Sketching the graph of a polynomial function
- 2. Solve for x and y intercepts Solving for the x and y intercepts is an important role step in graphing a polynomial function. These intercepts are used to determine the points the graphs intercepts or touches the x-axis and the y-axis. To find the x-intercept of a polynomial function: a. Factor the polynomial completely b. Let y be equal to zero c. Equate each factor to zero and solve for x To find the y-intercept: a. Let x be equal to zero and simplify
- 3. End of Behavior x4 – 5x2 + 4 Condition 1: An > 0 n is an odd number Q1 and Q3 Condition 3: An < 0 n is an odd number Q2 and Q4 Condition 2: An > 0 n is an even number Q1 and Q2 Condition 4: An < 0 n is an even number Q3 and Q4
- 4. No. of turning points The number of turning points is at most (n – 1) Multiplicity of roots If r is a zero of odd multiplicity, the graph of P(x) crosses the x – axis at r. If r is a zero of even multiplicity, the graph of P(x) is tangent to the x axis at r.
- 5. Since the roots are 1, -1, 2, and -2 the x-intercepts are (1, 0), (-1, 0), (2, 0), (-2,0). Make a table of values and assign values of x between these roots and also values of x higher than the bigger root (2) and lower than smaller root(-2). Then solve. x -2 -1 0 1 2 3 -3 0.5 -0.5 1.5 -1.5 y 0 0 4 0 0 40 40 2.81 2.81 -2.19 -2.19
- 6. Solve for y-intercept: y = x4 – 5x2 + 4 y = (0)4 – 5(0)2 + 4 y = 0 – 5(0) + 4 y = 0 – 0 + 4 y = 0 So, the y intercept is (0,4).
- 7. If x = 3 y = x4 – 5x2 + 4 = (3)4 – 5(3)2 + 4 = 81 – 5(9) + 4 = 81 – 45 + 4 = 36 + 4 = 40 If x = 0. 5 y = x4 – 5x2 + 4 = (0.5)4 – 5(0.5)2 + 4 = 0.0625 – 5(0.25) + 4 = 0.0625 – 1.25 + 4 = -1.1875 + 4 = 2.81 If x = -3 y = x4 – 5x2 + 4 = (-3)4 – 5(-3)2 + 4 = (81) – 5(9) + 4 = 81 – 45 + 4 = 40 If x = -0.5 y = x4 – 5x2 + 4 = (-0.5)4 – 5(0.5)2 + 4 = 0.0625 – 5(-0.25) + 4 = 0.0625 – 1.25 + 4 = 2.81
- 8. If x = 1.5 y = x4 – 5x2 + 4 = (1.5)4 – 5(1.5)2 + 4 = 5.0625 – 5(2.25) + 4 = 5.0625 – 11.25 + 4 = -6.1875 + 4 = -2.19 If x = -1.5 y = x4 – 5x2 + 4 = (-1.5)4 – 5(-1.5)2 + 4 = 5.0625 – 5(2.25) + 4 = 5.0625 – 11.25 + 4 = -6.1875 + 4 = -2.19 End of the behavior = Q1 and Q2 Leading term = x4 An > 0 ; (1 > 0) n is an even number (4) No. of turning points: (n – 1) = (n – 1) = 4 – 1 = 3