Direct & Inverse Relationships Lab
Patterns and Relationships in. Physics. Honors Physics. Page 2. Direct and Inverse Relationships. If you are going to understand science or physics in general. Two values are said to be in direct proportion when increase in one results in an increase in the other. Similarly, they are said to be in indirect proportion when. Science is all about describing relationships between different variables, and direct and inverse relationships are two of the most important.
We could write y is equal to 1x, then k is 1. We could write y is equal to 2x. We could write y is equal to negative 2x.
We are still varying directly. We could have y is equal to pi times x. We could have y is equal to negative pi times x. I don't want to beat a dead horse now.
I think you get the point. Any constant times x-- we are varying directly. And to understand this maybe a little bit more tangibly, let's think about what happens. And let's pick one of these scenarios. Well, I'll take a positive version and a negative version, just because it might not be completely intuitive.
So let's take the version of y is equal to 2x, and let's explore why we say they vary directly with each other. So let's pick a couple of values for x and see what the resulting y value would have to be. So if x is equal to 1, then y is 2 times 1, or is 2. If x is equal to 2, then y is 2 times 2, which is going to be equal to 4. So when we doubled x, when we went from 1 to so we doubled x-- the same thing happened to y.
So that's what it means when something varies directly. If we scale x up by a certain amount, we're going to scale up y by the same amount. If we scale down x by some amount, we would scale down y by the same amount.
And just to show you it works with all of these, let's try the situation with y is equal to negative 2x. I'll do it in magenta. Let's try y is equal to negative 3x. So once again, let me do my x and my y. When x is equal to 1, y is equal to negative 3 times 1, which is negative 3. When x is equal to 2, so negative 3 times 2 is negative 6. So notice, we multiplied. So if we scaled-- let me do that in that same green color. If we scale up x by it's a different green color, but it serves the purpose-- we're also scaling up y by 2.
To go from 1 to 2, you multiply it by 2. To go from negative 3 to negative 6, you're also multiplying by 2.
Intro to direct & inverse variation (video) | Khan Academy
So we grew by the same scaling factor. To go from negative 3 to negative 1, we also divide by 3. We also scale down by a factor of 3. So whatever direction you scale x in, you're going to have the same scaling direction as y. That's what it means to vary directly. Now, it's not always so clear. Sometimes it will be obfuscated. So let's take this example right over here.
And I'm saving this real estate for inverse variation in a second.
- Direct and inverse proportions
You could write it like this, or you could algebraically manipulate it. Or maybe you divide both sides by x, and then you divide both sides by y.
These three statements, these three equations, are all saying the same thing. So sometimes the direct variation isn't quite in your face.
Direct and inverse relationships - Math Central
But if you do this, what I did right here with any of these, you will get the exact same result. Or you could just try to manipulate it back to this form over here. And there's other ways we could do it. We could divide both sides of this equation by negative 3. And now, this is kind of an interesting case here because here, this is x varies directly with y.
Or we could say x is equal to some k times y. And in general, that's true. If y varies directly with x, then we can also say that x varies directly with y. It's not going to be the same constant. It's going to be essentially the inverse of that constant, but they're still directly varying. If you increase the independent variable x, such as the diameter of the circle or the height of the ball dropthe dependent variable increases too and vice-versa.
Sciencing Video Vault A direct relationship is linear.
Direct & Inverse Relationships Lab
Pi is always the same, so if you double the value of D, the value of C doubles too. The gradient of the graph tells you the value of the constant. Inverse Relationships Inverse relationships work differently. If you increase x, the value of y decreases. For example, if you move more quickly to your destination, your journey time will decrease.
Intro to direct & inverse variation
In this example, x is your speed and y is the journey time. Doubling your speed halves the journey time, and increasing the speed by ten times makes the journey time ten times shorter. Mathematically, this type of relationship has the form: As you start to increase x, y decreases really quickly, but as you continue increasing x the rate of decrease of y gets slower.
In this case, y is inversely related to x. At first an increase of 3 in x decreases y by 2, but then an increase of 6 in x only decreases y by 1. This is why inverse relationships are declining curves that get shallower the further you move along them.
The Difference In direct relationships, an increase in x leads to a correspondingly sized increase in y, and a decrease has the opposite effect.
This makes a straight-line graph.