To determine which graph represents a linear function with a slope of 0.5 and a y-intercept of 2, we can use the slope-intercept form of a linear equation, which is:
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept.
In this case, the slope \( m \) is 0.5 and the y-intercept \( b \) is 2. Therefore, the equation of the line we are looking for is:
\[ y = 0.5x + 2 \]
Next, we can check each of the options:
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First option: A line goes through points (-2, 0) and (0, 1):
- The y-intercept here is 1 (the value of y when x=0), which is not equal to 2.
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Second option: A line goes through points (0, 2) and (4, 0):
- The y-intercept is 2 (the value of y when x=0). Now, we need to find the slope.
- The slope \( m \) can be calculated using the two points: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
- Using points (0, 2) and (4, 0): \[ m = \frac{0 - 2}{4 - 0} = \frac{-2}{4} = -0.5 \]
- The slope is -0.5, which does not match.
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Third option: A line goes through points (0, 2) and (2, 3):
- The y-intercept is 2, which is correct. Now, we check the slope:
- Using points (0, 2) and (2, 3): \[ m = \frac{3 - 2}{2 - 0} = \frac{1}{2} = 0.5 \]
- The slope is 0.5, which matches.
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Fourth option: A line goes through points (-4, 0) and (0, 2):
- The y-intercept is 2, which is correct. Now we check the slope:
- Using points (-4, 0) and (0, 2): \[ m = \frac{2 - 0}{0 - (-4)} = \frac{2}{4} = \frac{1}{2} = 0.5 \]
- The slope is also 0.5, which matches.
Based on the analysis, the third option (a line going through points (0, 2) and (2, 3)) and the fourth option (a line going through points (-4, 0) and (0, 2)) are valid representations of the linear function with a slope of 0.5 and a y-intercept of 2. Therefore, both the third and fourth options are correct.
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