To find the equation described by the given points, we can start by observing the coordinates in the table:
- When \( x = 0 \), \( y = 10 \)
- When \( x = 1 \), \( y = -9 \)
- When \( x = 2 \), \( y = -8 \)
- When \( x = 3 \), \( y = -7 \)
We can look for a linear relationship since there is a consistent change in \( y \) as \( x \) changes.
Let's calculate the differences in \( y \):
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From \( (0, 10) \) to \( (1, -9) \): \[ -9 - 10 = -19 \]
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From \( (1, -9) \) to \( (2, -8) \): \[ -8 - (-9) = 1 \]
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From \( (2, -8) \) to \( (3, -7) \): \[ -7 - (-8) = 1 \]
Thus, the differences in \( y \) between each consecutive point suggest that after the first decrease, the \( y \)-values stabilize.
Let's plot the first two points to investigate the general slope:
- The first two points \( (0, 10) \) and \( (1, -9) \) provide the slope: \[ m = \frac{-9 - 10}{1 - 0} = \frac{-19}{1} = -19 \] However, based on the subsequent changes, the influence of other points can alter our slope consideration.
To find a linear function that fits the data, we'll use the points more strategically. Since the first point drops drastically when compared against the others, this suggests potentially a quadratic or piecewise function could fit if we extend the scope, especially since the later points stabilize around the value of -7.
However, concentrating on the linear comprehension, let’s take: \[ y = mx + c \] and solve for \(m\) and \(c\) using the first two points: Using \( (0, 10) \)
- Substitute \( x = 0 \): \[ 10 = m(0) + c \Rightarrow c = 10 \]
Using point \( (1, -9) \): 2. Substitute \( x = 1 \): \[ -9 = m(1) + 10 \Rightarrow m + 10 = -9 \Rightarrow m = -19 \]
Thus, a prospective linear function based on these data points takes the form: \[ y = -19x + 10 \] However, this does not satisfy all given values precisely due to subsequent values appearing to stabilize after the large jump.
To conclude, while not necessarily reflecting all data behavior, \( y = -19x + 10 \) is the most straightforward answer derived from the significant first point's influence against the latter linearly orientated outputs as those grow closer together.
More rigorous derivations might involve analyzing higher-degree polynomials or piecewise functions for intricate alignments across the provided (0,10), (1,-9), (2,-8), (3,-7) dataset.
Thus, the equation described in the table based on the function notation could reasonably approximate: \[ y = -19x + 10 \] up to \( x = 1 \) rather quickly adapt to distinct models for \( x \geq 2 \).