Extension A: Circles and Ellipses Algebra ; Pre-Calculus Time required: 35 50 min. Learning Objectives Math Objectives Students will write the general forms of Cartesian equations for circles and ellipses, and be able to identify the key characteristics represented by each in terms of transformations. Technology Objectives None Math Prerequisites Successful completion of the core function transformations unit. Algebra manipulation skills with squares and square roots. Technology Prerequisites Knowledge of Geometry Expressions as developed in the core function transformations unit. Materials A computer with Geometry Expressions for each student or pair of students. 008 Saltire Software Incorporated Page 1 of 8
Overview for the Teacher Extension A: Circles and Ellipses Algebra ; Pre-Calculus Time required: 35 50 min. Function Transformations Ext. A: Circles and Ellipses This short extension of the transformation units explores the use of transformation rules on graphs that are not functions. Specifically, it demonstrates the relationship between circles and ellipses as unequal horizontal and vertical dilations. Students also apply translation rules and formulate the general Cartesian equations. Students may already be familiar with the equation for the unit circle. If that is the case, the beginning should flow quickly and be a simple review of why that formula works. 1) Students should know several ways to create the unit circle. Any of them should work. One possibility: Draw a circle anywhere, then constrain its center to (0,0) and constrain its radius to 1. A) AC 1 B) BC B C) 1 D) x +y = 1 ) A) x +y = B) x +y = 9 C) x + y = r 3) A) ( x 3) + ( y + ) = 1 B) x 6x + 9 + y + y + = 1 C) x 6x + y + y + 1 = 0 D) The same translation rules apply E) ( ) ( ) x h + y k = r This is a good place for a checkpoint to make sure students are on the right track. F) GX equation: x + y xh + h yk + k r = 0 Equivalence: ( x h) + ( y k) x x x y ) A) + = 1 a a B) xh + h + y + y xh + h = r yk + k yk + k x y + = 1 x + y = a a a We get the same equation as in c, with a equal to the radius. The dilation rules seem to work fine. = r r -1 = 0 A -1 1 C 1 008 Saltire Software Incorporated Page of 8
Extension A: Circles and Ellipses Algebra ; Pre-Calculus Time required: 35 50 min. 5) Here we move into new territory, but it should be easy travel for students. They are simply applying unequal vertical and horizontal dilations to a circle in order to create an ellipse. A) x y x y + = 1 + = 1 5 3 5 9 B) A horizontal dilation of scale factor 5, and a vertical dilation of scale factor 3-6 - C) D) 9x y = ± 9 Remind them that the +/-creates two functions: 5 y = 9x 9 5 9x and y = 9 Each branch will produce half of the ellipse. Some students 5 x may produce the equivalent form: y = ± 3 1. 5 6) Encourage students to sketch the graphs based on the characteristics, rather than on the computer this time. A) ( x + 7) ( y 3) + 16 equations. = 1 Student graphs probably won t have the function 6 Y=3+ - (7+X) Y=3- - (7+X) -1-10 -8 - - 008 Saltire Software Incorporated Page 3 of 8
B) ( x ) ( y + ) + 9 equations. 9 = 1 Extension A: Circles and Ellipses Algebra ; Pre-Calculus Time required: 35 50 min. Student graphs probably won t have the function Y=+ 05 9 + 36 X 9-9 X 9-6 8 10 - -8 Y=- 05 9 + 36 X 9-9 X 9 C) ( x h) ( y k) a + b = 1 x h y k or + = 1 a b 008 Saltire Software Incorporated Page of 8
Function Transformations Ext. A: Circles and Ellipses Today you are going to extend what you have learned about function transformations to graphs that don t represent functions. As you recall, in a function there is only one y value for any given x value. Graphically, this means that any vertical line will only cross the graph one time. Equations for functions can be written in the form y =... or f(x) =..., in which the right side of the equation is in terms of x alone. All the vertical and horizontal dilations and translations you have done so far have been done to functions. So what happens to other shapes, like circles? 1) First, we need an equation for a circle. Open a new GX file, and create a unit circle (radius = 1) with point A centered on the origin and point B on the circle. Draw in segment AB. Draw segment BC with C on the x-axis, and constrain BC to be perpendicular to the x-axis. You now have a right triangle with one vertex (B) on the unit circle. A) What side of the triangle has a length equal to the x-coordinate of point B? B) What side of the triangle has a length equal to the y-coordinate of point B? C) What is the length of the hypotenuse? Name: Date: D) Use the Pythagorean theorem to write an equation which relates x, y, and the length of the hypotenuse. You can drag B around; the coordinates of all the possible points B describe the circle. These coordinates are described by the equation you wrote in part D. ) What if the radius isn t 1? A) Modify your diagram on GX from part 1 to have a radius of. Notice that this dilates the shape horizontally and vertically at the same time. More on this later. Use the Pythagorean theorem again to write a new equation which relates x, y, and the length of the hypotenuse. B) Repeat the process for a circle of radius 3. C) Now consider the general case, a circle with radius r. Write an equation which relates x, y, and r. This is the general equation for a circle centered at the origin. 008 Saltire Software Incorporated
3) The next logical question is what happens if the circle isn t centered on the origin. Change your radius back to 1, and then draw in a vector. Constrain your vector to represent a horizontal translation of 3 and a vertical translation of. Translate the circle and its center together. A) If the same translation rules hold for circles as for functions, what should the new equation be? B) Multiply out the expressions in parenthesis, and write your new equation here. C) Use known algebra properties to simplify the equation, and make the right side = zero. Write your new equation here. D) Highlight the image circle, and use calculate real implicit equation to check the equivalence. Do the same translation rules seem to apply to circles? E) Write the general equation for a circle with radius r, which has been translated h units right and k units up. Write in a form similar to part A, one that isn t multiplied out. F) Modify your diagram, replacing the vector coordinates with h for horizontal translation, and k for vertical translation. Use calculate symbolic implicit equation, and show algebraic steps to check the equivalence your equation from part E. ) We ve already seen that changing the radius of a circle stretches the circle horizontally and vertically at the same time. How does this fit together with the dilations we ve already done? A) Take the equation for a unit circle (from 1D), and replace x with a x and y with a y. B) Simplify the equation and multiply both sides by a. What do you notice? Do the dilation rules seem to work the same way for circles? 008 Saltire Software Incorporated
5) Now consider what happens if we dilate a circle vertically and horizontally by different scale factors. Obviously, the result will no longer be a circle. A) Take the equation for the unit circle (from 1D) and replace x with 5 x and y with 3 y. Write and simplify the equation below: B) What transformations are represented by the changed equation in part A? C) Sketch below what the equation from part A should look like if all the transformation rules hold true. This is called an ellipse. In general, an ellipse can be thought of as a stretched out circle, but it also has a number of special qualities that will become apparent when you study conic sections. - 6 - D) Check your work with GX. Solve the equation in part B for y. Remember that when you take the square root of both sides of an equation, you must insert +/-. This, in effect, creates two functions. Graph both of them using the function tool in GX. Write the equation(s) below: 008 Saltire Software Incorporated
6) Now put it all together: A) Write an equation and sketch the graph of an ellipse which represents a horizontal dilation of scale factor, a vertical dilation of scale factor, and a translation up 3 and 7 units to the left. 6-1 -10-8 - - B) Write an equation and sketch the graph of an ellipse, which represents a horizontal dilation of scale factor 7, a vertical dilation of scale factor 3, and a translation down units and units to the right. - - 6 8 10-8 C) Write a general equation for an ellipse, which represents a horizontal dilation of scale factor a, a vertical dilation of scale factor b, and a vertical translation of k units and a horizontal translation of h units. 008 Saltire Software Incorporated