# Scatterplots

## Data Analysis: Scatterplots

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

Always read the question carefully so that you can glean as much information from it as possible.  Identify the bottom line – what is it asking?  Then assess your options and choose the most efficient method to attack the problem.  When you have a solution, loop back to make sure that it matches your bottom line.

The scatterplot above shows the number of items purchased at a grocery store by 28 customers and the total cost of each purchase. How many of these 28 customers bought more than 10 items and spent less than $20? Bottom Line: # of people. Notice that this number must reflect those that meet 2 requirements: buying more than 10 things and spending less than$20.

Assess your Options:  When you have a graph, use the graph!  You can draw directly on it to help you visualize what you need.

Attack the Problem:  The dots represent each person.  Start with the first restriction.  If people must buy more than 10 items, then only the dots to the right of the 10 on the horizontal axis will be counted; those on the line do not count because they are equal to 10 rather than more than 10.  Draw a vertical line on the 10.  Now look at the second restriction.  If the people must spend less than $20, then they must be under the$20 hash mark on the vertical axis.  Draw a line at $20. Your graph should look like this: Count the number of dots in the lower right hand region that you created. Your answer is 4. Loop Back: Each dot represents a customer, a person, so you reached your bottom line. Look down at your answer choices. (A) Four (B) Five (C) Six (D) Seven (E) Eight The correct answer is (A). On sat.collegeboard.org, 66% 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 The correct answer is D. On sat.collegeboard.org, 35% of responses were correct. For more help with math, visit www.myknowsys.com! # Circle Graph Read the following SAT test question and then select the correct answer. When a math question involves a table or chart, read both the text of the question and the labels of the chart carefully. Identify the bottom line, and assess your options for solving the problem. Attack the problem to find the answer, and loop back to make sure that your answer addresses the bottom line. In a survey, a group of students from Westville High School were asked about their favorite movie genre. Each student in the group selected exactly one movie genre, and the data collected are summarized in the circle graph above. If 40 more students chose Action than Fiction, how many students were surveyed in total? Bottom Line: Total students = ? Assess your Options: You could plug in answer choices for the total and then take percentages of those to find out which answer would produce a difference of 40 between Action and Fiction. That would take a lot of steps! Instead, start with what you know and use what you know to write an equation. Attack the Problem: You know that 40 more students chose Action than Fiction. That means that Action – Fiction = 40. You also know the percentages for both Action and Fiction. Plug in the percentage from the chart and you will see that 30% (Action) – 14% (Fiction) = 16%. Now you need to combine the two things that you know, percents and actual numbers, into a single equation. You can write percents as decimals by moving the decimal two times to the left. What are these percents of? The uknown total number of students. For any unknown number, you can plug in the variable x. Now you have the equation .16x = 40. Solve for x by dividing each side by .16 and you will get the answer 250. Loop Back: What did x represent? The total number of students. That matches your bottom line, so you are ready to look down at your answer choices. (A) 100 (B) 150 (C) 200 (D) 250 (E) 300 The correct answer is (D). On sat.collegeboard.org, 50% of the responses were correct. For more help with SAT writing, visit www.myknowsys.com! # Graphs ## Link of the Day Your SAT essay should include a current event as one of your excellent examples to show that you are thinking critically about reality as you respond to the prompt. Sometimes "an event" can simply be the way that you use technology. Today Google is celebrating its 14th birthday. Does that seem bizarre to you? Read this article and think about how you could relate the current uses of Google to SAT writing questions. Previous questions have concerned the themes of change (whether it is for the better), success, mottos and motivation, discovery, questioning ideas, the overabundance of knowledge, planning, and creativity. If you select Google as one of the current events you want to write about, be sure to copy a few facts from this article and review them before you take the test. ## 9/27 Graphs Read the following SAT test question and then select the correct answer. When a question includes a graph, it is especially important to read both the text under the graph and the labels on the graph. Identify the bottom line and assess your options for reaching it. Ask yourself, "What could I do?" and then "What should I do?" Once you have selected an efficient method to solve the problem, attack the problem! Loop back to make sure that your answer addresses the bottom line. The histogram above shows the distribution of 31 black cherry trees, by height. For example, the leftmost bar represents the black cherry trees that are at least 60 feet, but not more than 65 feet, in height. Based on the histogram, which of the following can be the average (arithmetic mean) height of the 31 black cherry trees? Your bottom line is the average height of 31 trees, not the exact average, but what it could be. This histogram does not tell you the exact height of any of the trees, so how can you find their average heights? Look at that first bar. There are three trees that must be between 60 and 65 feet in height. If you assume that all of those trees are as short as possible (60 feet), you will find the lowest value that their average could possibly be. Find the lowest height that all of the trees could possibly be and then average those heights together. $\frac{3(60)+3(65)+8(70) + 10(7.5)+5(80)+2(85)}{31}\approx 72.74$ The lowest possible average for the heights of these trees is 72.74. Any answer lower than this will be wrong. Now, you could go back into your equation and plug in the highest possible value for each tree and average them again, but that will take a lot of time to retype into your calculator. Instead, you should think logically about the height of the trees. If you use the highest height that any tree can be, you are adding 5 to every single tree on the chart. That means that your final average will be 5 feet higher than your current average. $72.74 + 5 = 77.74$ You now have the highest and lowest possible averages of the heights of the trees. Since your bottom line asks which of the following answers could be the average, you must eliminate any answers that are not between 72.74 and 77.74. (A) 70 feet (B) 72 feet (C) 74 feet (D) 78 feet (E) 80 feet The correct answer is (C). On sat.collegeboard.org, 49% of the responses were correct. For more help with math, visit # Logic Read the following SAT test question and then select the correct answer. The Knowsys math method calls for you to read each question carefully, but you should be especially conscious of the possibility of misreading when your question involves a graph. Be sure that you understand each label on the graph and take the time to read any additional information given in the question. Identify the bottom line and asses your options for solving the problem in an efficient manner. Select your method, attack the problem, and loop back to make sure that the answer you found fits the bottom line, that it answers the question that you were asked. The bar graph above shows the number of people in attendance at each of the four meetings of the Maple Street Block Association that were held in 2011. Only members of the Block Association can attend the meetings, and no members joined or left the Block Association during 2011. Based on the bar graph, what is the least number of members the Maple Street Block Association could have had in 2011? Your bottom line is the least number of members belonging to the association. You must use logic to determine which information in this graph is relevant. The question is meant to mislead you, because as soon as you see the word “least,” you are likely to look for the smallest value on this bar graph. Before you jump to any conclusions, think about what the bar graph represents. At any of these four meetings, all of the members could have shown up, or only some of the members. In other words, it is possible for people to be absent, but it is not possible for people to be at these meetings without being members. All of the people present at any meeting must be members. The greatest number of people who came to any meeting is 72. Therefore, there cannot be fewer than 72 members in this club. Look down at your answer choices. (A) 61 (B) 65 (C) 67 (D) 72 (E) 268 The correct answer is (D). On sat.collegeboard.org, 43% of the responses were correct. For more help with SAT math questions, visit www.myknowsys.com! # Functions ## Link of the Day 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. The correct answer is (D). On sat.collegeboard.org, 39% of the answers were correct. For more help, visit www.myknowsys.com! # Graphing Equations ## Link of the Day 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! # Graphing Equations ## Link of the Day Since I shamefully allowed both Star Wars Day and Cinco de Mayo to pass without comment, I'll make up for it with two links. First, a brief history of Cinco de Mayo explains what the holiday is as well as who celebrates it, why, and how. Second, this article about Star Wars as a religious allegory reveals that no one is sure which (if any) religion is portrayed in Lucas' famous series. Happy Cinco de Mayo, and May the Fourth be with you! ## 5/6 Graphing Equations Read the following SAT test question and then select your answer. Always follow the Knowsys Method: Read carefully, getting information not only from the question itself but also from any charts, graphs, or figures included. Make careful note of labels and scales on all images. Identify the bottom line and copy it at the top of your scratch work. Assess your options, select the fastest way to solve the problem, and attack. Loop back to make sure you answered the question correctly. The function y=f(x), defined for $-1.5\leq x\leq 1.5$, is graphed above. For how many different values of x is f(x) = 0.2 First, write at the top of your scratch work # times f(x) = 0.2 = ? Next, assess your options. You could try plugging in 0.2, but you have nothing to plug it into. You could guess, but (while that is always an option on the test) that choice will not help you during practice. Your only option is to look at the graph and count. Make sure you pay attention to the labels on the graph. You're looking for points where f(x) = 0.2, so first draw a line at y = 0.2. Since the scale on this graph is 0.5, you need to draw a horizontal line a little less than halfway up to the first tick mark above the x-axis. Next, simply count how many times the two lines intersect. A) None B) One C) Two D) Three E) Four The answer is E. On sat.collegeboard.org, 39% of responses were correct. For more help with math, visit www.myknowsys.com! # Graphs ## Link of the Day Scholarships and college funding are a major concern for many families. No matter what your background, there is something for you among the myriad scholarships, grants, and funds available. Scholarships.com provides a free scholarship-hunting service to help you find your share of the money. The site also sends your information out to colleges so they can recruit you while you find money to pay for their services. ## 4/18 Graphs This question involves graphs, which mean you must take a slightly different approach to reading carefully. Skimming over the images to get to the words is extremely tempting, but the labels and trends easily visible on graphs can be instrumental in solving the problem. Look at the graph before you read the instructions in the question. First, the title is "PURCHASES AT A GROCERY STORE," so you know what the dots represent. Next, look at the axes. The X-axis displays the number of items by multiples of 5, and the Y-axis displays the total cost of each purchase. Finally, look at the content of the graph. This scatterplot shows that, in general, the more items people buy, the greater the cost of their purchase. Some people are able to get more groceries for their money, but not many people spend huge amounts of money on few items. Now that you know what the graph shows, look at the question. The scatterplot above shows the number of items purchased at a grocery store by 28 customers and the total cost of each purchase. How many of these 28 customers bought more than 10 items and spent less than$20?

The bottom line in this question has two parts: more than 10 and less than $20. Look at the scatterplot to find an answer that fills both requirements. First, focus on more than 10 items. The 10 on the X-axis indicates those who bought ten items or more. Visually draw a line upward from there. Every dot to the right of the line bought more than ten items. Keep in mind that since the problem specifies more than, the dots on the 10 line do not qualify. Next, focus on less than$20. Look at the Y-axis to find the $20 line. Extend that line to the right until it intersects the 10-item line. Points below this line represent purchases that totaled less than$20.

Now look at the lower-right quadrant of the grid you just created. The points in this area of the scatterplot represent purchases of more than 10 items that cost less than \$20. Count them, and you have your answer!

A) Four

B) Five

C) Six

D) Seven

E) Eight

On sat.collegeboard.org, 54% of responses were correct.

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

# Graphs

Graphs are actually a fairly new form of mathematical analysis, first systematically investigated in the 1930s. This link has a short and, hopefully, informative article on the history of graphs.

## 4/3 Graphs

Always take the time to carefully read the question, identify the bottom line, and assess your options. Then attack the problem and loop back to check your answer. Only after that should you look at the answer choices to select the correct one.

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

First, note the bottom line at the top of your scratch work.

k = ?

Next, assess your options. Getting from a slope to one part of a coordinate pair may take a few steps, so in this case the best way to start is to simply begin with what you know and move toward what you want.

You know that slope is rise over run, and you can calculate the total rise since you have both of the y coordinates for the two points. Using this information, you can set up a proportion.

7 - 1 = 6

$\frac{rise}{run}=\frac{3}{2}=\frac{6}{x}$

From here, you can either cross-multiply and solve for x or use multiples to determine its value (2*3=6, so 3*3=x). Either way, x = 4.

Now that you have solved for x, loop back to the bottom line to see if you are finished.

k = ?

You found the difference between the two x-values, but not the value of k. There is still one step left! Since x, the difference between the x coordinates of P and Q, is 4, simply add the value of the first x coordinate to find the value of k.

4 + 1 = 5

Now look at the answer choices

A) 4

B) 5

C) 6

D) 7

E) 8

On sat.collegeboard.org, 49% of responses were correct.

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

# Functions

Remember to read carefully! This problem includes a graph, which makes it doubly important to look closely and be cautious. Graphs can contain huge amounts of information, so read the question carefully to make sure you know what to look for.

The function f is graphed in the xy-plane above. If the function g is defined by  $g(x)=f(x)+4$, for how many values of x between -5 and 15 does g(x) equal 0?

First, read the question carefully. "for how many values of x between -5 and 15 does g(x) equal 0?" is long and complicated. Paraphrase it to pick out the most important parts. You are looking for how many times g(x)=0 when $-5< x< 15$. Unfortunately, g(x) is defined only in terms of f(x), making it impossible to solve. Is the problem hopeless?

Not at all! The test makers provided a graph of y=f(x), so you can translate (slide) that upward 4 units to get the graph of y=f(x)+4 or y=g(x). Next, consider what will happen to the graph when g(x)=0. Since the graph we're discussing is y=g(x), look for the point where y=0. There are three on the graph. Now loop back to make sure you answered the question correctly.

The bottom line only asked about points where x is between -5 and 15. One of our three points is to the left of -5, so eliminate it and two are left. Now look at the answer choices:

A) None

B) One

C) Two

D) Three

E) More than three

On sat.collegeboard.org, 45% of responses to this question were correct.

Want more help with math? Visit myknowsys.com!

# Graphs

## Mathematics: Standard Multiple Choice

The graph above shows the distribution of the number of days spent on business trips in 2010 by a group of employees of Company W. Based on the graph, what is the median number of days spent on business trips in 2010 for these employees?
The first thing you should do with any math question is read carefully. Always look closely at graphs; it is easy to make careless mistakes by misreading the labels or other information on a graph. This bar graph shows the number of employees who spent 20 or more days on business trips, separated by the number of days they spent traveling. Also look for key words in the question text; this one asks about the median number of days. Immediately ask yourself, "What is a median?" If you don't remember, make time to study math terminology between now and the SAT so that these words don't cost you time. The median of any set is the middle number when all members of the set are listed in order. This is not the same as an average (mean). The median of a set {1, 2, 3, 4, 5} is 3, and the median of a set {1, 2, 3, 4, 500} is also 3

After reading carefully, identify the Bottom Line. The last thing this problem asks is, "what is the median number of days spent on business trips in 2010 for these employees?" Put this in shorthand at the top of your scratch work.

median=?    or    m=?

How do you find the median? This is when you ask, "What could I do?" You have two options here: Write out "20, 20, 20, 20, 20, 21, 21, 21..." and then count to the middle, or determine the middle and then find its place on the graph. On the SAT, the long way is the wrong way, so use the second method.

First, calculate the total number of employees. 5+6+5+8+6+1=31 employees. Divide this by two to find the middle employee. Note that this method is different from the average formula, which would have required you to add up the total number of days all employees spent on business trips. If you lined these employees up along a hallway and then walked half the hallway, you would stop at the 15.5 mark, or the 16th employee. The 16th employee marks your median because there are 15 employees before her and 15 employees after her.

Finally, look at the graph to determine how many days the 16th employee spent traveling for work. The first column accounts for five employees, and the second column brings the total up to 11. The third column includes employees 12-16, inclusive, and each of these employees spent 22 days on business trips.

The answer is A, 22 days.

The College Board reports that 32% of those who attempted this question got it right.

Want more help with math? Go to myknowsys.com!