# Equation of a Line

## Link of the Day

How do you make sure that you have the best doctors and the best conditions for patients?  First there was a push for doctors to get more sleep.  Now there is a push to make sure that doctors are getting more hours to finish their work.  Take a look at the debate in this current event.  Write down the broad themes in this article, and the specific details that will make you sound informed.  Then try linking this current event to the following previous SAT essay prompts:  Is there always another explanation or another point of view?  Can success be disastrous?  Should people let their feelings guide them when they make important decisions?  Should people change their decisions when circumstances change, or is it best for them to stick with their original decisions?

## 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 for reaching the bottom line, and use the most efficient method to attack the problem.  When you have an answer, loop back to verify that your answer matches the bottom line.

If the graph of the function f is a line with slope 2, which of the following could be the equation of f?

Bottom Line: WOTF (which of the following)

Assess your Options:  For a “which of the following” question you should look at the answers choices, but not until you have used what you know about the equation of a line to decide what kind of equation you need to find.  Start with the information that you are given.

Attack the Problem:  Remember the generic equation for a line is y = mx + b.  In any equation, f(x) and y can mean the same thing.  The variable m is the slope of the line.  You know that your slope must be 2.  Plug that 2 into the equation.  You now have:

f(x) = 2x + b

(The variable b is the y-intercept.  You were not told anything about the y-intercept, so that could be any number.  All you need to do is match the part that you do know, the 2x.)

Loop Back:  You used all the information that you were given, so look down at your answer choices.

(A) f(x) = 4x - 2
(B) f(x) = 2x + 4
(C) f(x) = -2x – 2
(D) $f(x)=\frac{1}{2}x+2$
(E) $f(x)=-\frac{1}{2}x+\frac{1}{2}$

The correct answer is (B).

On sat.collegeboard.org, 64% 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 select the correct answer.

Approach every question the same way to minimize mistakes.  Start by reading the question carefully and identifying the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches your bottom line.

In the xy-plane, line l passes through the points (0, 0) and (2, 5). Line m is perpendicular to line l. What is the slope of line m?

Bottom line: slope m = ?

Assess your Options:  You could draw out a graph and solve this visually, but that is a waste of time if you know the formula to find the slope of a line.

Attack the Problem:  You are given the most information about line l, so start with that line.  You should have the formula for slope memorized:

$slope=\frac{rise}{run}=\frac{\Delta y}{\Delta x}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

It is easiest just to think about slope as the change in y-values over the change in x-values.  If you look up at the original points that you have been given, from zero the y-values go up to 5 and the x-values go up to 2.  You now have 5 over 2.

The slope of line l is $\frac{5}{2}$.

At this point, some students will think they are finished and select answer (D).  However, your bottom line was the slope of line m!  The problem tells you that line m is perpendicular to line l.  In order to find a perpendicular line, you must take the opposite reciprocal of the first line; in essence you must flip the sign (negative or positive) and the numbers (a fraction or whole number).

The slope of line m is $-\frac{2}{5}$.

Loop Back:  You solved for your bottom line, so look down at your answer choices.

(A) $-\frac{5}{2}$
(B) $-\frac{2}{5}$
(C) $\frac{2}{5}$
(D) $\frac{5}{2}$
(E) 5

The correct answer is (B).

On sat.collegeboard.org, 47% 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.

Read the question carefully and identify the bottom line.  Assess your options for solving the problem and use the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

What is the area of the triangle in the figure above?

Bottom Line: a =?  (What is the area?)

Assess your Options:  The best way to solve this problem is to use the formula for the area of a triangle.  You have already been given all the information that you need to solve the problem.

Attack the Problem:  Start with the formula for the area of a triangle.

$area =\frac{1}{2}(base)(height)$

The base of the triangle extends to the right of the origin (5 units).  The height of the triangle extends upwards from the origin (3 units).

$area =\frac{1}{2}(5)(3)$

Work with the easy numbers first: 5 times 3 is 15.  If you divide 15 by 2 you get 7.5.

Loop Back:  You solved for area, so you are ready to look down at the answer choices.

(A) 4.0
(B) 7.5
(C) 8.0
(D) 8.5
(E) 15.0

The correct answer is (B).

On sat.collegeboard.org, 75% of the responses were correct.

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

# Coordinate Geometry

## Link of the Day

Isn't it fascinating that no matter how long people study people, there is still more to learn?  Take a look at this current event article that endeavors to explain why women talk more than men.  Pick out the broad topics in this article.  How could you use the facts from this article to support a position on the following SAT essay prompts?

1. Do we need other people in order to understand ourselves?
2. Should heroes be defined as people who say what they think when we ourselves lack the courage to say it?
3. Are people best defined by what they do?

## 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 for reaching the bottom line and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that the answer matches the bottom line.

What is the equation of the line parallel to the x-axis and four units above the x-axis?

Bottom Line: equation of a line

Assess your Options:  You could look down at the answer choices, but if you look down without thinking first you will often confuse yourself.  Instead, use the information that you are given to write an equation.

Attack the Problem:  You know that you are dealing with an x-axis, which means you must use a normal xy-graph with a vertical y-axis and a horizontal x-axis.  Draw this on your paper.  Next, imagine 4 ticks on the y-axis and put a little dot four units above the x-axis.  Draw a horizontal line that is parallel to the x-axis.  Does that line ever leave y = 4?  No!  That is the equation of the line.

Note:  If you write x = 4, you create a vertical line.  Think about it this way: the x values change from negative infinity to positive infinity.  If you choose a single x value, the line along this value cannot be parallel to the x-axis because it is limited to a single value.

Loop Back:  You needed an equation of a line, and not necessarily one that mentioned x at all.  You found one.  Look down at your answer choices.

(A) x = -4
(B) x = 4
(C) y = -4
(D) y = 0
(E) y = 4

The correct answer is (E).

On sat.collegeboard.org, 67% 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.

Always be sure to read the question carefully and make a note of the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

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?

Bottom Line: slope of l = ?

Assess Your Options: You could select a number larger than 1, plug it in for the variable a, and then work the problem.  However, if you peek down at the answer choices, notice that some have a variable still in the problem.  It will take you longer to plug in a number than to work the problem using the variables.

Attack the Problem: Your bottom line is a slope, so use the formula for the slope of a line. The formula for slope of a line is:

$\frac{rise}{run}\: or\: \frac{\Delta y}{\Delta x}$

To find the change in y coordinates, subtract the first y-value from the second y-value.  Do the same with the x values:

$\frac{2a-0}{0-a}=\frac{2a}{-a}=-2$

The variable will cancel when you simplify the problem.  Your answer is -2.

Loop Back:  You found the slope of the line, so you are ready to look down at your answer choices.

(A) -2
(B)$-\frac{1}{2}$
(C) 2
(D) -2a
(E) 2a

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

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

# Coordinate Geometry

## Link of the Day:

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

Loop Back: You solved for k, so you are ready to look at your answer choices.

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

The correct answer is (B).

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

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

# Coordinate Geometry

## Link of the Day

Today is Veterans Day, a day set aside to thank those who have risked their lives to serve our country.  As you gather historical, current, and literary examples for your SAT essay, consider including an example involving soldiers.  Think about the courage that it takes to be willing to serve in such a capacity, and the reasons behind the choice to enlist.  Take a look at this article and think about how our lives are different due to the sacrifices of many veterans.

## 11/11 Coordinate Geometry

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

Use the same process for every math problem.  Read the problem carefully.  Identify the bottom line.  Assess your options, then choose the most efficient method to attack the problem.  Once you have worked the problem, loop back to verify that you have solved for the bottom line.

In the figure above, which quadrants contain pairs (x, y) that satisfy the condition  ?

Bottom line:  Where can you find an x and a y that work for this problem?

Assess your options:  This question concerns coordinate geometry, so you will have to use the facts that you know about graphing to answer the question.  You could pick specific points in each quadrant to see if they work, but simply knowing the properties of the graph should be enough to get you to the answer.

Attack the problem:  In order to divide a number by another number and get one, you need equal numbers.  To satisfy this condition, x and y must be equal to each other.  Ask yourself whether the x and y can be equal to each other in each quadrant.  In quadrant I, all the numbers are positive (+, +), so it is possible for x and y to equal each other and create a positive 1 after division.

Now think about the characteristics of quadrant II.  In quadrant II, all of the x values are negative and all of the y values are positive (-, +), so x and y cannot be equal.  When you divide a negative number by a positive number, you will get a negative number; there is no way to get a positive 1.  Quadrant II does not satisfy this condition.

In quadrant III x and y are both negative (-, -), so they could be equal.  If you divide a negative number by a negative number, the result will be positive.  You can get a positive number 1 in this quadrant.

In quadrant IV, the x values are positive while the y values are negative (+, -).  Once again x and y cannot be equal.  You cannot divide a positive by a negative and get a positive number, so quadrant IV does not satisfy this condition.

Loop Back:  You checked all four quadrants, so look down at your answer choices now.

(A) I only
(B) I and II only
(C) I and III only
(D) II and IV only
(E) I, II, III, and IV

The correct answer is (C).

On sat.collegeboard.org, 59% 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!

# Parabolas

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

Don’t let this question intimidate you just because it has a parabola.  Use the same method that you would use with any other math problem.  Read the question carefully, identify the bottom line, and choose an efficient method to solve the problem.  Then attack the problem and loop back to make sure that you solved for the bottom line.

The quadratic function f is graphed in the xy-plane above. If f(x) ≤ u for all values of x, which of the following could be the coordinates of point P?

Your bottom line is which values could be the coordinates of point P, so make a note of the bottom line on your paper, and start with what you know about this point.  You are told that f(x) ≤ u for all the values of x.  That is your y value, so that is just letting you know that nothing can be higher than u, which is on point P.  If you are looking for the highest point on a downward opening parabola, what are you actually looking for?  The vertex!

Think about it this way: as the parabola extends outward from the vertex, both sides stay an equal distance from the vertex. You have just examined the information given about the y-axis, so turn your attention to the x-axis.  You are given two x values that are of equal height on your parabola, so the x value of the vertex, P, must be exactly between them.  Your highest value is 4, so you might be tempted to halve 4 and get 2.  Just be sure to remember that the first point is not at zero, but at 1.  That means that your parabola has been shifted 1 unit to the right.  To find the midpoint, use the midpoint formula, which is simply an average of the two numbers that you have.

$\frac{1 +4}{2}=2.5$

You now have the x value of 2.5. You are not given any additional information about the limits of the y-axis, so loop back to the bottom line.  The question is not actually asking you to find both x and y coordinates.  Remember that your bottom line is what “could” be the coordinates, so this is probably enough information to find the correct answer.  Look down at your answer choices now.

(A) (2, 3.5)
(B) (2.25, 3.25)
(C) (2.5, 3)
(D) (2.75, 4)
(E) (3, 2.5)

All you know about the y value is that it must be greater than 0, so all of the y values will work, but only one of the answers has the x value of 2.5.

The correct answer is (C).

On sat.collegeboard.org, 53% of the responses were correct.

For more help with math, visit