Determine whether y varies directly with x. If so, find the constant of variation and write the function rule.
x, y
32, 16
34, 17
38, 19
What is the constant of variation? Select the correct choice and fill in any answer boxes in your choice below.
k=
To determine whether y varies directly with x, we need to check if the ratio of y to x remains constant for all pairs of values.
For the given values:
When x = 32, y = 16
The ratio of y to x is 16/32 = 0.5
When x = 34, y = 17
The ratio of y to x is 17/34 = 0.5
When x = 38, y = 19
The ratio of y to x is 19/38 = 0.5
Since the ratio of y to x is constant for all pairs of values, we can conclude that y varies directly with x.
To find the constant of variation, we can choose any pair of values and find the ratio of y to x:
Using the first pair (32, 16), the ratio of y to x is 16/32 = 0.5
Therefore, the constant of variation (k) is 0.5.
The function rule for this direct variation is y = kx. Substituting k = 0.5, the function rule is:
y = 0.5x