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Math 2 Discriminant Task MM2A4c NAME____________________ Graphing Calculator directions for graphing a parabola: Step 1: Step 2: Step 3: Step 4: Turn on your calculator. Press the Y = button right below the calculator’s screen Type in the function to be graphed Press the GRAPH button Sketch the graph of the following parabolas using the calculator. Don’t try to be exact. Show how the parabola interacts with the x-axis. 1. y = x2 + 2x – 8 2. y = x2 2x +3 3. y = x2 4x + 4 4. y = 2x2 5x 4 5. y = 4x2 + 20x 25 6. x2 + 3x + 5 What do you observe about the graphs?____________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ Now we are going to calculate the discriminant of each function. The discriminant of a function will tell the number of roots and the nature of the roots. A quadratic can have real or imaginary solutions. The discriminant of y = ax2 + bx + c is found using the formula b2 4ac. Fill in the chart using the same equations that were graphed on the front. Quadratic function Discriminant (b2 4ac) Number of x-intercepts 1. y = x2 + 2x – 8 2. y = x2 2x +3 3. y = x2 4x + 4 4. y = 2x2 5x 4 5. y = 4x2 + 20x 25 6. y = x2 + 3x + 5 Based on the results form the table, we can conclude the following: When D > 0, there is/are _______ x-intercept(s). When D = 0, there is/are _______ x-intercept(s). When D <0, there is/are _______ x-intercept(s). When the discriminant is greater than or equal to zero, that the parabola has either one or two real solutions. The x-intercepts are the solutions. When the discriminant is less than zero, then there are still solutions even though the parabola does not intercept the x-axis. The solutions are imaginary numbers. Every time the discriminant > 0, there are 2 x-intercepts and _________ real roots. These real roots are either rational or irrational. To determine if the real roots are rational: the discriminant is a __________ ____________ ___________. To determine if the real roots are irrational: the discriminant is positive but it is NOT a ___________ ___________. Every time the discriminant = 0, there is 1 x-intercept and __________ real root(s). It is a ____________ root. Every time the discriminant < 0, there are 0 x-intercepts and _________ real roots, instead there are _________ imaginary conjugate roots. Can you ever have 1 real root and 1 imaginary root for a quadratic equation?_______________________ Let’s put it all together: Discriminant # of x-intercepts Number of solutions Nature of the roots D > 0 and a perfect square D > 0 not a perfect square D=0 D<0 Complete the chart: # Quadratic Equation 1 f ( x) x 2 5 x 14 2 f ( x) x 2 3x 2 3 f ( x) x 2 6 x 58 4 f ( x) 4 x 2 12 x 9 5 f ( x) x 2 9 6 f ( x) 3x 2 11x 4 7 f ( x) 8 x 2 4 x 5 8 f ( x) 7 x 2 2 x 9 9 f ( x) 25 x 2 20 x 4 10 f ( x) 5 x 2 3x 1 Discriminant # of xintercepts # of solutions Nature of the roots Use the graph to describe the discriminant (hint use: D > 0, D = 0, D < 0), number of x-intercepts and the nature of the roots for the equation graphed. 11. 12. Discriminant: ______________________ Discriminant: ______________________ # of x-intercepts: ___________________ # of x-intercepts: ___________________ Nature of roots:_____________________ Nature of roots:_____________________ 13. 14. Discriminant: ______________________ Discriminant: ______________________ # of x-intercepts: ___________________ # of x-intercepts: ___________________ Nature of roots:_____________________ Nature of roots:_____________________