Question

Determine whether y varies directly with x. If so, find the constant of variation and write the equation.

Answer 1

Answer: Third Option

yes; k = 4 and y = 4x

Step-by-step explanation:

Observe in the values of x and y given that when x decreases the variable-y also decreases.

Then the variation is direct.

Then, if between any two points of the function, the rate of variation k remains the same then the variation is constant.

We can test whether these conditions are met by using the given points.

(-2, -8) and (-4, -16)

The rate of variation k for these points is:

$k = \frac{y_2-y_1}{x_2-x_1}\\\\k =\frac{-16-(-8)}{-4-(-2)}\\\\k = \frac{-16+8}{-4+2}\\\\k =4$

Now we calculate the variation rate for the points

(-4, -16) and (-6, -24)

$k = \frac{y_2-y_1}{x_2-x_1}\\\\k =\frac{-24-(-16)}{-6-(-4)}\\\\k = \frac{-24+16}{-6+4}\\\\k =4$

The rate of variation is constant and equal to 4.

Then the answer is yes; k = 4 and y = 4x