To determine which students correctly wrote an equation to represent the linear function based on the provided table of values, we can start by identifying the slope and y-intercept from the data points given.
The points from the table are:
- (-2, -8.5)
- (4, -7)
- (12, -5)
Step 1: Calculate the Slope (m)
The slope \( m \) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
We can use two points for finding the slope. Let's use the points (4, -7) and (12, -5): \[ m = \frac{-5 - (-7)}{12 - 4} = \frac{-5 + 7}{12 - 4} = \frac{2}{8} = \frac{1}{4} \]
Step 2: Find the Equation
We can use point-slope form \( y - y_1 = m(x - x_1) \) to find the equation. Let's use the point (4, -7): \[ y - (-7) = \frac{1}{4}(x - 4) \] This simplifies to: \[ y + 7 = \frac{1}{4}(x - 4) \] This is equivalent to: \[ y + 7 = \frac{1}{4}x - 1 \] Bringing it to slope-intercept form, \( y = mx + b \): \[ y = \frac{1}{4}x - 8 \]
Step 3: Check Each Student's Equation
Susan: \[ y + 7 = 4(x - 4) \] This does not match our derived equation; it seems incorrect.
Joseph: \[ x - 4y = 32 \] Rearranging gives us: \[ 4y = x - 32 \implies y = \frac{1}{4}x + 8 \] This does not match the derived equation, so he is incorrect.
Alice: Alice’s equation appears incomplete, but if it implies the slope of \( -\frac{1}{8} \), it also does not match our derived slope of \( \frac{1}{4} \).
Conclusion
None of the students wrote the correct equation based on the analysis of the data. Therefore, the correct answer is:
None of the given answer choices (A, B, C, D) is correct based on the equations provided by the students.
If you are required to give an answer based on the options presented, it would be implied that none of them are correct. However, if this were a test scenario or practice problem, ensure to review the students' equations and match against the derived equation thoroughly.
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