To identify the graph that represents the equation \( y = 4x \), we need to analyze what this equation means. The equation suggests that for every unit increase in \( x \), \( y \) increases by 4 units.
The slope of the line is 4, and the y-intercept is 0 (the line passes through the origin (0, 0)). We can also calculate a few points:
- When \( x = 0 \), \( y = 4(0) = 0 \). So the point is (0, 0).
- When \( x = 1 \), \( y = 4(1) = 4 \). So the point is (1, 4).
- When \( x = 2 \), \( y = 4(2) = 8 \). So the point is (2, 8).
- When \( x = 3 \), \( y = 4(3) = 12 \). So the point is (3, 12).
Now, let's look at the options provided:
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Option 1: A line goes through points (0, 40) and (5, 45) - This does not match the equation since it has a much higher y-value.
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Option 2: A line goes through points (0, 0) and (1, 4) - This matches our calculations. (0, 0) is the y-intercept, and (1, 4) correctly reflects that when \( x = 1 \), \( y = 4 \).
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Option 3: A line goes through points (0, 4) and (1, 5) - This does not match since the y-intercept is not 0 and the slope is not 4.
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Option 4: A line goes through points (0, 4) and (4, 0) - This does not match since the slope is negative rather than positive.
Given this analysis, Option 2 is the correct graph that represents the relationship defined by the equation \( y = 4x \).