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4.4 Transforming Circles Specific Curriculum Outcomes E13 analyze and translate between symbolic, graphic, and written representation
of circles and ellipses E14 translate between different forms of equation of circles and ellipses E11 write proofs using various axiomatic systems and assess the validity of
deductive arguments E3 write the equations of circles and ellipses in transformational form and as
mapping rules to visualize and sketch graphs E16 demonstrate an understanding of the transformational relationship between
a circle and an ellipse E4 apply the properties of circles E15 solve problems involving the equations and characteristics of circles and
ellipses Assumed Prior Knowledge q understanding of transformations for other graphs q ability to complete the square, and balance an equation q understanding inequalities q rewriting an equation to isolate a variable q using a graphing calculator to graph equations q familiarity with appropriate window setting on a graphing calculator
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Investigation 6 The Equation of a Circle
Purpose: Determine the equation for a circle using coordinate geometry. Procedure: A&B: A circle with radius 10 and centre (0, 0) on a coordinate grid is show. A point is given on the circle with integral coordinates, A(6, 8). Draw the radius to point A.
10
5
-5
-10
-10 10B
A
C
C: Draw a vertical line segment AB to the x-axis 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.
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F: Repeat steps B-D for point Q(-8, -6). G: Write an equation for the coordinates of ALL points satisfying a circle with
the centre at the origin and a radius of 10. H: What is the radius of x2 + y2 = 256? Investigation Questions: 1. x2 + y2 = 49
What is the radius? Sketch the circle.
Does the point (4, 5) lie on the circle? If not, where does it lie? Justify.
Describe the circle with equation x2 + y2 = 49 in terms of a transformation of the unit circle, x2 + y2 = 1.
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
4
2. x2 + y2 = 64
What is the radius? Sketch the circle. Do the points Q(-4, 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) à
The Equation of a Circle
x2 + y2 = r2 centre (0, 0); r = radius
(x, y) à (rx, ry) horizontal stretch r units vertical stretch r units
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
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Investigation 7 Transformations of Circles
Purpose: Examine circles under various transformations: Procedure: A: Write an equation for the centre ring on page 254.
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 – 10) 2 + y2 = 16. 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.
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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 – 12) 2 + (y + 7) 2 = 2
Transformations of Circles
Centre (h, k), radius = r Standard Form: (x – h) 2 + (y – k) 2 = r2
Transformations Form: ( ) ( ) 111 22
=
−+
− ky
rhx
r
(x, y) à (rx + h, ry + k) h.s. = r v.s. = r h.t. = h v.t. = k
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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 x2 + y2 = 1.
-4 -2 2 4
-4
-2
2
4
B: Graph (½x) 2 + y2 = 1 by stretching the original equation horizontally.
-4 -2 2 4
-4
-2
2
4
C: What are the x-intercepts of the graph in step B?
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D: Graph x2 + (31 y) 2 = 1
by stretching the original graph vertically.
-4 -2 2 4
-4
-2
2
4
E: What are the y-intercepts of the graph in step D? F: Graph (½x)2 + (
31 y) 2 = 1.
-4 -2 2 4
-4
-2
2
4
How does this compare to the graph of x2 + y2 = 1? Express this new equation as a mapping of the equation x2 + y2 = 1.
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G: Use your results from step F to graph (31 x) 2 + (½y) 2 = 1.
-4 -2 2 4
-4
-2
2
4
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:
-10 -5 5 10
6
4
2
-2
-4
-6
minor axis = 10
major axis = 20
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.
(½x) 2 + y2 = 1 x2 + (31 y) 2 = 1 (½x) 2+ (
31 y) 2 =1 (
31 x) 2 + (½y) 2 =1
Major Minor
J: How can you tell, by inspecting the equation, whether the major axis is
vertical or horizontal?
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K: Graph 141
51 22
=
+
yx .
State the intercepts and the lengths of the major and minor axes.
-4 -2 2 4
-4
-2
2
4
L: Graph 171
72 22
=
+
yx .
State the intercepts and the lengths of the major and minor axes.
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
11
M: Graph 141
51 22
=
+
yx .
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
Translate this ellipse to produce the graph of ( ) ( ) 1341
251 22
=
−+
− yx .
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
( ) ( ) 111 22
=
−+
− ky
bhx
a
centre (h, k) length of horizontal axis = 2a length of vertical axis = 2b
(x, y) à (ax + h, by + k)
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