Final answer:
To find the value of r for the function y=g(x) that is parallel to f(x)=3x, we use the slope-intercept form with a slope of 3 and y-intercept of 5. The equation for g(x) becomes y=3x+5, and plugging in x=2 gives us y=11, so r=11.
Step-by-step explanation:
The student's question about the function f(x)=3x is related to the concept of slope and parallels lines in the xy-plane.
Since the graph of y=f(x) is parallel to the graph of the linear function y=g(x), and given that g(0)=5 which represents the y-intercept, we know two points on the function g(x).
The first is (0,5), and with the information that the slope of g(x) must be the same as f(x), which is 3 (because parallel lines have the same slope), we can find the second point using the second given value g(2)=r.
To find the value of r, we 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. Since g(x) has the same slope as f(x), which is 3, and its y-intercept is 5, the equation for g(x) is y = 3x + 5.
Plugging in x = 2 into this equation gives us y = 3(2) + 5 = 6 + 5 = 11, so r is 11.