# Coordinate Geometry

## Geometry:  Coordinate Geometry

Read the following SAT test question and then select the correct answer.

Use the same method for all the math questions on the SAT.  First, read the question carefully to avoid making mistakes.  Identify the bottom line and assess your options for reaching it.  Next, choose an efficient method to attack the problem.  Finally, loop back to make sure that your answer addresses the bottom line.  Many problems have multiple steps.

If the graph of the function f in the xy-plane contains the points (0, -9), (1, -4), and (3, 0), which of the following CANNOT be true?

Bottom Line:  You are looking for something false.

Assess your Options:  You could try drawing an xy-plane and graphing the points to help you visualize the question, but your graph may be inaccurate without graph paper.  Instead, try to find the relationship between the three points.

Attack the problem:  To find the relationship between these points, you will need to find the slope of the line between each point.  The formula for slope is:

Then check the slope of the line between (1, -4) and (3, 0):

$\frac{0--4}{3-1}= \frac{0+4}{2}=2$

The function in this problem has a very steep slope between the first two points, but becomes less steep between the second two.  This is a “which of the following” question, so start with answer (E) as you work through your answer choices.

(A) The graph of f has a maximum value.
(B) y ≤ 0 for all points (x, y) on the graph of f.
(C) The graph of f is symmetric with respect to a line.
(D) The graph of f is a line.
(E) The graph of f is a parabola.

(E) The function could be a downward facing parabola if it continues to the right.  You are only given three points, but there could be many more points on this function.

(D)  In geometry, a line is always straight, without any curves.  Notice that there are different slopes connecting the three points.  You cannot draw one straight line through all three of these points, so this choice cannot be true.

Loop Back:  Your goal was to find an answer choice that was false.  You did so, so you are finished!  If you have extra time, you can check the other answer choices and see that they are all possible, depending on how you draw the rest of the function.  (E), (C), (B), and (A) could all describe a downward facing parabola with the equation y = -(x – 3)².

The correct answer choice is (D).

On sat.collegeboard.org, 42% of the responses were correct.

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