NAME _______________________________ AIM: How are the x-­‐intercepts helpful in our Quadratic Functions? Fatima and Frank have been exploring different representations of quadratic equations. Fatima says, “I know that values of a, b and c in the equation determine what the graph will look like. For example, I can determine the line of symmetry, vertex and y-­‐intercept of the parabola just by looking at its equation.” Frank questions, “I wonder if it’s possible to easily determine the x-­‐intercepts.” Fatima responds, “It might be helpful for us to analyze the table and graph of several quadratic equations to see if we can make sense of their x-­‐intercepts.” LAUNCH: Match the quadratic equations to the corresponding graphs below and complete the table with the appropriate information. y = x2 – 6x + 8 ______________________ Equation Number of x – intercepts Coordinates of x-­‐
intercept(s) y = x2 + 4x + 3 ______________________ y = -­‐x2 – 2x + 1 y = -­‐x2 – 2x -­‐ 3 ______________________ ______________________ What observation can be made about the coordinates of any x-­‐intercept? Using that information, determine the x-­‐intercept(s) of each quadratic equation given the three tables below. Circle the points. y = x2 – 6x + 8 y = x2 + 4x + 3 y = -­‐x2 – 2x + 1 y = -­‐x2 – 2x – 3 Fatima and Frank (and hopefully you too) have started to notice that the x-­‐intercepts of these quadratic equations are always occurring where the y-­‐value is zero. Since the y-­‐value is always zero at the x-­‐intercept, we can start calling these zeros instead of “x-­‐intercepts.” Practice a few. a) y = x2 + 4x + 4 b) y = x2 + 8x + 12 c) y = -­‐ x2 – 6x – 5 d) y = x2 -­‐ 6x -­‐ 1 What did we find out about the table and the graph in exercise d? What do you think about that? Fatima is curious about equation d, y = x2 -­‐ 6x + 8. Frank recalls that there was a third way to determine the line of symmetry and vertex of a quadratic equation – algebraically. He wonders if that might also work for finding the x-­‐intercept. Use Frank’s idea to set up an equation that we could solve to determine the x-­‐intercept(s) for the quadratic function y = x2 -­‐ 6x -­‐ 1. (What equation would we set up?) Try to solve that equation. What happens?