To find the transformation needed for a different line equation to become \(y = \frac{2}{3}x\), we can analyze the potential line you are starting with and how its transformation results in the slope and intercept of the target line.
The given points \((0, 2)\), \((3, 4)\), \((6, 6)\), and \((9, 8)\) suggest that there is a linear relationship existing, more specifically, the equation of the line can be interpreted.
To derive the equation of the line that passes through these points, we start from the standard slope-intercept form \(y = mx + b\).
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Determining the slope (m): The rise over run (slope) can be calculated using the points:
- From \((0, 2)\) to \((3, 4)\): the rise is \(4 - 2 = 2\) and the run is \(3 - 0 = 3\). Thus, the slope is \(\frac{2}{3}\).
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Determining the y-intercept (b): The line begins at \( y = 2 \) when \( x = 0 \), indicating the y-intercept is \(2\).
So the equation of the line that goes through these points is \(y = \frac{2}{3}x + 2\).
To transform this equation to become \(y = \frac{2}{3}x\), we need to adjust the y-intercept:
- Starting with \(y = \frac{2}{3}x + 2\),
- We want to decrease the y-intercept from \(2\) to \(0\).
That means we need to shift the graph down by \(2\) to eliminate the \(+2\) and make the line touch the origin.
Thus, the correct transformation needed for the line to become \(y = \frac{2}{3}x\) is:
Shift the graph down by 2.
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