b The variable y varies inversely with x when y= kx, where k≠0.
A
a The value of y doubles.
B
b The value of y is cut in half.
Practice makes perfect
a We are asked to explain how the value of y changes if y varies directly with x and the value of x is doubled. First, let's recall the definition of direct variation.
The variable y varies directly with x when y=kx, where k≠0.
The constant k is called the constant of variation for a direct variation. We are not given an exact value of k which means we are considering any direct variation equation.
y=kx, k≠0
Now, we are ready to review how y changes when x is doubled. To do this, we will substitute 2x for x into the direct variation equation and compare the value of y before and after the substitution.
Direct Variation Equation
y= kx
Substitute
y=k( 2x)
Simplify
y=2kx
Compare
y=2 kx
We can see that when x is doubled, the value of y also doubles. This is true for any direct variation equation. Here are some examples for different values of k.
k
x
y=kx
Double x
y=k(2x)
Did y Double?
2
1
2
2
4
4=2 ( 2) ✓
3
- 2
- 6
- 4
- 12
- 12=2 ( - 6) ✓
- 4
3
- 12
6
- 24
- 24=2 ( - 12) ✓
b This time we are asked to explain how the value of y changes if y varies inversely with x and the value of x is doubled. Let's recall the definition of inverse variation.
The variable y varies inversely with x when y=k/x, where k≠0.
The constant k is called the constant of variation for an inverse variation. Again, we are not given an exact value of k which means we are considering any inverse variation equation.
y=k/x, k≠0
Let's review how y changes when x is doubled. To do this, we will substitute 2x for x into the inverse variation equation and compare the value of y before and after the substitution.
Inverse Variation Equation
y= k/x
Substitute
y=k/2x
Rewrite
y=1/2*k/x
Compare
y=1/2* k/x
We can see that when x is doubled, the value of y is cut in half. This is true for any inverse variation equation. Below we show a few examples for different values of k.
