PSAT 8/9 Prep with 2 practice tests - Princeton Review 2020

Chapter 12 Advanced math
Part III PSAT 8/9 Prep

There will be 6 questions on the PSAT 8/9 that test what College Board calls “Passport to Advanced Math.” This category includes topics such as functions, quadratics, and growth and decay. If you’ve learned these topics already in school, great! You’ll have a step up on the PSAT 8/9. If not, fear not—this chapter will give you the foundation needed for tackling these questions on the PSAT 8/9.

FUNCTIONS

In the Math Basics chapter, we looked at some concepts related to the xy-plane. Here, we will look at some more complicated topics involving functions and graphs. The functions on the PSAT 8/9 mostly look like this:

f(x) = x² + 6x + 24

Most questions of this type will give you a specific value to plug in for x and then ask you to find the value of the function for that x. Each function is just a set of instructions that tells you what to do to x—or the number you plug in for x—in order to find the corresponding value for f(x) (a fancy name for y). Just plug your information into that equation and follow the instructions.

Just Follow the Instructions

Functions are like recipes. Each one is just a set of directions for you to follow. The College Board provides the ingredients and you work your magic.

Let’s try an easy one.

4.What is the value of g(9) for the function g defined by g(x) = x + 4?

A) 4

B) 5

C) 9

D)13

Here’s How to Crack It

The questions asks for the value of g(9) for the given function g. The value in the parentheses is the x value of the function, and g(x) is the resulting y value. Plug in 9 for x in the function to get g(9) = 9 + 4 = 13. The correct answer is (D).

One way the PSAT 8/9 can make functions more complicated is to give you two functions to deal with together. If you approach these problems one piece at a time, they will be easier to handle.

Here’s an example.

12.Given the functions above, what is the value of h(2) + k(2)?

Here’s How to Crack It

The question asks for the value of h(2) + k(2), given the definitions of the two functions. This is really no more complicated than a question dealing with a single function. Just plug in the value for x and work the question in bite-sized pieces. Start with h(2). Plug in x = 2 to get . Now plug x = 2 into the k function to get . Finally, add the values of h(2) and k(2) to get . The correct answer is 21.

Sometimes the PSAT 8/9 will use a word problem to describe a function, and then ask you to “build a function” that describes that real-world situation.

3.Vacay Homes Co. rents out vacation homes for periods ranging from 7 to 21 days for $115 per day plus a summer discount of $245. When 7 ≤ d ≤ 21, which of the following functions B shows the amount of money a customer pays to Vacay Homes Co.?

A)B(d) = 21d — 245

B)B(d) = 115d — 245

C)B(d) = 115d + 2,170

D)B(d) = 245d — 115

Here’s How to Crack It

The question asks for the function that represents the amount a customer must pay. Translate the information in the question in bite-sized pieces and use Process of Elimination. The cost per day is $115, and d represents day. Therefore, the correct function must include the term 115d. You can eliminate (A) and (D), as those have other coefficients for the d term. The question also mentions a discount, which will reduce the cost of the rental for the summer. The correct function must feature subtraction to show this reduction of cost. Now you can eliminate (C), which adds an additional fee. The correct answer is (B).

QUADRATIC EQUATIONS

Ah, quadratics. You’re likely to see several questions on the PSAT 8/9 that require you to expand, factor, or solve quadratics. You may even need to find the vertex of a parabola or the points of intersection of a quadratic and a line. So let’s review, starting with the basics.

Expanding

Most often you’ll be asked to expand an expression simply by multiplying it out. When working with an expression of the form (x + 3)(x + 4), multiply it out using the following rule:

FOIL = First Outer Inner Last

Start with the first figure in each set of parentheses: x × x = x².

Now do the two outer figures: x × 4 = 4x.

Next, the two inner figures: 3 × x = 3x.

Finally, the last figure in each set of parentheses: 3 × 4 = 12.

Add them all together, and we get x² + 4x + 3x + 12, or x² + 7x + 12.

Factoring

If you ever see an expression of the form x² + 7x + 12 on the PSAT 8/9, there is a good chance that factoring it will be the key to cracking it.

The key to factoring is figuring out what pair of numbers will multiply to give you the constant term (12, in this case) and add up to the coefficient of the x term (7, in this question).

Let’s try an example:

x² + 7x + 12

Step 1: Draw two sets of parentheses next to each other and fill an x into the left side of each. That’s what gives us our x² term.

(x )(x )

Step 2: 12 can be factored a number of ways: 1 × 12, 2 × 6, or 3 × 4. Which of these adds up to 7? 3 and 4, so place a 3 on the right side of one parenthesis and a 4 in the other.

(x 3)(x 4)

Step 3: Now we need to figure out what the correct signs should be. They should both be positive in this case, because that will sum to 7 and multiply to 12, so fill plus signs into each parenthesis.

(x + 3)(x + 4)

If you want to double-check your work, try expanding out (x + 3)(x + 4) using FOIL and you’ll get the original expression.

Now try the following problem.

11.When the expression x² — 4x — 32 is factored to (x — 8)(x + h), where h is a constant, what is the value of h?

Here’s How to Crack It

The question asks for the value of h in the factored form of a quadratic. When the PSAT 8/9 gives you a factored quadratic, you almost always need to use FOIL to multiply it out in order to answer the question. Doing so on the factors (x — 8)(x + h) results in x² + hx — 8x — 8h. Set this equal to the expression given in standard form to get x² — 4x — 32 = x² + hx — 8x — 8h. The constants on either side of the equation (the numbers with no x terms) must be equal, so —32 = —8h. Divide both sides by —8 to get 4 = h.

Don’t forget that you can plug in on a question like this instead. If you let x = 2, the given expression becomes (2)² — 4(2) — 32 = 4 — 8 — 32 = —36. The factored expression becomes (2 — 8)(2 + h) = (—6)(2 + h), and this is equal to the —36 from the original expression. Therefore, (—6)(2 + h) = —36, and dividing both sides by —6 results in 2 + h = 6. Subtracting 2 from both sides shows once again that h = 4. No matter how you approach this one, the correct answer is 4.

Solving Quadratic Equations

Sometimes you’ll want to factor to solve an equation. In this case, there will be two possible values for x, called the roots of the equation. To solve for x, use the following steps:

Step 1: Make sure that the equation is set equal to zero.

StepStep 2: Factor the equation.

StepStep 3: Set each parenthetical expression equal to zero. So if you have (x + 2)(x — 7) = 0, you get (x + 2) = 0 and (x — 7) = 0. When you solve for each, you get x = —2 and x = 7. Therefore, —2 and 7 are the solutions or roots of the equation.

Try the following problem.

9.Which of the following is a solution to the equation v² — 11v + 28 = 0?

A)—11

B) 0

C) 4

D) 28

Here’s How to Crack It

The question asks for the solution to a quadratic, which is the value of v that will make the equation true. Now follow the steps:

1. Set the equation to zero. (This has already been done.)

2. Factor the left side to get (v — 7)(v — 4) = 0.

3. Set each factor equal to zero to get v = 4 and v = 7. Since only one of these is an option, the correct answer is (C).

An alternative approach to this question is to plug in the answers. Starting with (B), plug v = 0 into the equation to get 0² — 11(0) + 28 = 0, or 28 = 0. Since this is not true, eliminate (B). It is difficult to know which direction to go after this, so just pick a direction and try it out. Try (C). Plug v = 4 into the equation to (4)² — 11(4) + 28 = 0, or 16 — 44 + 28 = 0. This simplifies to 0 = 0, so you find that the correct answer is (C) using this method as well.

Sometimes, quadratic equations will be tested with word problems. Let’s look at this type of question.

5.A biologist studying a population of birds on an island finds that the equation P = —0.2w² + 10w + 120 can be used to model the number of birds in the population, P, for each number of weeks, w, since she began her study. Which of the following values of w is most useful in determining the number of birds in the population when she began her study?

A)w = 0

B)w = 10

C)w = 20

D)w = 120

Here’s How to Crack It

The question asks about the population when the biologist began her study. The time of her study is measured in weeks, represented by w. When she began the study, she hadn’t put in any time yet, so the value for w would be 0. This matches the value in (A). Another option, given that calculator use is allowed, would be graphing the equation as y = —0.2x² + 10x + 120 to see what the graph looks like. If you are a visual learner, this may help you to better see what is going on with the population at the different times listed in the answer choices. Either way, the correct answer is (A).

SOLVING RATIONAL EQUATIONS

Since you are not always allowed to use your calculator on the PSAT 8/9, there will be some instances in which you will need to solve an equation algebraically. Even on the sections in which calculator use is permitted, you may find it faster and more effective to use your mathematical skills to efficiently answer a question.

Here is an example.

2.Given the proportion , which of the following is the value of s?

A)20

B)36

C)60

D)70

Here’s How to Crack It

The question asks for the value of s in the given proportion. You are asked for a specific value and given numbers in the answer choices, so Plugging In the Answers is an option. However, if the answer you start with doesn’t work, it may not be clear which direction to go next, and you may end up doing a bunch of unnecessary work. Instead, ones like this are faster to solve through cross-multiplication. Doing so results in (s)(10) = (120)(3), which simplifies to 10s = 360. Divide both sides of the equation by 10 to get s = 36. The correct answer is (B).

ABSOLUTE VALUES

Absolute value is just a measure of the distance between a number and 0. Since distances are always positive, the absolute value of a number is also always positive. The absolute value of a number is written as |x|.

When solving for the value of a variable inside the absolute value bars, it is important to remember that variable could be either positive or negative. For example, if |x| = 2, then x = 2 or x = —2 since both 2 and —2 are a distance of 2 from 0.

Here’s an example.

PSAT 8/9 Smoke and Mirrors

When you’re asked to solve an equation involving an absolute value, it is very likely that the correct answer will be the negative result. Why? Because the test-writers know that you are less likely to think about the negative result! Another way to avoid mistakes is to do all the math inside the absolute value symbols first, and then make the result positive.

4.For |2x — 3| = 5, which of the following is a possible value of x?

A)—4

B)—1

C) 0

D) 1

Here’s How to Crack It

The question asks for the value of x that is a solution to the absolute value. You could solve this by taking the part inside the absolute value symbol and setting it equal to 5 and —5, then solving both equations, but there is a lot of room for sign errors on that path. Instead, try PITA. Label the answers as “x” and start with (B), —1. The equation becomes |2(—1) —3| = 5, which simplifies to |—2 —3| = 5. This further simplifies to |—5| = 5, which is true. You can stop there—the correct answer is (B).

GROWTH AND DECAY

Another aspect of percent questions may relate to things that increase or decrease by a certain percent over time. This is known as “growth and decay.” Real-world examples include population growth, radioactive decay, and credit payments, to name a few. While Plugging In can help on these, it is also useful to know the growth and decay formula.

When the growth or decay rate is a percent of the total population:

final amount = original amount (1 ± rate)^{number of changes}

Let’s see how this formula can make quick work of an otherwise tedious question.

y = 300(1.06)^x

6.The equation above describes the total money in an interest-generating savings account x months after it was opened. Which of the following best describes the meaning of the value of y for x = 0?

A)The account had a value of $0 when it was initially opened.

B)The account will have a value of $300 when it has been open for 300 months.

C)The account had a value of $300 when it was initially opened.

D)The account will have a value of $0 when it has been open for 300 months.

Here’s How to Crack It

The question asks for the meaning of the value of y when x = 0. The x in the equation corresponds to the number of changes in the growth and decay formula. If that value is 0, there have been no changes, and the account only contains the original amount. In this case, the original amount was $300, as that is the value in front of the parentheses. This corresponds to the statement in (C). If you forget the formula for growth and decay, never fear! It is unlikely to come up in anything more complicated than this question, and these can be solved with POE and Plugging In, too. The value of x indicates the number of months, so if x = 0, you can eliminate (B) and (D), as those refer to 300 months. The difference between (A) and (C) is the initial value of the account. Plugging x = 0 into the equation results in y = 300(1.06)⁰. Any number raised to the power of 0 becomes 1, so this becomes y = 300(1) = 300. Thus, the value of the account when x = 0 is $300, not $0. Either way, the correct answer is (C).

ANALYSIS IN SCIENCE

If some of the questions you’ve seen so far are reminding you of science class, you’re not crazy. One of the cross-test scores the PSAT 8/9 aims to measure is called Analysis in Science. This means that questions on science-based ideas will show up in Reading and Writing passages and also in Math questions.

One way this concept will be tested is through word problems. Many of the strategies we’ve already discussed, such as translating or Plugging In, will help you to answer these questions, regardless of the scientific context.

Here’s an example.

23.A chemist uses different amounts of organic polymer and silica to produce certain thicknesses of a gel. The equation 3.9P — 0.89S = 40.15 represents one such combination, where P and S represent the amounts of organic polymer and silica, respectively, in milligrams. If the chemist uses 16 milligrams of organic polymer, how many milligrams of silica are used?

Here’s How to Crack It

The question asks how many milligrams of silica are used. Don’t let the scientific context of the question throw you off. There is an equation and a given value, so focus on those. The value of 16 milligrams refers to the polymer, and the question states that the amount of polymer is represent by P. Plug P = 16 into the equation to get 3.9(16) — 0.89S = 40.15. This simplifies to 62.4 — 0.89S = 40.15, then you can subtract 62.4 from both sides to get —0.89S = —22.25. Divide both sides of the equation by —0.89 to get S = 25. This is the amount of silica, so the correct answer is 25.

Sometimes, you will be asked science questions based on a chart or graph. In those cases, carefully look up the numbers in question, do the required calculations, and eliminate answers that aren’t true.

Let’s look at one.

16.The scatterplot below shows how India’s annual rainfall, in centimeters (cm), changed over time after the year 2000.

How many years after 2000 was the year in which India had the greatest annual rainfall?

A) 5

B)10

C)70

D)85

Here’s How to Crack It

The question asks about the year that India had the greatest rainfall. Start with Process of Elimination. The values for years on the horizontal axis only go from 0 to 10, so the correct answer must be in this range. Eliminate (C) and (D), which are answers you might get if you read the wrong axis. The amount of rainfall is measured on the vertical axis, so the greatest rainfall will be the dot closest to the top of the graph. This occurs right on the gridline for year 5, but make sure this represents 5 years after 2000. The question and the axis label both say after 2000, so there is no trick there. The correct answer is (A).

You may also be asked to graph the data presented in a table. Your knowledge of graphing in the xy-plane should help you with most of those. If anything gets too tricky, consider skipping it and spending your time on something else.

Advanced Math Drill

Answers can be found in Part IV.

3.The table below displays coordinate pairs of a function.

Of the equations listed, which of the following correctly displays the relationship between the values of x and f(x) in the table?

A)f(x) = —5x

B)f(x) = —4x + 2

C)f(x) = —2x — 4

D)f(x) = 2x + 2

7.Which of the following functions represents the parabola graphed in the xy-plane above?

A)g(x) = —(x — 4)²

B)g(x) = —x² — 4

C)g(x) = (x — 4)²

D)g(x) = x² — 4

q(x) = — x + 12

12.Given the function above, what value of c will make q(c) equal to 6?

5.There are 210 members of a fitness center. A poll answered by a random sample of 30 members found that 7 of those members own exercise equipment at home. Approximately how many of the total members of the fitness center can be expected to own exercise equipment at home?

A)187

B) 49

C) 37

D) 30

10.The data table above shows the distribution of distance from the Sun for nine planets, including the dwarf planet Pluto, in Earth’s solar system. If an astronomer randomly selects one of these planets to investigate further, what is the probability that the planet’s distance from the Sun is greater than 999 million miles?

A)

B)

C)

D)

13.A group of students observed that there was a relationship between the hours a petri dish was left to incubate and the resulting number of cells in the petri dish. The graph below displays their findings as well as a best-fit line drawn by one of the students.

Which of the following is the correct approximate average hourly increase in the number of cells, based on the best-fit line in the graph?

A) 0.06

B) 0.43

C) 6.00

D)92.00

Summary

o Given a function, you put an x value in and get an f(x) or y-value out.

o Look for ways to use Plugging In and PITA on function questions.

o For questions about the graphs of functions, remember that f(x) = y.

o If the graph contains a labeled point or the question gives you a point, plug it into the equations in the answers and eliminate any that aren’t true.

o To find a point of intersection, set the equations equal, plug a given point into both equations to see if it works, or graph the equations on your calculator when it is allowed.

o To solve rational equations, look for ways to use PITA or just solve them by cross-multiplying.

o The absolute value of a number is the positive distance from zero, or practically, making the thing inside the | | sign positive. Everything inside the | | is equal to the positive and the negative value of the expression to which it is equal. Also remember that | | work like (); you need to complete all the operations inside the | | before you can make the value positive.

o Growth and decay are given by the formula final amount = original amount(1 ± rate)^{number of changes}.

o Analysis in Science questions may seem weird, but they can usually be handled with the same strategies as those used for other math questions. Plug in or translate, read the chart or text carefully, and always use Process of Elimination to get rid of answers that don’t match the data or don’t make sense.