The system's state space and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v. Countable state space Continuous or general state space Discrete-time discrete-time Markov chain on a countable or finite state space Harris chain Markov chain on a general state space Continuous-time Any continuous stochastic process with the Markov property, e. Usually the term "Markov chain" is reserved for a process with a discrete set of times, i.

We will assign a number to a line, which we call slope, that will give us a measure of the "steepness" or "direction" of the line. It is often convenient to use a special notation to distinguish between the rectan- gular coordinates of two different points.

We can designate one pair of coordinates by x1, y1 read "x sub one, y sub one"associated with a point P1, and a second pair of coordinates by x2, y2associated with a second point P2, as shown in Figure 7.

Note in Figure 7. The ratio of the vertical change to the horizontal change is called the slope of the line containing the points P1 and P2.

This ratio is usually designated by m.

Thus, Example 1 Find the slope of the line containing the two points with coordinates -4, 2 and 3, 5 as shown in the figure at the right. Solution We designate 3, 5 as x2, y2 and -4, 2 as x1, y1. Substituting into Equation 1 yields Note that we get the same result if we subsitute -4 and 2 for x2 and y2 and 3 and 5 for x1 and y1 Lines with various slopes are shown in Figure 7.

Slopes of the lines that go up to the right are positive Figure 7. And note Figure 7.

However, is undefined, so that a vertical line does not have a slope. In this case, These lines will never intersect and are called parallel lines.

Now consider the lines shown in Figure 7. In this case, These lines meet to form a right angle and are called perpendicular lines.

In general, if two lines have slopes and m2: If we denote any other point on the line as P x, y See Figure 7.

In general let us say we know a line passes through a point P1 x1, y1 and has slope m. If we denote any other point on the line as P x, y see Figure 7. In Equation 2m, x1 and y1 are known and x and y are variables that represent the coordinates of any point on the line.

Thus, whenever we know the slope of a line and a point on the line, we can find the equation of the line by using Equation 2. Example 1 A line has slope -2 and passes through point 2, 4. Find the equation of the line. The slope and y-intercept can be obtained directly from an equation in this form.

Example 2 If a line has the equation then the slope of the line must be -2 and the y-intercept must be 8. Solution We first solve for y in terms of x by adding -2x to each member. We say that the variable y varies directly as x.Improve your math knowledge with free questions in "Write inequalities from graphs" and thousands of other math skills.

A beautiful, free online graphing calculator from leslutinsduphoenix.com The original A Maths Dictionary for Kids is an animated, interactive online math dictionary for students which explains over common mathematical terms and math words in simple language with definitions, examples, activities, practice and calculators.

A Maths Dictionary for Kids Quick Reference is a device friendly html version with definitions and detailed examples for over math words.

Writing, Solving, and Graphing Inequalities in One Variable. Learning Objective · Solve algebraic inequalities in one variable using a combination of the properties of inequality. · . The graph of a first-degree equation in two variables is a straight line.

That is, every ordered pair that is a solution of the equation has a graph that lies in a line, and every point in the line is associated with an ordered pair that is a solution of the equation.

Answer to The graph of g(x) is the result of translating the graph of f(x) = {1/2}x three units to the left. What is the equation of g(x)?

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