What are the different types of mathematical relationships?
Direct variation describes a simple relationship between two variables. We say y varies directly with x (or as x, in some textbooks) if. Inverse or Indirect Variation refers to relationships of two Variation, like we saw above for a Direct Variation. Mathematical Relationships in Science. measuring The points fall close enough to the straight line to conclude that this is a linear or direct relationship.
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. 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.
Direct, Inverse, Joint and Combined Variation
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. 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. 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.
What is indirect relationship? definition and meaning - klokkenluideronline.info
Now with that said, so much said, about direct variation, let's explore inverse variation a little bit. Inverse variation-- the general form, if we use the same variables. And it always doesn't have to be y and x. It could be an a and a b. It could be a m and an n. If I said m varies directly with n, we would say m is equal to some constant times n. Now let's do inverse variation. So let me draw you a bunch of examples. And let's explore this, the inverse variation, the same way that we explored the direct variation.
And let me do that same table over here. In an inverse relationship, an increase in one quantity leads to a corresponding decrease in the other. Faster travel means a shorter journey time.
How Does y Vary with x? Scientists and mathematicians dealing with direct and inverse relationships are answering the general question, how does y vary with x? Here, x and y stand in for two variables that could be basically anything. By convention, x is the independent variable and y is the dependent variable.
So the value of y depends on the value of x, not the other way around, and the mathematician has some control over x for example, she can choose the height from which to drop the ball. When there is a direct or inverse relationship, x and y are proportional to each other in some way. Direct Relationships A direct relationship is proportional in the sense that when one variable increases, so does the other. Using the example from the last section, the higher from which you drop a ball, the higher it bounces back up.
A circle with a bigger diameter will have a bigger circumference. Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems. Sine Wave Relationship The graphs of the sine and cosine functions are sinusoids of different phases.
The sine wave or sinusoid is a mathematical curve that describes a smooth repetitive oscillation. It is named after the function sine, of which it is the graph.
Lab Activities and Resources What are Mathematical Relationships What is a mathematical relationship and what are the different types of mathematical relationships that apply to the laboratory exercises in the following activities. What is the relationship between how much a spring stretches and the force pulling on the spring?
- Intro to direct & inverse variation
What is the relationship between the mass of a ball and its volume assuming a constant density? What is the relationship between the intensity of a beam of light and the distance from a light source? What is a the relationship between how the distance travels and the time in travel for an accelerating object?
What is the relationship between how much light passes through a Polaroid filter and the angle the filter is rotated? What is the relationship between current, voltage when there is a constant resistance in an electric circuit.
Radioactive Decay- - Problem: What is the relationship between the decay of radioactive material and the time allowed for the decay? Water Pressure - - Problem: What is the relationship between water pressure and depth of water? What is the relationship between the height a ball bounces and the number of times it has bounced?