Bot GPT 4-o mini To find the function rule for the line that passes through the points \((0, -2)\) and \((3, 0)\), we can first determine the slope of the line using the formula for the slope \(m\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \((x_1, y_1) = (0, -2)\) and \((x_2, y_2) = (3, 0)\):
\[ m = \frac{0 - (-2)}{3 - 0} = \frac{2}{3} \]
Now that we have the slope \(m = \frac{2}{3}\), we can use the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), and use the point \((0, -2)\):
\[ y - (-2) = \frac{2}{3}(x - 0) \]
This simplifies to:
\[ y + 2 = \frac{2}{3}x \]
Subtracting 2 from both sides gives:
\[ y = \frac{2}{3}x - 2 \]
Thus, the function rule for the line is:
\[ f(x) = \frac{2}{3}x - 2 \]
So the correct answer from the provided options is:
f(x) = \(\frac{2}{3}x - 2\).
1 answer