if y varies directly as the square of x and x is doubled, how will y change? Explain reasoning.

Question

This is a really confusing question. I really want to understand the steps to doing this problem and finding the answer. Thank you.

Answer 1

First, if y varies as the square of x, that means:

 

y = kx²

 

where k is the constant of proportionality. When thinking about change, I like to use subscripts to represent before and after. We will use a subscript of 1 to denote before (or the initial value of a variable) and a subscript of 2 to represent after (or the final value of a variable). So, y₁ is the initial and y₂ is the final. Also, x₁ is the initial and x₂ is the final. k does not change, so it doesn't get any subscripts. Thus, we can write two equations, because the above equation must hold before and after.

 

y₁ = k(x₁)²

y₂ = k(x₂)²

 

This is not obvious, but after a few times it becomes natural. We can now eliminate k. An easy way to do that is to divide the second equation by the first to get:

 

y₂ / y₁  =  (x₂)² / (x₁)²

 

The right hand side of the equation can be written in another form, to get:

 

y₂ / y₁ = ( x₂ / x₁ )²

 

Okay... so why did I do this? Because we are talking about factors of change, not absolute changes. So, y₂/y₁ represents the factor change of y and x₂/x₁ represents the factor change of x. Thus, when the prompt says x is doubled, we can understand that to mean that:

 

x₂/x₁ = 2

 

And we can determine the factor change of y by substituting 2 for the the factor change of x.

 

y₂ / y₁ = (2)² = 4

 

So, when x is doubled, y is quadrupled.

Answer 2

y = kx² where k is a constant (definition of y varying directly with x²)

 

Let y₁ = kx²

 

Now replace x with 2x:

 

y₂ = k(2x)² = 4kx² = 4y₁

 

Note the value of y is quadrupled when x is doubled.