# Coordinate Geometry

A new year symbolizes a new start for many.  Although the world is essentially the same as it was before the clock struck midnight, there is a new optimism about the future.  People want to focus on goals such as peace and prosperity.  Read this current event about an unexpected gesture from North Korea, and then ask yourself what you can expect from 2013.  What themes can you identify in this article that are likely to be part of an SAT essay question?

## Geometry: Coordinate Geometry

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

Always read the question carefully and identify the bottom line.  Assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it addresses the bottom line.

In the figure, the slope of the line through points P and Q is $\frac{3}{2}$. What is the value of k?

Bottom Line: k = ?

Assess Your Options:  You could start from the point (1, 1) and use the slope to find new points, hoping that by adding 3 to the y value and 2 to the x value you will reach a point that contains a 7 y value.  Unfortunately, it is very easy to make a mistake using this method, such as adding the y change to the x value or vice versa.  Instead, use the information that you are given, the slope, to write an equation.

Attack the problem:  Although you are given the slope, you also know how the slope was obtained.  Think about it:  The slope is rise over run or the change in y over the change in x
$slope=\frac{rise}{run}=\frac{\bigtriangleup y}{\bigtriangleup x}=\frac{y_{2}\, -\, y_{1}}{x_{2}\, -\, x_{1}}$
You know two different y values, and two different x values, so you can plug in all the information that you know for the slope.
$slope =\frac{y_{2}\, -\, y_{1}}{x_{2}\, -\, x_{1}}=\frac{7-1}{k-1}$
Now you need to set this formula for slope equal to the value for slope that you were given in the problem, isolate the variable k, and solve for it.
$\frac{7-1}{k-1}=\frac{3}{2}$
$\frac{6}{k-1}=\frac{3}{2}$
3(k – 1) = 6 × 2
3k – 3 = 12
3k = 15
k = 5

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

On sat.collegeboard.org, 51% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

# 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.

For more help with SAT math, visit www.myknowsys.com!

# Slope

Write it Down! This infographic linked today from www.coolsiteoftheday.com discusses the importance of taking notes, a few different methods, and the potential benefits and drawbacks of taking notes digitally or the old--fashioned way. Did you know that your brain actually processes information differently while you're taking notes? This is a good resource to bookmark and revisit when you notice that your class notes are less than helpful--it might be time to try out a different method.

## 5/15 Slope

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

Remember to read carefully, identify the bottom line, assess your options, attack the problem, and loop back. When you use this method, you will get more problems right and you will move faster through the test.

In the xy-plane, line l passes through the points (a, 0) and (0, 2a), where a > 1. What is the slope of line l?

First, read carefully. You have two points on a line, which means you can visualize that line if you wish. Picking a number for a might make that easier if the variable trips you up. Next, identify the bottom line. The question asks for the slope of line l, so at the top of your scratch work write "slope = ?"

Now assess your options. Since you need to find the slope of the line, a good place to start is with the formula for slope: rise over run. There are two choices here; you can use a as a variable or you can pick a number for a. Using a directly involves fewer steps because you don't need to plug in the value, but manipulating the variable can be confusing for some and can cost time. Which tool you choose to solve the problem is up to your personal preference.

Either way, the first step in the problem is to set up your formula. Since a must by greater than 1, I'll use 2.

$\frac{rise}{run}=\frac{2a-0}{0-a}$                                                             $\frac{rise}{run}=\frac{2(2)-0}{0-(2)}$

$\frac{rise}{run}=\frac{2a}{-a}$                                                                    $\frac{rise}{run}=\frac{4}{-2}$

$\frac{rise}{run}=-2$                                                                    $\frac{rise}{run}=-2$

Now loop back to make sure that you answered the right question. Your bottom line asks for the slope, so you found the change in y-coordinates (rise) and the change in x-coordinates (run), divided one by the other and reduced. That is the slope, so -2 is the answer you need.

A) -2

B) $\frac{-1}{2}$

C) 2

D) -2a

E) 2a