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Cubic curve sketching

y. y. x. x. Cubic curve sketching. General form. General shape:. a < 0. a > 0. Characteristics: usually 2 humps, may or may not be asymmetrical Important point to note: If there are 3 factors, then there will be 3 x-intercepts

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Cubic curve sketching

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  1. y y x x Cubic curve sketching General form General shape: a < 0 a > 0 Characteristics: usually 2 humps, may or may not be asymmetrical Important point to note: If there are 3 factors, then there will be 3 x-intercepts Just like in quadratic curve, when there are 2 factors, there are 2 x-intercepts.

  2. y x Getting the factors from the curve: quadratic y=f(x) -1 3 In order to get the x-intercepts, we are actually solving for x  x = -1 and x = 3 means (x +1)(x – 3) = 0 We can get the equation of the curve from its x-intercepts!!

  3. y x Getting the factors from the curve: cubic 4 -3 -1 In order to get the x-intercepts, we are actually solving for x  x = -1, -3 and x = 4 means (x +1)(x + 3)(x – 4) = 0 Again, we can get the equation of the curve from its x-intercepts!!

  4. Graphing Cubic Polynomials (Optional) • Identify various types of cubic curves (Refer to Excel Applet for Cubic Curves) • Sketch simple cubic curves • Form equation of cubic polynomial from sketch

  5. Graphing Cubic Polynomials The real roots of the polynomial equation P(x) = 0 are given by the values of the intercepts of the function y = P(x) with the x-axis. Nature of roots: 3 real and distinct  x = x1, x = x2 and x= x3 are the solutions. 2 real and equal and 1 real and distinct 1 real and 2 complex roots

  6. 10 5 0 -4 -2 0 2 4 -5 -10 Definition of Cubic Function A cubic function is a polynomial function of the form ax3 + bx2 + cx + d, where a, b, c and d are constants and a cannot be 0. Example 1: y = x3

  7. 10 5 0 -4 -2 0 2 4 -5 -10 Use the excel applet to investigate Example 2: y = x3 – 5x2 + 2x + 8

  8. 10 5 0 -2 -1 0 1 2 3 4 -5 -10 Use the excel applet to investigate Example 3: y = x3 – x2 - x +1

  9. Graphing Cubic Polynomials How to graph a cubic function? Example : y = x3 – 2x2 –x + 2 (Note: a > 0) Step 1: Check if y can be factorise into 3 linear factors y = (x + 1)(x -2)(x -1) (Sometimes, you may get 1 linear factor and a quadratic factor that cannot be factorised When this happens – use the quadratic formula to solve for x. If it cannot be solved, then there will be 2 complex roots and 1 real root) Step 2: Set y = 0, x = -1, x = 2, x = 1

  10. y = x3 – 2x2 –x + 2 Graphing Cubic Polynomials Step 3: Finding the y-intercept. When x = 0, y = 2.  (0, 2)

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