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10.3 - Circles

10.3 - Circles. Circles – Warm Up. Simplify. 1. 16 2. 49 3. 20 4. 48 5. 72. Find the missing value to complete the square. 6. x 2 – 2 x + 7. x 2 + 4 x + 8. x 2 – 6 x +. Find the missing value to complete the square.

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10.3 - Circles

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  1. 10.3 - Circles

  2. Circles – Warm Up Simplify. 1. 16 2. 49 3. 20 4. 48 5. 72 Find the missing value to complete the square. 6.x2 – 2x + 7.x2 + 4x + 8.x2 – 6x + Find the missing value to complete the square. 6.x2 – 2x + 7.x2 + 4x + 8.x2 – 6x +

  3. Solutions 1. 16 = 4 2. 49 = 7 3. 20 = 4  5 = 2 5 4. 48 = 16  3 = 4 3 5. 72 = 36  2 = 6 2 6. x2 – 2x + ; c = = – = (–1)2 = 1 7. x2 + 4x + ; c = = = 22 = 4 8. x2 – 6x + ; c = = – = (–3)2 = 9 b 2 2 2 2 2 b 2 b 2 4 2 2 2 6 2 b 2 b 2 b 2 b 2 2 2

  4. CIRCLE TERMS Definition: A circle is an infinite number of points a set distance away from a center (x – h)² + (y – k)² = r² r (h, k ) C=(h , k) r

  5. Check: Solve the equation for y and enter both functions into your graphing calculator. (x – 3)2 + (y + 2)2 = 9 (y + 2)2 = 9 – (x – 3)2 y + 2 = ± 9 – (x – 3)2 y = –2 ± 9 – (x – 3)2 Circles Write an equation of a circle with center (3, –2) and radius 3. (x – h)2 + (y – k)2 = r2Use the standard form of the equation of a circle. (x – 3)2 + (y – (–2))2 = 32Substitute 3 for h, –2 for k, and 3 for r. (x – 3)2 + (y + 2)2 = 9 Simplify.

  6. Circles Write an equation for the translation of x2 + y2 = 16 two units right and one unit down. (x – h)2 + (y – k)2 = r2Use the standard form of the equation of a circle. (x – 2)2 + (y – (–1))2 = 16 Substitute 2 for h, –1 for k, and 16 for r2. (x – 2)2 + (y + 1)2 = 16 Simplify. The equation is (x – 2)2 + (y + 1)2 = 16.

  7. WRITE and GRAPH • A) write the equation of the circle in standard form • x² + y² - 4x + 8y + 11 = 0 • Group the x and y terms • x² - 4x + y² + 8y + 11 = 0 • Complete the square for x/y • x² - 4x + 4 + y² + 8y + 16 = -11 + 4 + 16 • (x – 2)² + (y + 4)² = 9 • YAY! Standard Form! • B) GRAPH • Plot Center (2,-4) • Radius = 3

  8. WRITE and GRAPH • A) write the equation of the circle in standard form • 4x² + 4y² + 36y + 5 = 0 • Group the x and y terms • 4x² + 4y² + 36y + 5 = 0 • Complete the square for x/y • 4x² + 4(y² + 9y) = -5 • 4x² + 4(y² + 9y + 81/4) = -5 + 81 • 4x² + 4(y + 9/2)² = 76 • x² + (y + 9/2)² = 19 • YAY! Standard Form! • B) GRAPH • Plot Center (0 , -9/2) • Radius = √19 = 4.5

  9. WRITING EQUATIONSWrite the EQ of a circle that has a center of (-5,7) and passes through (7,3) • Plot your info • Need to find values for h, k, and r • (h , k) = (-5 , 7) • How do we find r? • Use distance formula with C and P. • Plug into formula • (x – h)² + (y – k)² = r² • (x + 5)² + (y – 7)² = (4√10)² • (x + 5)² + (y – 7)² = 160 C = (-5,7) P = (7,3) radius

  10. Let’s Try OneWrite the EQ of a circle that has endpoints of the diameter at (-4,2) and passes through (4,-6) • Plot your info • Need to find values for h, k, and r • How do we find (h,k)? • Use midpoint formula • (h , k) = (0 , -2) • How do we find r? • Use dist form with C and B. • Plug into formula • (x – h)² + (y – k)² = r² • (x)² + (y + 2)² = 32 A = (-4,2) radius B = (4,-6) Hint: Where is the center? How do you find it?

  11. Suppose the equation of a circle is (x – 5)² + (y + 2)² = 9 • Write the equation of the new circle given that: • A) The center of the circle moved up 4 spots and left 5: • (x – 0)² + (y – 2)² = 9 Center moved from (5,-2)  (0,2) • B) The center of the circle moved down 3 spots and right 6: • (x – 11)² + (y + 5)² = 9 Center moved from (5,-2)  (11,-5)

  12. Let‘s Try One Find the center and radius of the circle with equation (x + 4)2 + (y – 2)2 = 36. (x – h)2 + (y – k)2 = r 2Use the standard form. (x + 4)2 + (y – 2)2 = 36 Write the equation. (x – (–4))2 + (y – 2)2 = 62Rewrite the equation in standard form. h = –4k = 2r = 6 Find h, k, and r. The center of the circle is (–4, 2). The radius is 6.

  13. Draw the center (3, –1) and radius 2. Draw a smooth curve. Let’s Try One Graph (x – 3)2 + (y + 1)2 = 4. (x – h)2 + (y – k)2 = r 2Find the center and radius of the circle. (x – 3)2 + (y – (–1))2 = 4 h = 3k = –1r 2 = 4, or r = 2

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