On the graph, you see this challenge’s two base functions, the straight line y=x+6 and the parabola y=x^2+6x+9. You can see the intersection of these two functions and pick the correct solution from the rest. But we want you solve the rest of the questions by using your logic, deduction, and spatial reasoning skills.

By now, you should have some notes or graphs that describe how the vertex of the base parabola changes as the coefficients of the parabola change. In other words, if you see a change in the parabola’s coefficients, you should able to mentally plot the new parabola on the graph. And we assume that your math skills are already strong enough to imagine straight lines on a graph.

As you are mentally plotting the new straight line or parabola (or both), determine approximate values for the intersections. Then go to the list of answers and find a solution with those approximate values. Drag it over, and go to the next answer.

You will likely rely a lot on the decimal approximations, but sometimes the value of the radical in the exact solution may offer clues to the right solution. Use whatever logic, deduction, and spatial reasoning to get the right answers.

You will build more skills if you leave the “Check Answer” button alone until you are certain you have answered all seven problems correctly.

And if your skills for logic, deduction, and spatial reasoning are not strong enough, you can build these skills by mathematically solving this challenge and seeing how the numbers create their own patterns. Return to this website in a few months to test your logic skills.