CHAPTER 18 Passport to Advanced Math - SAT Suite of ...

1 CHAPTER 18. Passport to Advanced Math Passport to Advanced Math questions include topics that are especially important for students to master before studying Advanced math. Chief among these topics is the understanding of the structure of expressions and the ability to analyze, manipulate, and rewrite these expressions. These questions also include reasoning with more REMEMBER. complex equations and interpreting and building functions. 16 of the 58 questions (28%) on the SAT Math Test are Passport to Heart of algebra questions focus on the mastery of linear equations, Advanced Math questions. systems of linear equations, and linear functions. In contrast, Passport to Advanced Math questions focus on the ability to work with and analyze more complex equations.

2 The questions may require you to demonstrate procedural skill in adding, subtracting, and multiplying polynomials and in factoring polynomials. You may be required to work with expressions involving exponentials, integer and rational exponents, radicals, or fractions with a variable in the denominator. The questions may ask you to solve quadratic, radical, rational, polynomial, or absolute value equations. They may also ask you to solve a system consisting of a linear equation and a nonlinear equation. You may be required to manipulate an equation in several variables to isolate a quantity of interest. Some questions in Passport to Advanced Math will ask you to build a quadratic or exponential function or an equation that describes a context or to interpret the function, the graph of the function, or the solution to the equation in terms of the context.

3 Passport to Advanced Math questions may assess your ability to recognize structure. Expressions and equations that appear complex may use repeated terms or repeated expressions. By noticing these patterns, the complexity of a problem can be reduced. Structure may be used to factor or otherwise rewrite an expression, to solve a quadratic or other equation, or to draw conclusions about the context represented by an expression, equation, or function. You may be asked to identify or derive the form of an expression, equation, or function that reveals information about the expression, equation, or function or the context it represents. 227. PART 3 | Math Passport to Advanced Math questions also assess your understanding of functions and their graphs.

4 A question may require you to demonstrate your understanding of function notation, including interpreting an expression where the argument of a function is an expression rather than a variable. The questions may assess your understanding of how the algebraic properties of a function relate to the geometric characteristics of its graph. Passport to Advanced Math questions include both multiple-choice questions and student-produced response questions. Some of these questions are in the no-calculator portion, where the use of a calculator is not permitted, and others are in the calculator portion, where the use of a calculator is permitted. When you can use a calculator, you must decide whether using your calculator is an effective strategy for that particular question.

5 Passport to Advanced Math is one of the three SAT Math Test subscores, reported on a scale of 1 to 15. Let's consider the content and skills assessed by Passport to Advanced Math questions. Operations with Polynomials and Rewriting Expressions Questions on the SAT Math Test may assess your ability to add, subtract, and multiply polynomials. Example 1. ( x 2 + bx 2)(x + 3) = x 3 + 6x 2 + 7x 6. In the equation above, b is a constant. If the equation is true for all values of x, what is the value of b? A) 2. B) 3. C) 7. REMEMBER D) 9. Passport to Advanced Math questions build on the knowledge To find the value of b, use the distributive property to expand the left-hand and skills tested on Heart of algebra side of the equation and then collect like terms so that the left-hand side questions.

6 Develop proficiency with is in the same form as the right-hand side. Heart of algebra questions before (x 2 + bx 2)(x + 3) = (x 3 + bx 2 2x ) + (3x 2 + 3bx 6). tackling Passport to Advanced Math questions. x 3 + (3 + b ) x 2 + (3b 2) x 6. =. 228. CHAPTER 18 | Passport to Advanced Math Since the two polynomials are equal for all values of x, the coefficient of matching powers of x should be the same. Therefore, comparing the coefficients of x 3 + (3 + b ) x 2 + (3b 2) x 6 and x 3 + 6x 2 + 7x 6. reveals that 3 + b = 6 and 3b 2 = 7. Solving either of these equations gives b = 3, which is choice B. Questions may also ask you to use structure to rewrite expressions. The expression may be of a particular type, such as a difference of squares, or it may require insightful analysis.

7 Example 2. Which of the following is equivalent to 16s4 4t 2? A) 4(s 2 t )(4s 2 + t ). B). PRACTICE AT. C) 4(2s 2 t )(2s 2 + t ). Passport to Advanced Math D) (8s 2 2t )(8s 2 + 2t ) questions require a high comfort level working with quadratic A closer look reveals that the given equation follows the difference of equations and expressions, two perfect squares pattern, x 2 y 2, which factors as (x y ) (x + y ). including multiplying polynomials The expression 16s 4 4t 2 is also the difference of two squares: and factoring. Recognizing classic 2 2. 16s 4 4t 2 = (4s 2) (2t ) . Therefore, it can be factored as quadratic patterns such as x 2 y 2 =. 2 2. (4s 2) (2t ) = (4s 2 2t ) (4s 2 + 2t ).

8 This expression can be rewritten as (x y )(x + y ) can also improve your (4s 2 2t ) (4s 2 + 2t ) = 2(2s 2 t ) (2) (2s 2 + t ) = 4(2s 2 t ) (2s 2 + t ), which is speed and accuracy. choice C. Alternatively, a 4 could be factored out of the given equation: 4(4s 4 t 2). The expression inside the parentheses is a difference of two squares. Therefore, it can be further factored as 4(2s 2 + t )(2s 2 t ). Example 3. Which expression is equivalent to xy 2 + 2xy 2 + 3xy? A) 2xy 2 + 3xy B) 3xy 2 + 3xy C) 6xy 4. D) 6xy 5. There are three terms in the expression, the first two of which are like terms. The like terms can be added together by adding their coefficients: xy 2 + 2xy 2 + 3xy = (xy 2 + 2xy 2) + 3xy, which is equivalent to 3xy 2 + 3xy.

9 Therefore choice B is correct. 229. PART 3 | Math Quadratic Functions and Equations Questions in Passport to Advanced Math may require you to build a quadratic function or an equation to represent a context. Example 4. A car is traveling at x feet per second. The driver sees a red light ahead, and after seconds reaction time, the driver applies the brake. After the brake is applied, the car takes _ x seconds to stop, during which time the average speed 24. x feet per second. If the car travels 165 feet from the time the of the car is _. 2. PRACTICE AT driver saw the red light to the time it comes to a complete stop, which of the following equations can be used to find the value of x?

10 Example 4 requires careful A) x 2 + 48x 3,960 = 0. translation of a word problem into an algebraic equation. It pays to be B) x 2 + 48x 7,920 = 0. deliberate and methodical when C) x 2 + 72x 3,960 = 0. translating word problems into D) x 2 + 72x 7,920 = 0. equations on the SAT. During the reaction time, the car is still traveling at x feet per second, so it travels a total of feet. The average speed of x x the car during the _ -second braking interval is _ feet per second, 24 2. x _ x x2. _. ( )( ) _. so over this interval, the car travels = feet. Since the 2 24 48. total distance the car travels from the time the driver saw the red light to the time it comes to a complete stop is 165 feet, you have the x2.