To find the equation of the linear function in slope-intercept form (y = mx + b), we need to determine the slope (m) and the y-intercept (b) from the provided points in the table.
Let's start by determining the slope (m) using two points:
- Point 1: (11, 99)
- Point 2: (22, 1414)
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates into the formula:
\[ m = \frac{1414 - 99}{22 - 11} = \frac{1315}{11} \approx 119.54545454545455 \]
Now, we can also use other pairs of points to confirm the slope:
Using the points (22, 1414) and (33, 1919):
\[ m = \frac{1919 - 1414}{33 - 22} = \frac{505}{11} \approx 45.909090909090914 \]
Using the points (33, 1919) and (44, 2424):
\[ m = \frac{2424 - 1919}{44 - 33} = \frac{505}{11} \approx 45.909090909090914 \]
However, it seems there is a pattern in the y-values since they increase substantially with each x-step, indicating a possible relationship.
To better understand the relationship, we can check the ratios or differences:
- From \(x=11\) to \(x=22\): Change in \(y\) is \(1414 - 99 = 1315\)
- From \(x=22\) to \(x=33\): Change in \(y\) is \(1919 - 1414 = 505\)
- From \(x=33\) to \(x=44\): Change in \(y\) is \(2424 - 1919 = 505\)
It appears that this is not a linear function since the differences are not consistent across the x-values.
To find a linear approximation and assuming a simple linear representation still, we can attempt to calculate further. However, we should observe that the values do not fit a simple linear model (y = mx + b).
We conclude that based on the data presented, we cannot define a unique linear function from the inconsistent spacing seen with the y-values. The values indicate possible computational error or misrepresentation leading us to report as a linear equation maintaining to create complexity in functional requirements—therefore we must dig deeper to rationalize the equation, yet not solve it entirely as a direct linear function.
Please share more details or specify further how you wish to define functional representation if linear is required despite nature. The observed function would need proper investigation or verification against polynomial sequences or similar structures (quadratic, exponential, etc.).
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