To determine which graph represents the equation \( y = \frac{1}{2}x + 2 \), we can evaluate the y-intercept and the slope of the line.
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Y-intercept: The y-intercept occurs when \( x = 0 \).
- Plugging \( x = 0 \) into the equation: \[ y = \frac{1}{2}(0) + 2 = 2 \] This tells us that the line intersects the y-axis at the point (0, 2).
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Slope: The slope of the line is \( \frac{1}{2} \), which means that for every increase of 1 in \( x \), \( y \) increases by \(\frac{1}{2}\).
Now, let's analyze the options given:
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Graph of a line passing through the points (-2, -2) and (0, -1):
- At \( x = 0 \), \( y \) is -1, which does not match our y-intercept of 2.
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Graph of a line passing through the points (-2, 0) and (0, 1):
- At \( x = 0 \), \( y = 1\), which does not match our y-intercept of 2.
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Graph of a line passing through the points (-2, -3) and (0, -2):
- At \( x = 0 \), \( y = -2\), which does not match our y-intercept of 2.
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Graph of a line passing through the points (-4, 0) and (0, 2):
- At \( x = 0 \), \( y = 2\), which matches our y-intercept.
- The slope can also be checked; from (-4, 0) to (0, 2):
- Change in y = \( 2 - 0 = 2\)
- Change in x = \( 0 - (-4) = 4\)
- Slope = \( \frac{2}{4} = \frac{1}{2} \)
Thus, the correct answer is:
Graph of a line passing through the points (-4, 0) and (0, 2).