# Equation of a Line

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) $f(x)=-\frac{1}{2}x+\frac{1}{2}$

On sat.collegeboard.org, 64% of the responses were correct.

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

# Functions

Is there always another explanation or point of view?  Before you answer this released SAT essay prompt, check out this article that is part current event and part historical example with a literary connection thrown in just for fun.  Richard the III was a real king who is best known as a villain in Shakespeare’s work.  Read about what happened to him and why he is appearing in the news now.  There are far too many themes in this article to name them all, so come up with about a dozen ways you could connect this example to an essay prompt.  Then memorize a few of the most interesting facts so that you can use them to support your opinion on any of the themes that show up in your SAT essay prompt.

Note: The identity of King Richard the III has been confirmed.  Read here for details.

## Algebra: Functions

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

Always read the problem carefully, identify the bottom line, and assess your options for solving the problem before you attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

The function y = f(x), defined for -1.5 ≤ x ≤ 1.5, is graphed above. For how many different values of is f(x) = 0.2?

Bottom Line: # times f(x) = .2

Assess your Options:  Some students will skip this problem, thinking that it requires a lot of time to somehow write a formula for the function from the graph.  However, once you know what you are looking at, this is one of the easiest and fastest problems on the test!  All that you have to do is read the graph!

Attack the Problem:  You know that f(x) = .2 is the same thing as y = .2.  Anytime you see f(x), you can just substitute a y for f(x) if that clarifies the problem in your head.  If y is constant, you know that it will be a horizontal line at .2.  Draw that line on your graph.

Anywhere that the line crosses the function f(x), that function is equal to .2.  Count up the number of intersections between the line that you drew and the original function.  There are four.  That means that f(x) = .2 four times.

(A) None
(B) One
(C) Two
(D) Three
(E) Four

On sat.collegeboard.org, 39% of the responses were correct.

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

# Functions

## Algebra: Functions

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

Always use the same process for math problems on the SAT.  Read 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 be sure it matches your bottom line.

If the function f is defined by , where 0 < a < b < c, for which of the following values of x is f undefined?

I. a
II. b
III. c

Bottom Line: For which value(s) of x is f undefined?

Assess your Options: You could pick numbers, but that will get confusing with three variables.  You could just start plugging in the variables a, b, and c for x and then simplify the function, but you will end up wasting time.  Time is precious on the SAT!  Start with the information that you are given and think about it logically.

Attack the Problem:  Always think about the information that you are given before you jump into the problem.  The inequality that you are given simply tells you that all of your variables are positive numbers.  A function or a fraction is undefined whenever it is divided by zero because you cannot divide by zero.

Think about it logically:  do you care what is on the top of the fraction?  No!  Focus on the bottom of the fraction.  How can you make x c = 0?  The variable that you are changing in this problem is x.  If you set x = to c, then cc = 0.

Note:  You do not know whether a or b is equal to c, so you cannot assume that ac or bc would equal 0.  If you plug those variables in, you still have a lot of variables on the bottom!

Loop Back:  You found the only answer that will work out of the three that you were given.  Look down at your answer choices.

(A) None
(B) I only
(C) III only
(D) I and II only
(E) I, II, and III

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

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

# Functions

## Algebra: Functions

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

Read each question carefully and identify the bottom line to avoid making careless mistakes.  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.

If f(x) = x + ax, and $a =\frac{7}{2}$ what is $f(\frac{3}{2})$?

Bottom Line$f(\frac{3}{2})=?$

Assess your Options:  You could use your graphing calculator to solve this problem, but it would probably take you more time to type in the fractions than to just solve the problem.  You are given a value for each variable in the problem so all you need to do is plug them in.

Attack the Problem:  Start by plugging in the value of a to the function that you were given.

$f(x)=x+ax$
$f(x)=x+\frac{7}{2}x$

Simplify the problem by adding.  Remember that the first x is a whole 1x, but that you must have like terms before you can add fractions.

$f(x)=\frac{2}{2}x+\frac{7}{2}x$
$f(x)=\frac{9}{2}x$

Now solve your function by plugging in the value for x that you were given.

$f(\frac{3}{2})=(\frac{9}{2}\)(\frac{3}{2})$
$f(\frac{3}{2})=\frac{27}{4}$

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

(A) $\frac{1}{3}$
(B) $\frac{3}{2}$
(C) $\frac{7}{2}$
(D)$\frac{21}{4}$
(E) $\frac{27}{4}$

On sat.collegeboard.org, 61% of the responses were correct.

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

# Functions

## Algebra: Functions

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

Use the same method for every math question on the SAT.  Start by reading the question carefully and identifying the bottom line; what do you need to find?  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that it matches the bottom line.

In the xy-plane, the graph of the line with equation y = a intersects the graph of the quadratic function f(x) = x² - 6x + 8 in exactly one point. What is the value of a?

Bottom Line: a = ?

Assess your Options:  You could just try plugging this into your calculator, but if you do not think carefully about what you are doing, you are likely to answer a question that was not asked.  Instead, think through every piece of information that you were given in this problem.

Attack the Problem:  What kind of graph is the function that you are given?  A parabola!  You know this because it has an x².  Picture a parabola in your mind (you know that this is a normal, upward-facing parabola because there is no negative before the x²).  Draw a u-shaped parabola on the xy-axis as part of your scratch work.

Now think about the fact that when y equals a certain number, it creates a vertical line. No matter what y equals, that vertical line will only ever intercept the graph at one point. That's not very useful! However, try flipping the given equation on its head: consider a = y. Remember that a =  is just like x =  and will create a horizontal line. Depending on what x equals, the horizontal line might cross the graph at two points, at no point at all, or at exactly one point--the vertex. You know that you must find the vertex of the parabola, so solve your function for x by setting your polynomial equal to zero and finding the roots of the equation:

x² - 6x + 8 = 0
(x – 2)(x – 4) = 0
(x – 2) = 0 and (x – 4) = 0
x = 2 and x = 4

You just found the two places where the parabola crosses the x-axis: 2 and 4.  All parabolas are symmetrical.  That means that the vertex must be halfway between these two numbers at x = 3.  You found the x value of the vertex, but you need the y value.

Plug in 3 for the x in your original equation:

f(x) = x² - 6x + 8
f(3) = (3)² - 6(3) +8
f(3) = 9 – 18 + 8
f(3) = -1

Loop Back:  When you solve a function for the f(x), you solve for y.  In this problem, you are told that y = a.  You have solved for a, so you are ready to look down at your answer choices.

(A) -3
(B) -1
(C) 1
(D) 3
(E) 4

On sat.collegeboard.org, 31% of the responses were correct.

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

# Graphing Functions

Algebra: Graphing Functions:

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

Always use the Knowsys Method on all math questions. This will help you think systematically and avoid careless mistakes. First, read the entire question carefully. Identify the bottom line and note it at the top of your scratch work. Next assess your options: What could I do? What should I do? Choose the most efficient method to attack the problem, and loop back to make sure that your answer matches the bottom line you were looking for.

$y=-2x^{2}+bx+5$

In the xy-plane, the graph of the equation above assumes its maximum value at x = 2. What is the value of b?

First, consider your bottom line. "What is the value of b?" At the top of your scratch work, write b = ?

Next, start assessing your options. What does it mean that the chart "assumes its maximum" at x = 2? Look at all the parts of the function. The highest power is 2, so you know that this is a quadratic function and that the chart will have a parabola. Since the coefficient of that variable is -2, you also know that the parabola will open downward. If the graph's maximum value is located at x = 2, you know that the vertex of the parabola will be somewhere to the right of the origin, on the vertical line two spaces to the right of the y-axis.

What can you do with that knowledge? Think about how you can move a parabola to the right of the origin. You might remember the formula $f(x - h)^{2}+k$. If you've forgotten, the point (h, k) represents the vertex of the parabola. You need to combine this with the function you were originally given.

$y=-2(x - 2)^{2}+5+k$

Next, use FOIL and the Distributive Property to square the binomial (x - 2) and multiply in the coefficient.

$y=-2x^{2}+8x-8+5+k$

At this point, you should stop and double-check your bottom line. You don't need to worry about solving for k, x, or y because you have already solved what your bottom line was asking: the value of b. Always keep your bottom line in mind so you remember to loop back and so you can be sure you answer what was asked.

Now that you know that b = 8, look at the answer choices:

(A) -8
(B) -4
(C) 4
(D) 8
(E) 10

On sat.collegeboard.org, 35% of responses were correct.

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

# Functions

How do medical breakthroughs happen?  How do you feel about animal testing?  A few dogs that were once paralyzed are now walking again with the aid of some cells from healthy dogs.  Scientists recognize that they are not ready to apply their findings to humans with spinal cord injuries, but they are hopeful about the future.  This development would make a great current event example for your SAT essay.  If you choose to use it as a current event, take notes detailing the facts in this article and how you could apply them to a wide variety of topics.

## Algebra: Functions

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

Take the time to read math questions carefully because this will save you from wasting time as you solve the problem.  Start by identifying the bottom line and assessing your options for reaching it.  Choose the most efficient method to attack the problem.  When you think you have an answer, loop back to make sure that it matches the bottom line.
• f(2n) = 2f(n) for all integers n
• f(4) = 4
If f is a function defined for all positive integers n, and f satisfies the two conditions above, which of the following could be the definition of f?

Bottom line: f(n) = ?

Assess your options:  You could start by plugging in f(4) to see which of your answer choices results in the number 4, then check any that do against the first condition.  This is the method recommended by collegeboard.com.  However, this requires multiple steps as you compare each answer choice to both conditions.  Instead, take a moment to think logically about the two conditions that you are given.

Attack the problem:  Start with the second condition because it is already in a format that is easy to use.  If f(4) = 4 and you can use the same variable to represent numbers that are the same,  that is the same as saying that f(n) = n.  Now look at the first condition and think about it logically.  If you multiply the variable within the function by 2, that gives you the same number as multiplying the result of the function by 2.  In order for those two numbers to be the same, the final result of the function has to match the number that is plugged into the function.  In other words, f(n) = n.  Both conditions give you the same definition of the function.

Loop Back:  You found a simple way to define both of the conditions for f(n), so look down at your answer choices.

(A)  f(n) = n - 2
(B)  f(n) = n
(C)  f(n) = 2n
(D)  f(n) = 4
(E)  f(n) = 2n – 4

On sat.collegeboard.org, 46% of the responses were correct.

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

# Functions

## 10/18 Functions

For every SAT math problem, read the problem carefully so that you know exactly what information you are given.  Then identify the bottom line, the information that you must find.  Assess your options for solving the problem, and choose the most efficient method to get to the answer.  Attack the problem to find the answer, and loop back to your bottom line to make sure that your answer matches what you were supposed to find.

A manager estimates that if the company charges p dollars for their new product, where 0 ≤ p ≤ 200, then the revenue from the product will be r(p) = 2,000p – 10p² dollars each week. According to this model, for which of the following values of p would the company’s weekly revenue for the product be the greatest?

Bottom Line:  Which of the following values of p will result in the greatest revenue?

Assess your options:  You could work backwards by plugging in all of the answer choices to r(p) = 2,000p – 10p², but that will take time.  Instead, use what you know about functions to determine the answer.

Attack the problem:  You know what the graph of x² looks like: a parabola that makes a “u.”  What happens to that graph when it is -x²?  That “u” turns upside-down and the parabola looks like a hill.  That is what you have for your function r(p) = 2,000p – 10p².  Now simplify your function by pulling out the numbers and variables that your two terms have in common so that r(p) = 10p(200 – p).  If you set each part of this equation equal to zero, you will find where the parabola crosses the x-axis.  If 10p = 0 and 200 – p = 0, then p = 0 and 200.  The parabola crosses the x-axis at 0 and 200.  That makes sense because you were told in the problem that 0 ≤ p ≤ 200.  Think about the characteristics of parabolas once more.  All parabolas are symmetrical.  Where will your greatest value for the revenue be?  It will be at the top of that “hill” exactly between 0 and 200.  What is the midpoint between 0 and 200?  100.

Loop back: Your bottom line was the value of p that would have the greatest revenue.  Although your function used r(p) rather than f(x),  that p value had to be on the x-axis.  You solved for the bottom line, so you are ready to look down at the answer choices.

(A)  10
(B)  20
(C)  50
(D)  100
(E)  200

On sat.collegeboard.org, 37% of the responses were correct.

For more help with math, visit

# Functions

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

All math problems require the same approach.  Read the question carefully and identify the bottom line that you must find.  Assess your options for working the problem, choose the most efficient strategy, and attack the problem.  Always make sure that your answer addresses the bottom line, especially in problems with multiple variables.

If the function f is defined by f(x) = 2x + 3, and if f(a) = 11, what is the value of a?

This question may look complicated because it uses functions, but it can be worked very quickly.  You are given a function using the variable x:
f(x) = 2x + 3

However, your bottom line requires you to solve for another variable: a. Think about the first information you are given in terms of a by substituting an a for every x:
f(a) = 2a + 3

Now plug in your last piece of information, f(a) = 11, and solve the problem using simple arithmetic.
11 = 2a + 3
8 = 2a
4 = a

Your bottom line was to solve for a so you are finished!  Take a look at the answer choices.

(A) 4
(B) 7
(C) 11
(D) 17
(E)25

On sat.collegeboard.org, 61% of the responses were correct.

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

# Functions

Many schools in the United States participate in campaigns to keep children from smoking.  However, there are countries that are taking even greater measures to make smoking unattractive.  Cigarette packaging in Australia will no longer display colorful logos, but instead will display images depicting the dangers of smoking.  As you read this article, think about whether or not you agree with these measures, and then think about the themes that might relate this current event to an SAT essay topic.

Also, if you are a senior who dreads the college application process, take a look at this checklist and remember to breathe in the next few months!

## 8/16 Functions

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

Read each math question carefully so that you can identify exactly what is being asked of you.  Once you have identified the bottom line, assess your options to find an efficient way to solve the problem.  Finally, attack the problem, solve it, and loop back to make sure that your answer addresses the bottom line that you were asked to find.

Which of the following could be the equation of the function graphed in the xy-plane above?

You have been given a graph, and you must find the equation that has been graphed.  You could plug all of the answer choices into your calculator, but that would take a long time and you risk making a typo.  Instead, break the graph down into its most basic components.  What shape that you have often seen does this graph most resemble?  It looks like a parabola opening upwards, so you know that f(x) = x² will be part of your equation.

Picture the f(x) = x² parabola in your mind.  It passes through the origin at (0,0).  However, the graph in this problem would extend past the point (0,0) into the negative numbers if you continued the basic curve of the parabola.  To translate the function down on the graph, you would need to subtract a number from the original function.  Now you have f(x) = x² - n, where n = any number.

There is one more step.  The basic curve of the normal parabola has been reflected across the x-axis in this problem so that all the values of the parabola are now positive.  What can you do to make sure that all of the numbers in a function are positive?  Take the absolute value of the function.  Now you have f(x) = |x² - n|.  Look down at your answer choices.

(A) y =  (-x)² + 1
(B) y = -x² + 1
(C) y = |x² + 1|
(D) y = |x² - 1|
(E) y = |(x – 1)²|

(A), (B), and (C) cannot be the answers because they all add to the equation and would result in a parabola that has been shifted above the x-axis.  (E) will not be symmetric to the y-axis, and the graph that you have remains symmetric to the y-axis; it has not been shifted to the right or the left. The (x – 1)² part of the equation in (E) shifts the entire parabola away from its original position on the y-axis.  (D) is the only answer that matches the equation you wrote for this graph.

On sat.collegeboard.org, 39% of the answers were correct.

For more help, visit www.myknowsys.com!

# Graphing Equations

Most of you are probably familiar with the book Alice in Wonderland. The story is a fantastic and whimsical one, but on the surface it would seem to be nothing but nonsense written for children. However, as with most literary works that stand the test of time there is more to Alice in Wonderland than meets the eye. Alice in Wonderland was written by Charles Dodgeson (who used the pen name Lewis Carroll). Dodgeson was a math teacher in Oxford, England, and many of the strange things that happen to Alice during her adventures in Wonderland were actually written as satire to refer to and criticize the newly emerging mathematical theories. You can read more about the math behind Alice in Wonderland here.

## Mathematics: Graphing Equations

Always follow the Knowsys Method for math problems. It will save you time not only on the SAT, but also on math tests and quizzes in your school classes. Read the question carefully and identify the bottom line. Assess your options. Ask what you could do and then what you should do, and solve the problem quickly and efficiently once you have decided on a strategy. Finally, loop back to double check that you answered the question correctly.

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

Remember, the first step to any problem is to read it carefully. That seems obvious, but it is often where students make their first mistake. When you glance at this question, it may seem a little confusing at first. If that is the case, make sure that you take your time and reread the question carefully. We are trying to find the equation of a line that is parallel to the x-axis. That means that the line will run horizontally. In other words, the slope (which is the rise over the run) will be zero. The line will also be 4 units above the x-axis. That means that the y intercept will be 4. At this point we could check each answer and graph the different equations until we find one that fits. That might seem like a tempting choice but remember, we want to solve the problem as quickly as possible (to leave time for the other problems in the section). It would be much faster to remember the slope intercept form of the equation for a line, y = mx + b. In this case the slope (m) is zero and the y-intercept (b) is 4. Our equation, then, must be y = 4. Now we look at the answers below to find the one that matches our prediction.

(A) x = 4

(B) x = -4

(C) y = -4

(D) y = 0

(E) y = 4

We can see that E matches our prediction exactly, and that is the correct answer.

On sat.collegeboard.org, 66% of responses were correct.

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

# Functions

How to manage time and adapt to college life is a major concern for many students. A site full of College Study Tips could be a great help! This website has "college study tips, college study skills, study guides and tricks to help you manage your time, take better notes, study more effectively, improve memory, take tests, and handle the stresses of college life."

## 4/30 Functions

Always try to solve the problem as if the answer choices weren't there. This has two benefits: it makes you more efficient at multiple-choice questions, and it makes you more confident on grid-in questions. First, read carefully and look for important information. Identify the bottom line, then assess your options and select the most efficient way to solve the problem. Attack the problem, solve it quickly, and loop back to ensure that the answer you found matches the bottom line. Finally, check the answer choices and select the correct answer.

If the function f is defined by $f(x)=\frac{(x-a)(x-b)}{(x-c)}$, where 0 < a < b < c, for which of the following values of x is f undefined?

I. a
II. b
III. c

Read carefully and focus on the bottom line.

f undefined when x = ?

The most important clue here is the word "undefined." What does "undefined" mean in a math problem? It means that you have attempted to do something impossible, such as dividing by zero. In order for this function to divide by zero, x - c must equal zero because it is the denominator. Given that, there is only one possible value for x. Any number minus itself equals zero, so if x = c, then x - c = 0. Any other answer choice would yield a positive or negative number for the denominator, and the function would not be undefined. Only choice III makes the function undefined.

Now look at the answer choices:

A) None

B) I only

C) III only

D) I and II only

E) I, II, and III

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

Want more help with math? Visit www.myknowsys.com!

# Fractions

College can seem intimidating, and all the more so because it practically involves a different language. Learn the terms around college admission, and you will be able to navigate the process much more easily.

## 4/24 Fractions

Remember to always follow the Knowsys Method when approaching math problems: First, read carefully. Then identify the bottom line and assess your options. Next, choose the most efficient method and attack the problem! Finally, loop back to make sure you have the correct answer.

If f(x) = x + ax, and $a=\frac{7}{2}$, what is $f(\frac{3}{2})$?

First, as always, identify the bottom line.

$f(\frac{3}{2})=?$

Next, assess your options. You have variables in the question and numbers in the answer choices, so you could work backward from the answer choices. However, there are also numbers in the question, so you could plug in those numbers to see what that gives you. The second method will be far more efficient.

First, plug in the numbers. $a=\frac{7}{2}$ and $x=\frac{3}{2}$, so plug each of those in where appropriate.

$f(\frac{3}{2})=\frac{3}{2}+\frac{7}{2}(\frac{3}{2})$

Next, following the Order of Operations (PEMDAS), multiply the last two numbers together. When multiplying fractions, simply multiply the numerators and then the denominators.

$f(\frac{3}{2})=\frac{3}{2}+\frac{21}{4}$

Obviously, the next thing to do is to add the fractions together. Remember that when adding fractions, the denominators must match, and you can ONLY add the numerators. You'll need to do something to make the denominators the same.

$f(\frac{3}{2})=\frac{3}{2}(\frac{2}{2})+\frac{21}{4}$

$f(\frac{3}{2})=\frac{6}{4}+\frac{21}{4}$

$f(\frac{3}{2})=\frac{27}{4}$

Now loop back. Is $f(\frac{3}{2})$ what you actually needed to find? Yes, it is! Now look at the answer choices.

$A)\frac{1}{3}$

$B)\frac{3}{2}$

$C)\frac{7}{2}$

$D)\frac{21}{4}$

$E)\frac{27}{4}$

On sat.collegeboard.org, 57% of responses were correct.

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

# Functions

Remember that your goal is to get as many questions right as possible, not to answer as many as you can. In some cases, leaving a question blank can be better for your score than guessing, but best of all is to solve the problems efficiently and confidently enough that you can answer every question right. Solve each problem as if it did not have answer choices by reading carefully, noting the bottom line, assessing your options, attacking the problem, and looping back to double-check your answer.

In the xy - plane, line l is perpendicular to the graph of the function f(x) = 5x - 2. Line l could be the graph of which of the following functions?

Your bottom line in this case is "Which of the following?" In WOTF questions, always check E first and work your way back up to A. WOTF questions are much more likely than normal questions to have D or E coded as the correct answer. (The exception to this is Roman Numeral questions, in which the actual answer choices will say things like A) I and II, B) II only, etc. These are random.)

Next, assess your options. You could guess, and since this is a "which of the following" question, you would be relatively safe in guessing D or E. Or you could work from what you know to make a prediction about the correct answer.

You should know that the slope of any line is the negative reciprocal of the line perpendicular to it. That is, if the first line has a slope of m, the second line has a slope of $-\frac{1}{m}$. You should also know that slop-intercept the formula for a line is y = mx + b. Since the equation in the problem is f(x) = 5x - 2, the slope of the given line is 5. Therefore, the slope of the perpendicular line is $-\frac{1}{5}$.

Now look at the answer choices:

A) g(x) = -5x

B) $g(x)=-\frac{1}{5}x$

C) g(x) = x - 2

D) $g(x)=\frac{1}{5}x$

E) g(x) = 5x

The only option that has the correct slope is B. That is the correct answer. Remember that rules of thumb, like starting with E on WOTF questions, are not always reliable.

On sat.collegeboard.org, 62% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

# Equation of a Line

The BBC is a great source for international news or simply a different perspective on American news. Look at this site or other news sites in the last week or two before you take the SAT to find your five current events examples. Even if you don't normally keep up with the news, looking like you do can increase your score!

## 3/25 Equation of a Line

Remember to follow the Knowsys Method and note the bottom line, assess your options, attack the problem, and loop back to check the question before you select your answer.

A line segment containing the points (0,0) and (12,8) will also contain the point

The bottom line here is which answer choice lies on the given line.

There are multiple ways to solve this problem. You could find the equation of the line, or, since the line goes through the origin, you could use ratios to find a point that has the same relationship between its x and y coordinates.

First, reduce the coordinates to lowest terms. You can arrange them in a ratio format if you wish; whether you prefer $\frac{x}{y}$ or $\frac{y}{x}$ does not matter. You might use $\frac{y}{x}=\frac{8}{12}$ since "rise over run" is also the formula for slope. Reduce this to its lowest terms and then check the answer choices for multiples.

$\frac{y}{x}=\frac{8}{12}=\frac{2}{3}$

Now convert it back to (x,y) format. Make sure the x and y go in the right places.

(3,2)

A) (2,3)

B) (2,4)

C) (3,2)

D) (3,4)

E) (4,2)

On sat.collegeboard.org, 58% of responses were correct.

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

# Functions

history.com The History Channel's website is full of fascinating articles and videos. Watch their current series, "Full Metal Jousting," and learn some math facts about St. Patrick's Day.

## 3/22 Functions

Remember to follow the Knowsys Method: Read carefully, identify the bottom line, and assess your options. After you ask, "What could I do?" and "What should I do?" attack the problem, then loop back to check whether you answered the question correctly. Finally, take a look at the answer choices and select the correct one.

The graph of y=f(x) is shown above. If $0\leq t\leq 5$, and if (t,v) is on the graph of f, which of the following must be true?

A glance at the answer choices shows that they all have v, so v=? will do for an imperfect bottom line. Now look at the question again. If t is somewhere from 0 to 5, what does that say about v? Note that t is the x value, and v is the y value, so you need to focus on the part of the graph from x=0 to x=5. In that range, the smallest y value is 5 and the greatest is 10. Now look at the answer choices.

$A) -10\leq v\leq -5$

$B)-5\leq v\leq 0$

$C) 0\leq v\leq 5$

$D) 5\leq v\leq 10$

$E)10\leq v\leq 15$