends $3 per day on coffee.
1 answer