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Tangents and Normals

Tangents and Normals. A ‘sketch’ requires: General shape Interception with axes Turning points A RULER FOR AXES!. Sketch the graph of y = x 2 – 4x – 12. 8. 6. 4. General shape. 2. 6. 8. 4. -2. 2. -4. (0, -12). x = 0. y-axis. -2. (x-6)(x+2), x=6, x=-2. factorise. y = 0.

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Tangents and Normals

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  1. Tangents and Normals

  2. A ‘sketch’ requires: • General shape • Interception with axes • Turning points • A RULER FOR AXES! Sketch the graph of y = x2 – 4x – 12 8 6 4 General shape 2 6 8 4 -2 2 -4 (0, -12) x = 0 y-axis -2 (x-6)(x+2), x=6, x=-2 factorise y = 0 x-axis -4 -6 Minimum (2, -16) -8 Find the gradient of the graph when x = -2 and when x = 4. -10 -12 -14

  3. To find a gradient of a curve, find y = x2 – 4x – 12 8 = 6 - 4 2x 4 2 When x = -2 6 8 4 -2 2 -4 = 2(-2) – 4 = -8 -2 -4 When x = 4 -6 -8 = 2(4) – 4 = 4 -10 -12 -14

  4. What is the gradient of the NORMAL when x = -2 and when x = 4? 8 = TANGENT -8 When x = -2 6 NORMAL 4 2 = TANGENT 4 When x = 4 6 8 4 -2 2 -4 -2 NORMAL -4 -6 -8 -10 -12 -14

  5. The curve C has equation , x 0. • The point P on C has x-coordinate 1. • Find the equation of the tangent to C at P. • This tangent meets the x-axis at k. Find the coordinate of k. • When x = 1…

  6. (a) x1 = 1, y1 = 8, m = 3 • y – y1 = m(x – x1) • y – 8 = 3(x – 1) • y – 8 = 3x – 3 • y = 3x + 5 • y = 0 • At k we know… • (b) y = 3x + 5 • y = 3x + 5 • 0 = 3x + 5 • -5 = 3x • -5/3= x • k (-5/3, 0)

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