q=0 is one value for q. b) The greater values for q are q>20. c) There are infinitely many points of intersection, and their coordinates can be expressed as (x,0), where x can be any real number. Let's analyze the quadratic function f(x)=4x^2 −qx 25 and consider the condition that the vertex is on the x-axis. The x-coordinate of the vertex for a quadratic function ax^2 bx c is given by x=− b/2a In our case, a=4, b=−q. (a) To have the vertex on the x-axis, the x-coordinate of the vertex must be 0. Therefore, we set − b/2a =0: (−q)/2(4)​ =0 Solving this equation gives us: q/8 =0 Therefore, q=0 is one value for q. Now, let's consider the case where there are two real solutions for the quadratic equation, meaning the parabola intersects the x-axis at two distinct points. For this to happen, the discriminant (b^2 −4ac) must be greater than 0. (b) The discriminant is given by b^2 −4ac: (−q)^2 −4(4)(25)>0 Solving this inequality gives us: q^2 −400>0 q^2 >400 q>20 or q<−20 So, the greater values for q are q>20. (c) For the greater value of q, solve f(x)=0: 4x^2 −qx 25=0 Now, use the quadratic formula x= −b± √b^2 −4ac/2a ​to find the roots. In this case: x= q± √q^2 −4(4)(25)/ 2(4) For q>20, the quadratic equation will have two distinct real solutions. You can find the coordinates of the points of intersection of the two graphs by setting the two quadratic functions equal to each other and solving for x: 4x^2 −qx 25=4x^2 −qx 25 This simplifies to: 0=0 Since this is always true, it means that the two graphs coincide and intersect at every point. Therefore, any x-value will satisfy this equation. So, there are infinitely many points of intersection, and their coordinates can be expressed as (x,0), where x can be any real number. Question 2. Consider two different quadratic functions of the form f(x)=4x^2-qx 25. The graph of each function has its vertex on the x-axis. Sketch the two graphs on the same diagra