More Formulas: circles Elementary Education 12

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1 More Formulas: circles Elementary Education 12 As the title indicates, this week we are interested in circles. Before we begin, please take some time to define a circle: Due to the geometric shape of circles, they play an important role in our society. Depending on the industry that one works in, understanding the effects of a circle may play a daily role. This might include anything from designing storage systems to understanding circular farming methods. In today s activity, we will develop an equation for a circle in general and then take a deeper look at formulas. To begin with, let s take a look at an equation that few people forget, the Pythagorean Formula for finding lengths of sides of a right triangle: This appears to be a very easy formula to remember and is primarily due to its simplistic relation with our alphabet (our a, b, c s). It is really too bad that more formulas don t have a simplistic pneumonic device to make remembering them this easy, but since they don t, we just have to build our skills to make it easier to remember those formulas which become important to us. From the Pythagorean formula it is easy to build the formula for finding the distance between two points:. Consider the two points on a graph: If we desire to find the distance between the two points, we can easily build a right triangle.

2 We can find the length of the two sides of the triangle by considering that only the x value changes on the horizontal axis and only the y-value changes on the vertical axis so that we have: Next, we substitute these lengths into the Pythagorean Formula: Activity: Please explain why we stated we would only consider the positive! Should we always only consider the positive? The next substitution is to replace the C with a d to represent distance: We take the square root of both sides, and only worry about the positive: And since the desired goal is to find, the distance between two points, we typically write:

3 Find the distance between the following points. 1) 2) 3) 4) Now we are ready for the circle. How does your definition that you stated at the beginning of today s work compare to this definition? A circle is the set of all points in a plane which are an equal distance (Radius) from a single point (Center). How could you restate your definition so that the mathematical definition is completely part of your definition and a third grader can understand it?

4 Based on this definition, there are three items which are needed to determine the location of a circle: The set of all points. We will label these points. The radius. We will label this The center point. This is a specific point that we will label. The use of (h,k) for the center of circles has a long history and is based on a topic in math called conic sections. Thus, we may see (h,k) being used for a specific purpose, when talking about parabolas, ellipses and other conic sections. Hopefully you can recognize that the relationship between the center and the point on the circle is the distance formula with the distance being. Next, it is generally accepted that when discussing the circle we square both sides of the distance formula so that equation of a circle is: If we know the center, (3, -6) and the radius, 4, we can write the equation of the specific circle as follows: Notice that the radius is squared. Graph the following circles. 5) 6)

5 7) 8) So far this semester, we have discussed several shapes with respect to the graphs. The formula problems 9-12 list the general form of an equation. Discuss the shapes that will occur when the information is given to graph these equations. Include information about directions: 9) 10) 11) 12) This problem has been briefly discussed and you will need to have your answer checked. The point of the last four problems is to show that if we have an understanding for what a graph is going to look like before we begin to draw the graph, it is easier to graph correctly. Now let s consider a population growth problem where the initial population rate is 6% compounded monthly. Recall that the compound formula is: and the growth Filling in our known information, we have:

6 What do we do if we want to graph this function with on the vertical axis (normally the y-axis) and on the horizontal axis (normally the x-axis). What does this graph look like? When we do not know what a graph will look like, we need to return to the t-chart. Fill in the following t-chart for the first ten years (including year 0 makes a total of 11 years). Then graph the function So what is this shape? Does it have one? If we examine the equation closely, we recognize that the variable,, which is used on the horizontal axis is in the exponent of the equation. Thus, this type of equation is called an exponential function. And like other named equations it has certain characteristics to look for when graphing. The general formula of an exponential function is: Why are the restrictions placed on? First, if is negative, a situation will occur where will be part of the equation and one cannot have the square root of a negative number when working An exponential equation is used whenever the growth rate or the decay rate is considered to be at a constant percent. In this case the savings account grows at a constant percent rate and thus is best described by an exponential function. Recall that a line is used whenever the growth rate is constant. Thus, we now have a graphical representation when the growth is constant (linear) or constant percentage growth (exponential).

7 with real numbers. The other case where creates, which is just a constant function. We have left for you to explain what happens when values of. and why this is not included in acceptable At this time, experiment with changing the values for a, b, c and d and see if you can determine what happens to the shape of the graph when these values change? Describe the changes below. As defined above, an exponential equation is used any time there is a constant percent change. This means that the exponential function is a mainstay of science to study growth and decay.

Pure Math 30: Explained!

Pure Math 30: Explained! www.puremath30.com 5 Conics Lesson Part I - Circles Circles: The standard form of a circle is given by the equation (x - h) +(y - k) = r, where (h, k) is the centre of the circle and r is the radius. Example