Now, what determines horizontal stresses? We have seen that total vertical stress is mostly a function of overburden and depth. The maximum principal stress in the horizontal case is and the minimum horizontal stress is, such that If vertical stress is a principal stress, then the two other principal stresses are horizontal. Total vertical stress may not be a principal stress, although in most cases it is. These are three independent normal stresses in directions all perpendicular to each other.Ī stress is a principal stress if there is no shear stress on the plane in which it is applied. The state of stress can be fully defined by the “principal stresses”. Stresses in horizontal direction are very often different to the stress in vertical direction. Vertical (effective) stress is not enough to define the state of stress in a solid. 5 Ideal orientation of open-mode fractures Subsurface Stresses and Previous: 2.2 Non-hydrostatic pore pressure Contents We can work around this by factoring inside the function.Next: 2.4 Problems Up: 2. This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch or compress a graph before shifting. To solve for x, we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression. What input to g would produce that output? In other words, what value of x will allow g\left(x\right)=f\left(2x 3\right)=12? We would need 2x 3=7. When we write g\left(x\right)=f\left(2x 3\right), for example, we have to think about how the inputs to the function g relate to the inputs to the function f. Horizontal transformations are a little trickier to think about. In other words, multiplication before addition. Given the output value of f\left(x\right), we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. When we see an expression such as 2f\left(x\right) 3, which transformation should we start with? The answer here follows nicely from the order of operations. For example, vertically shifting by 3 and then vertically stretching by a factor of 2 does not create the same graph as vertically stretching by a factor of 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first. When combining transformations, it is very important to consider the order of the transformations. This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching. Our new population, R, will progress in 1 hour the same amount as the original population P does in 2 hours, and in 2 hours, the new population R will progress as much as the original population P does in 4 hours. Let’s let our original population be P and our new population be R. Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. If the constant is between 0 and 1, we get a horizontal stretch if the constant is greater than 1, we get a horizontal compression of the function. When we multiply a function’s input by a positive constant, we get a function whose graph is stretched horizontally away from or compressed horizontally toward the vertical axis in relation to the graph of the original function. Notice that we are changing the inside of a function. Now we consider the changes that occur to a function if we multiply the input of an original function f\left(x\right) by some constant.
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