To understand the relationship between the functions \( f(x) = 2x \) and \( f(x) = 2x + 3 \), let’s analyze them:
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\( f(x) = 2x \):
- This is a linear function with a slope of 2 and a y-intercept at \( (0, 0) \). The graph is a straight line passing through the origin.
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\( f(x) = 2x + 3 \):
- This is also a linear function with the same slope of 2 but a different y-intercept, which is at \( (0, 3) \). This means that the line is parallel to the line of \( f(x) = 2x \) but shifted upwards by 3 units.
Graphically, both lines will have the same steepness (the same slope of 2) but will differ in their vertical position due to the constant term (3) in the second function.
In a graph of these two functions:
- The line for \( f(x) = 2x \) will pass through the origin.
- The line for \( f(x) = 2x + 3 \) will be parallel to the first line and will intersect the y-axis at 3.
So, the correct graph should show two parallel lines, one passing through the origin and the other crossing the y-axis at 3. If you are looking at options for graphs, pick the one that matches this description.