Sol 1 The x-coordinate is given by , and then the y-coordinate is given by .
Sol 2 The x-coordinate of the vertex is given by ; so (since ) the maximum value is given by .
Sol 3 Since 5 and 1 are the x-intercepts of the graph, for some number . Since the graph of is symmetric around the vertical line passing through its vertex, the x-coordinate of the vertex is 3 (the average of 5 and 1). Therefore the minimum value of is , so and or .
Sol 4 Since the vertex is at , the parabola has an equation of the form or . Since the parabola passes through the point , substituting and gives , so and , so .
Sol 5 Substituting in and replacing by gives the quadratic function . Then the t-coordinate of the vertex of the graph of is given by , so the minimum value of is given by Therefore the minimum value of is also 9, and this value is attained when so .
Sol 6 Since the roles of and are interchanged, the y-coordinate of the vertex is given by and the x-coordinate is given by .
Sol 7 Since is the minimum value of , the graph of has an equation of the form or with . Since , substituting and gives and therefore . Therefore , so or .
Sol 8 Since the line is non-vertical and passes through the point , it has an equation of the form or . Since the line intersects the parabola exactly once, the equation or has exactly one solution. Therefore the discriminant is equal to 0, so ,so the line has equation or .
Sol 9 (If the horizontal line intersects the graph of only once, then will correspond to the maximum value or minimum value of .) Setting gives , so and therefore . This equation will have exactly one solution if the discriminant is zero, so . Therefore has a maximum value of 3, since has no solution.
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