2. From General Form: y = ax 2 + bx + c # of x-intercepts determined by the, D =

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1 Alg2H 5-3 Using the Discriminant, x-intercepts, and the Quadratic Formula WK#6 Lesson / Homework --Complete without calculator Read p.181-p.186. Textbook required for reference as well as to check some answers DETERMINING THE NUMBER OF X-INTERCEPTS (ROOTS) 1. From Vertex Form: y k = a(x h) 2 # of x-intercepts determined by the location of vertex and sign of a. a) If k > 0 and a < 0, the vertex is the x-axis and the graph goes. If k < 0 and a > 0, the vertex is the x-axis and the graph goes. Both result in x-intercept(s) b) If k = 0 and a < 0, the vertex is the x-axis and the graph goes. If k = 0 and a > 0, the vertex is the x-axis and the graph goes. Both result in x-intercept(s) c) If k > 0 and a > 0, the vertex is the x-axis and the graph goes. If k < 0 and a < 0, the vertex is the x-axis and the graph goes. Both result in x-intercept(s) 2. From General Form: y = ax 2 + bx + c # of x-intercepts determined by the, D = x-intercepts are points where y = 0. Therefore they are the same as the roots of the equation: 0 = ax 2 + bx + c which can be solved by using the Quadratic formula: x =. Which part of Quad formula determines the number and type of solutions? If D > 0, there are real x-intercepts (or roots). If D is a perfect square, then they are. If D is not a perfect square, then they are. If D = 0, there is real x-intercept (or root) which is and the same as the of the function. If D < 0, there are real x-intercepts or roots. (The roots are.) 1

2 SAMPLE PROBLEMS: (Complete without calculator) Using General Form: 1) y = -5x 2 + 5x +30 WK#6 a) Quickly determine x-coordinate of vertex and use it to determine the y-coordinate of the vertex. Remember: Quick way to determine Vertex: V(h,k) = V((, f( )) How could you decide the # of x-intercepts from the information so far? b) Use discriminant to determine the # of x-intercepts and if their value is rational or irrational: c) Find the x-intercepts by factoring, if possible, or use quadratic formula if not possible. State answers in exact simplified form. (Note: If irrational, radical form is the exact form.) (How can the discriminant be used to decide if a quadratic is factorable? ) d) Sketch the graph of the function. Label coordinates of vertex, axis of symmetry, x-intercepts, y-intercept and symmetric point to y-intercept. 2

3 2. y = 2x 2 8x + 2 (Complete without calculator) WK#6 a) Quickly determine x-coordinate of vertex and use it to determine the y-coordinate of the vertex. How could you decide the # of x-intercepts from the information so far? b) Use discriminant to determine the # of x-intercepts and if their value is rational or irrational: c) Find the x-intercepts by factoring, if possible, or use quadratic formula if not possible. State answers in exact simplified form. (Note: If irrational, radical form is the exact form.) (How can the discriminant be used to decide if a quadratic is factorable? ) d) Sketch the graph of the function. Label coordinates of vertex, axis of symmetry, x-intercepts, y-intercept and symmetric point to y-intercept. 3

4 3. y = 16x x 25 (Complete without a calculator) WK#6 a) Quickly determine x-coordinate of vertex and use it to determine the y-coordinate of the vertex. How could you decide the # of x-intercepts from the information so far? b) Use discriminant to determine the # of x-intercepts and if their value is rational or irrational: c) Find the x-intercepts by factoring, if possible, or use quadratic formula if not possible. State answers in exact simplified form. (Note: If irrational, radical form is the exact form.) (How can the discriminant be used to decide if a quadratic is factorable? ) d) Sketch the graph of the function. Label coordinates of vertex, axis of symmetry, x-intercepts, y-intercept and symmetric point to y-intercept. 4

5 Using Vertex form, Find the x-intercepts in exact simplified form and the coordinates of the vertex: (Complete without calculator) (Refer to Wk#4, problem #6) WK#6 4. y + 32 = 25(x 3)2 Using Intercept Form, sketch a graph of the function by Finding the x-intercepts, the coordinates of the vertex, y-intercept and the symmetric pt: (Complete without calculator) (Refer to Wk#4, problem #5) 5. y = -7(x 3)( x + 4) Problems 6-7, follow these directions: a) Transform each equation to vertex form by completing the square. b) State the coordinates of the vertex and check by using the quick method. 6) y = 2x2 7x ) y = -3x2 4x + 5 5

6 Problems 8-11, follow these directions: WK#6 a) From the general form, quickly determine the x-coordinate of the vertex and use it to determine the y-coordinate of the vertex. b) Find the value of the discriminant to determine the number of x-intercepts and if their value is rational or irrational. c) If they exist, find the x-intercepts in exact simplified (radical) form. Use factoring, if possible. Otherwise use the quadratic formula. d) Sketch the graph of each function. Label coordinates of vertex, axis of symmetry, x and y-intercepts, if they exist and symmetric point to y-intercept. 8) y= -4x2 8x + 12 (Answer similar to Exercise 5-3: #35 but y values multiplied by 4) 9) y = -4x2 + 4x 1 (Check answer with Exercise 5-3: #39) 10) y = x2 + 2x ) y = x2 + 2x 5 (Check answer with Exercise 5-3: #41) (Check answer with Exercise 5-3: #43) 6