# How to write absolute value equations as piecewise equations for circles

Power Series — In this section we give a brief review of some of the basics of power series. Here is the work for this equation. The Wave Equation — In this section we do a partial derivation of the wave equation which can be used to find the one dimensional displacement of a vibrating string.

Note as well that while we example mechanical vibrations in this section a simple change of notation and corresponding change in what the quantities represent can move this into almost any other engineering field. We do not work a great many examples in this section.

Graphs of Absolute Value Functions Note that you can put absolute values in your Graphing Calculator and even graph them. Admittedly, it would take a calculator to determine just what those numbers are, but they are numbers and so we can do the same thing here.

The student uses the process skills to understand geometric relationships and apply theorems and equations about circles.

We will also take a look at direction fields and how they can be used to determine some of the behavior of solutions to differential equations. In particular we will model an object connected to a spring and moving up and down.

This section will also introduce the idea of using a substitution to help us solve differential equations.

Step Functions — In this section we introduce the step or Heaviside function. Summary of Separation of Variables — In this final section we give a quick summary of the method of separation of variables for solving partial differential equations.

As students work I make observations to help me decide how to proceed with the investigation. If you are looking for the original first edition black coverplease go here.

We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table.

We will restrict ourselves to systems of two linear differential equations for the purposes of the discussion but many of the techniques will extend to larger systems of linear differential equations.

Linear Equations — In this section we solve linear first order differential equations, i. Now we have to separate the equations. With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes.

The student uses the process skills to recognize characteristics and dimensional changes of two- and three-dimensional figures.

In this case our answer is all real numbers, since an absolute value is always positive. Piecewise Function Word Problems Problem: We get the first equation by just taking away the absolute value sign away on the left.

In addition, we will discuss reduction of order, fundamentals of sets of solutions, Wronskian and mechanical vibrations. We will also define the even extension for a function and work several examples finding the Fourier Cosine Series for a function. The piecewise function is: We work a wide variety of examples illustrating the many guidelines for making the initial guess of the form of the particular solution that is needed for the method.

Students will select appropriate tools such as real objects, manipulatives, paper and pencil, and technology and techniques such as mental math, estimation, and number sense to solve problems.

In proof and congruence, students will use deductive reasoning to justify, prove and apply theorems about geometric figures. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Both ln7 and ln9 are just numbers. This means we can write this absolute value function as a piecewise function. Make sign chart of the expression inside the absolute value.

3. Rewrite the equation without the absolute value as a piecewise function. For each interval where the Write as a piecewise. Learn how to solve absolute value equations and how to graph absolute value functions. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

Section Lines, Circles and Piecewise Functions. We’re not really going to do any graphing in this section. In fact, this section is here only to acknowledge that we’ve already looked at these equations and functions in the previous chapter.

Determine where the graph is positive vs. negative (reflected). Substitute values into the argument to see if the outcome is positive or negative. N Number and Quantity. N-RN The Real Number System. Extend the properties of exponents to rational exponents.

N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Section Solving Exponential Equations.

Now that we’ve seen the definitions of exponential and logarithm functions we need to start thinking about how to solve equations involving them.

How to write absolute value equations as piecewise equations for circles
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Solving Absolute Value Equations and Inequalities – She Loves Math