4.4 Transforming Circles


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1 Specific Curriculum Outcomes. Transforming Circles E13 E1 E11 E3 E1 E E15 analyze and translate between symbolic, graphic, and written representation of circles and ellipses translate between different forms of equation of circles and ellipses write proofs using various axiomatic systems and assess the validity of deductive arguments write the equations of circles and ellipses in transformational form and as mapping rules to visualize and sketch graphs demonstrate an understanding of the transformational relationship between a circle and an ellipse apply the properties of circles solve problems involving the equations and characteristics of circles and ellipses Assumed Prior Knowledge understanding of transformations for other graphs ability to complete the square, and balance an equation understanding inequalities rewriting an equation to isolate a variable using a graphing calculator to graph equations familiarity with appropriate window setting on a graphing calculator 1
2 Investigation The Equation of a Circle Purpose: Determine the equation for a circle using coordinate geometry. Procedure: A&B: A circle with radius and centre (0, 0) on a coordinate grid is show. A point is given on the circle with integral coordinates, A(, 8). Draw the radius to point A. A 5  C B 5  C: Draw a vertical line segment AB to the xaxis and a horizontal segment BC, forming ABC. What is the measure of CBA? Determine the length of each side of the triangle. D: Which side of the triangle is the radius? Label this side r. E: Write a formula relating the three sides of the triangle.
3 F: Repeat steps BD for point Q(8, ). G: Write an equation for the coordinates of ALL points satisfying a circle with the centre at the origin and a radius of. H: What is the radius of x + y = 5? Investigation Questions: 1. x + y = 9 What is the radius? Sketch the circle. Does the point (, 5) lie on the circle? If not, where does it lie? Justify Describe the circle with equation x + y = 9 in terms of a transformation of the unit circle, x + y = 1. 3
4 . x + y = What is the radius? Sketch the circle. Do the points Q(, 7) and R(0, 8) lie on the circle? If not, where do they lie? Justify Write a mapping rule to express the circle as a transformation of the unit circle. (x, y) The Equation of a Circle x + y = r centre (0, 0); r = radius (x, y) (rx, ry) horizontal stretch r units vertical stretch r units
5 Investigation 7 Transformations of Circles Purpose: Examine circles under various transformations: Procedure: A: Write an equation for the centre ring on page 5. Express the ring as a mapping rule of the unit circle. B: How many units and in what direction has ring A shifted? What are the coordinates of the centre of ring A? C: The equation for ring A is (x ) + y = 1. Write the equation for ring B. Express ring A as a mapping of the unit circle. Repeat for ring B. D: What is the radius of ring C? How many units and in what direction has ring C shifted? What are the coordinates of the centre of ring C? Write the equation for ring C. Express ring C as a mapping of the unit circle. E: Write the equations for Rings D, E, and F. 5
6 F: Write a general equation for a circle with centre (h, k). Write a general mapping rule for circles as mappings of the unit circle. G: What is the centre and the radius of (x 1) + (y + 7) = Transformations of Circles Centre (h, k), radius = r Standard Form: (x h) + (y k) = r 1 1 r r (x, y) (rx + h, ry + k) h.s. = r v.s. = r h.t. = h v.t. = k Transformations Form: ( x h) + ( y k) = 1
7 Investigation 8 The Equation of an Ellipse Purpose: Explore the equation of an ellipse as a transformation of a circle. Procedure: A: Graph the equation x + y = B: Graph (½x) + y = 1 by stretching the original equation horizontally C: What are the xintercepts of the graph in step B? 7
8 D: Graph x + ( 3 1 y) = 1 by stretching the original graph vertically E: What are the yintercepts of the graph in step D? F: Graph (½x) + ( 3 1 y) = How does this compare to the graph of x + y = 1? Express this new equation as a mapping of the equation x + y = 1. 8
9 G: Use your results from step F to graph ( 3 1 x) + (½y) = H: How is an ellipse the same as a circle? How is it different? ellipse: the curve obtained by expanding or compressing a circle in perpendicular directions I: minor axis = major axis = C major axis: the longest chord in a ellipse minor axis: the chord through the centre perpendicular to the major axis  State the length of the major and minor axes for each ellipse from steps B to G. Major Minor (½x) + y = 1 x + ( 3 1 y) = 1 (½x) + ( 3 1 y) =1 ( 3 1 x) + (½y) =1 J: How can you tell, by inspecting the equation, whether the major axis is vertical or horizontal? 9
10 1 1 K: Graph x + y = 1. 5 State the intercepts and the lengths of the major and minor axes L: Graph x + y = State the intercepts and the lengths of the major and minor axes
11 1 1 M: Graph x + y = x 5. Translate this ellipse to produce the graph of ( ) + ( y 3) = 1 What is the centre of the new ellipse? Express this ellipse as the image of a mapping of the unit circle. The Equation of the Ellipse 1 1 ( x h) + ( y k ) = 1 a b centre (h, k) length of horizontal axis = a length of vertical axis = b (x, y) (ax + h, by + k) 11
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