12. Determine whether y varies directly with x. If so, find the constant of variation and write the function rule.
x - 1, 2, 5
y - 1, 2, 5
you cannot tell by looking at that table??
Really???
Look again.
To determine if y varies directly with x, we need to check if the ratios of corresponding y and x values are equal. Let's find the ratios:
For the first pair: y/x = 1/1 = 1
For the second pair: y/x = 2/2 = 1
For the third pair: y/x = 5/5 = 1
Since the ratios are equal for all the pairs, we can conclude that y varies directly with x.
To find the constant of variation, we can choose any pair and take the ratio. Let's use the first pair:
k = y/x = 1/1 = 1
Therefore, the constant of variation is 1.
Now, we can write the function rule by putting the value of k into the equation:
y = kx
y = 1x
So, the function rule for this scenario is y = x.
To determine whether y varies directly with x, we need to check if the ratio of y to x is constant for all values of x and y. Let's calculate the ratio for each pair of values:
For the first pair (x = 1, y = 1):
Ratio = y/x = 1/1 = 1
For the second pair (x = 2, y = 2):
Ratio = y/x = 2/2 = 1
For the third pair (x = 5, y = 5):
Ratio = y/x = 5/5 = 1
Since the ratio of y to x is always 1, we can conclude that y varies directly with x.
Now, let's find the constant of variation. The constant of variation represents the ratio between y and x. In this case, the constant of variation is 1.
Finally, let's write the function rule. Since y varies directly with x, the function rule will be of the form y = kx, where k represents the constant of variation. In this case, the function rule can be written as y = 1x or simply y = x.