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M3U4D11 Warm Up A rocket is launched into the air so that its height, in feet, after t seconds is modeled by the equation: h(t) = -16t2+75t+50 When and what is the maximum height of the rocket and when will it hit the ground? Max height 137.891 feet at 2.344 sec. and it will hit the ground at 5.279 sec. Homework Check: ON DOCUMENT CAMERA REMEMBER: Standard Form y = ax2 + bx + c and Vertex Form y = a(x-h)2 + k QUIZ !!! M3U4D10b Graphing Polynomial Functions OBJ: Use the process of completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graphs. M3U4D11 Systems of NonLinear Equations OBJ: Solve polynomial equations and systems of polynomial equations approximately by using technology to graph the functions they define. (A-REI.11) Systems of Nonlinear Equations and Their Solutions A system of two nonlinear equations in two variables contains at least one equation that cannot be expressed in the form Ax + By = C. Here are two examples: x2 = 2y + 10 3x – y = 9 y = x2 + 3 x2 + y2 = 9 A solution to a nonlinear system in two variables is an ordered pair of real numbers that satisfies all equations in the system. The solution set to the system is the set of all such ordered pairs. Example 1 – Substitution: Two-Solution Case • Solve the system of equations. • 3x2 + 4x – y = 7 Equation 1 • 2x – y = –1 Equation 2 • Solution: • Begin by solving for y in Equation 2 to obtain y = 2x + 1. • Next, substitute this expression for y into Equation 1 and solve for x. • • 3x2 + 4x – (2x + 1) = 7 Substitute 2x + 1 for y in Equation 1. 3x2 + 2x – 1 = 7 Simplify. Example 1 – Solution • 3x2 + 2x – 8 = 0 • (3x – 4)(x + 2) = 0 cont’d Write in general form. Factor. Solve for x. • Back-substituting these values of x to solve for the • corresponding values of y produces the solutions • and Graphical Approach to Finding Solutions • A system of two equations in two unknowns can have exactly one solution, more than one solution, or no solution. • By using a graphical method, you can gain insight about the number of solutions and the location(s) of the solution(s) of a system of equations by graphing each of the equations in the same coordinate plane. • The solutions of the system correspond to the points of intersection of the graphs. Graphical Approach to Finding Solutions • For instance, the two equations in Figure 6.1 graph as two lines with a single point of intersection; the two equations in Figure 6.2 graph as a parabola and a line with two points of intersection; and the two equations in Figure 6.3 graph as a line and a parabola that have no points of intersection. One intersection point Figure 6.1 Two intersection points Figure 6.2 No intersection points Figure 6.3 M3U4D11 Systems of Nonlinear Equations Notes p. 35 COMPLETE TOGETHER Classwork 2 M3U4D11 Systems of Nonlinear Equations Notes #1-10 P. 36-37 Homework Finish CW