Friday, October 29, 2021

Geometry Problems of the Day (Geometry Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, June 2012

Part IV: A correct answer will receive 6 credits. Partial credit is possible.

38. On the set of axes below, solve the system of equations graphically and state the coordinates of all points in the solution.

y = (x - 2)2 - 3
2y + 16 = 4x


I'm actually surprised that there wasn't a second part to this question.

There is a parabola in vertex form. And a linear equation not quite in either Standard nor Slope-Intercept form. They could have 2, 1 or 0 solutions. Always label the solutions according to your graph, even if your graph is (or you think may be) incorrect.

The quadratic equation can be put into your calculator as is and you can find the points to plot. Or you can plot the Vertex at (2,-3) and start graphing from there. If you realize that the y values will increase by +1, +3, +5, etc., then you know that the next few points are at (3,-2), (4,1), (5,6) and (6,13) which is off the top of the graph. Since the parabola has symmetry, there are also points at (1,-2), (0,1), (-1,6), and then off the top of the graph.

Rewrite the linear equation so you can enter it into your calculator:

2y + 16 = 4x
2y = 4x - 16
y = 2x - 8

You have a slope of 2 and a y-intercept of -8. You can plot points at (0,-8), (1,-6), (2,-4), etc.

Your graph should look like the one below. Remember to label which line is which, using the original equations, and label the solutions.

There is only one solution at (3,-2). The line is tangent to the parabola, but depending upon how well you sketch the parabola, it might look like it crosses it twice. It doesn't. If it didn't, you wouldn't be able to read the solutions from a graph. You would need to do it algebraically. If you need to convince yourself, your calculator will show you that there is only one intersection point, as will algebra.

End of Exam.

More to come. Comments and questions welcome.

More Regents problems.

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