1. What is the maximum number of intersections for 2 different parabola? 2. When you have multiple inequalities, what area represents the answer?

Let's address each question step by step. ### 1. Maximum Number of Intersections for 2 Different Parabolas To find the maximum number of intersections between two different parabolas, we can analyze the equations of the parabolas. Assume the equations of the two parabolas are given by: \[ y = a_1x^2 + b_1x + c_1 \] \[ y = a_2x^2 + b_2x + c_2 \] To find the points of intersection, we set the two equations equal to each other: \[ a_1x^2 + b_1x + c_1 = a_2x^2 + b_2x + c_2 \] Rearranging this gives us a quadratic equation: \[ (a_1 - a_2)x^2 + (b_1 - b_2)x + (c_1 - c_2) = 0 \] This is a quadratic equation in the standard form \(Ax^2 + Bx + C = 0\), where: - \(A = a_1 - a_2\) - \(B = b_1 - b_2\) - \(C = c_1 - c_2\) A quadratic equation can have at most 2 real solutions (intersections) depending on the discriminant \(D\): \[ D = B^2 - 4AC \] - If \(D > 0\), there are 2 distinct real solutions (2 intersections). - If \(D = 0\), there is 1 real solution (1 intersection). - If \(D < 0\), there are no real solutions (0 intersections). Thus, the maximum number of intersections for 2 different parabolas is **2**. ### 2. Area Representing the Answer for Multiple Inequalities When dealing with multiple inequalities, the solution set is typically represented as a region in the coordinate plane. The area that represents the answer is the intersection of the regions defined by each inequality. For example, consider the following inequalities: 1. \(y < f_1(x)\) 2. \(y > f_2(x)\) The area that satisfies both inequalities is the region where the graph of \(f_1(x)\) is above \(f_2(x)\) and below the line defined by \(f_1(x)\). To find this area: - Graph each inequality on the same coordinate plane. - Identify the regions that satisfy each inequality. - The solution set is the overlapping area where all inequalities are satisfied. In summary, the area representing the answer to multiple inequalities is the intersection of the regions defined by each inequality in the coordinate plane.