410 McGRAW-HILL’S SAT Concept Review 1 1. A sequence is simply a list of numbers, each of which is called a “term.” 2. If the sequence repeats every six terms, you can find the 115th term by finding the remainder when 115 is divided by 6. Since 115 ÷ 6 equals 19 with a remainder of 1, the 115th term will be the same as the first term. 3. Begin by finding the sum of the repeating pattern. Next, determine how many times the pattern oc- curs in the first 32 terms: 32 ÷ 4 = 8 times. Then multiply the sum of the pattern by 8 to obtain the sum. 4. Count the number of negative terms in each repeti- tion of the pattern, then find how many times the pattern repeats in the first 36 terms. Since 36 ÷ 5 = 7 with a remainder of 1, the pattern repeats 7 times and is 1 term into the eighth repetition. Multiply the number of negative terms per repetition by 7, and if the first term of the sequence is negative, add 1 to the total. 5. This is a geometric sequence. Each term is the previous one times 3 (1 × 3 = 3; 3 × 3 = 9, etc.). The first term of the sequence is 3 Ϫ2 , and the 30th term is 3 Ϫ2 × 3 29 = 3 27 . 6. The first term is 4. Second: 4(2) + 8 = 16. Third: 16(2) + 8 = 40. Fourth: 40(2) + 8 = 88. Fifth: 88(2) + 8 = 184. Sixth: 184(2) + 8 = 376. 7. The pattern repeats every five terms, and each rep- etition contains two vowels. Since 143 ÷ 5 = 28 with a remainder of 3, the first 143 letters contain 28 × 2 = 56 vowels plus the one vowel in the first three letters of the word SCORE, for a total of 56 + 1 = 57. 8. Work backwards: x was found by subtracting 3 from the second term and dividing by 2. Therefore, mul- tiply x by 2 and add 3 to get the second term: 2x + 3. Repeat to find the first term: 2(2x + 3) + 3 = 4x + 9. 9. The integers 1 through 9 represent the first 9 digits, and 10 through 19 represent the next 20 digits. Each integer thereafter contains 2 digits. 26 represents the 42nd and 43rd digits, so 2 is the 44th digit. Answer Key 1: Sequences SAT Practice 1 1. D The first term of the sequence is x. The second term is 2(x) Ϫ 3 = 2x Ϫ 3. The third term is 2(2x Ϫ 3) Ϫ 3 = 4x Ϫ 6 Ϫ 3 = 4x Ϫ 9. The fourth term is 2(4x Ϫ 9) Ϫ 3 = 8x Ϫ 18 Ϫ 3 = 8x Ϫ 21. The fifth term is 2(8x Ϫ 21) Ϫ 3 = 16x Ϫ 42 Ϫ 3 = 16x Ϫ 45. 2. C Each term in the sequence is the previous term times 2. The first term, 1 ⁄ 8 , is equal to 2 Ϫ3 . To find the value of the 13th term, multiply the first term by 2 twelve times or by 2 12 to get your answer. 2 Ϫ3 × 2 12 = 2 Ϫ3 + 12 = 2 9 3. 15 The first term is 400, after which each term is 20 less than 1 ⁄ 2 the previous term. The second term is 1 ⁄ 2 (400) Ϫ 20 = 180. The third term is 1 ⁄ 2 (180) Ϫ 20 = 70. The fourth term is 1 ⁄ 2 (70) Ϫ 20 = 15. 4. C The sequence contains a repeating six-term pattern: 148285. To find out how many times the pattern repeats in the first 500 terms, divide 500 by 6: 500 ÷ 6 = 83 1 ⁄ 3 . By the 500th term, the pattern has repeated 83 full times and is 1 ⁄ 3 of the way through the 84th repetition. Each repetition of the pattern contains two odd digits, so in the 83 full repetitions there are 83 × 2 = 166 odd digits. In the first 1 ⁄ 3 of the pattern there is one odd digit. Therefore there are 166 + 1 = 167 odd digits. 5. B There will be 44 + 45 + 46 = 135 6s between the 44th and 47th appearances of 5. 6. B In this arithmetic sequence you must add 6 to each term. To get from the 1st to the 104th term you will add 103 terms, or 103 6s. The value of the 104th term is thus Ϫ5 + (103)(6) = 613. 7. B 3 1 = 3; 3 2 = 9; 3 3 = 27; 3 4 = 81; 3 5 = 243; 3 6 = 729. The units digits repeat in the pattern 3, 9, 7, 1, 3, 9, 7, 1, . . ., and 36 ÷ 4 = nine full repetitions. Since it goes in evenly, it must fall on the last term of the pattern, which is 1. 8. 70 The pattern alternates back and forth between 210 and 70. Each odd-numbered term is 210 and each even-numbered term is 70, so the 24th term is 70. 9. 5/8 or .625 The first term of the sequence is 640. Each term thereafter is 1 ⁄ 4 of the immediately pre- ceding term. The first six terms are 640, 160, 40, 10, 2.5, .625 (.625 = 5/8). 10. D The third term of the sequence is 4 ÷ 2 = 2. The fourth term is 2 + 8 = 10. The fifth term is 10 ÷ 2 = 5. The sixth term is 5 + 8 = 13. The seventh term is 13 ÷ 2 = 6.5, which is the first noninteger term. 11. D In this problem, only the signs of the terms matter. The first term is negative and the second is positive. The third term is (Ϫ)(+) = Ϫ. The fourth term is (+)(Ϫ) = Ϫ. The fifth term is (Ϫ)(Ϫ) =+. The sixth term is (Ϫ)(+) = Ϫ. The first six terms of the sequence are: Ϫ, +, Ϫ, Ϫ, +, Ϫ. The pattern Ϫ, +, Ϫ repeats every three terms. In the first 90 terms, the pattern repeats 90 ÷ 3 = 30 times. Each repetition contains two negative numbers, so in 30 full repetitions there are 30 × 2 = 60 negative numbers. CHAPTER 11 / ESSENTIAL ALGEBRA 2 SKILLS 411 412 McGRAW-HILL’S SAT Lesson 2: Functions What Is a Function? A function is any set of instructions for turning an input number (usually called x) into an output number (usually called y). For instance, f(x) = 3x + 2 is a function that takes any input x and multiplies it by 3 and then adds 2. The result is the output, which we call f(x) or y. If f(x) = 3x + 2, what is f(2h)? In the expression f(2h), the 2h represents the input to the function f. So just substitute 2h for x in the equation and simplify: f(2h) = 3(2h) + 2 = 6h +2. Functions as Equations, Tables, or Graphs The SAT usually represents a function in one of three ways: as an equation, as a table of inputs and outputs, or as a graph on the xy-plane. Make sure that you can work with all three represen- tations. For instance, know how to use a table to verify an equation or a graph, or how to use an equation to create or verify a graph. Linear Functions A linear function is any function whose graph is a line. The equations of linear functions always have the form f(x) = mx + b, where m is the slope of the line, and b is where the line intersects the y-axis. (For more on slopes, see Chapter 10, Lesson 4.) The function f(x) = 3x + 2 is linear with a slope of 3 and a y-intercept of 2. It can also be represented with a table of x and y (or f(x)) values that work in the equation: also that the y-intercept is the output to the function when the input is 0. Now we can take this table of values and plot each ordered pair as a point on the xy-plane, and the result is the graph of a line: x f (x) –2 2 3 4 1 –1 0 –4 5 8 11 14 –1 2 y x 1 f(x) = 3x + 2 1 Quadratic Functions The graph of a quadratic function is always a parabola with a vertical axis of symmetry. The equations of quadratic functions always have the form f(x) = ax 2 + bx + c, where c is the y-intercept. When a (the coefficient of x 2 ) is positive, the parabola is “open up,” and when a is negative, it is “open down.” y x 1 f(x) = –x 2 + 4x – 3 1 The graph above represents the function y = Ϫx 2 + 4x Ϫ 3. Notice that it is an “open down” parabola with an axis of symmetry through its vertex at x = 2. The figure above shows the graph of the function f in the xy-plane. If f(0) = f(b), which of the following could be the value of b? (A) Ϫ3 (B) Ϫ2 (C) 2 (D) 3 (E) 4 Although this can be solved algebraically, you should be able to solve this problem more simply just by inspecting the graph, which clearly shows that f(0) = Ϫ3. (You can plug x = 0 into the equation to verify.) Since this point is two units from the axis of symmetry, its reflection is two units on the other side of the axis, which is the point (4, Ϫ3). Notice several important things about this table. First, as in every linear function, when the x values are “evenly spaced,” the y values are also “evenly spaced.” In this table, whenever the x value increases by 1, the y value increases by 3, which is the slope of the line and the coefficient of x in the equation. Notice CHAPTER 11 / ESSENTIAL ALGEBRA 2 SKILLS 413 Concept Review 2: Functions 1. What is a function? 2. What are the three basic ways of representing a function? 3. What is the general form of the equation of a linear function, and what does the equation tell you about the graph? 4. How can you determine the slope of a linear function from a table of its inputs and outputs? 5. How can you determine the slope of a linear function from its graph? 6. What is the general form of the equation of a quadratic function? 7. What kind of symmetry does the graph of a quadratic function have? 414 McGRAW-HILL’S SAT SAT Practice 2: Functions y x 1 1 y = f(x) y = g(x) 1. The graphs of functions f and g for values of x between Ϫ3 and 3 are shown above. Which of the following describes the set of all x for which g(x) ≥ f(x)? (A) x ≥ Ϫ3 (B) Ϫ3 ≤ x ≤ Ϫ1 or 2 ≤ x ≤ 3 (C) Ϫ1 ≤ x ≤ 2 (D) 1 ≤ x ≤ 6 (E) 3 ≤ x ≤ 5 2. If f(x) = x + 2 and f(g(1)) = 6, which of the following could be g(x)? (A) 3x (B) x + 3 (C) x Ϫ3 (D) 2x + 1 (E) 2x Ϫ1 3. What is the least possible value of (x + 2) 2 if Ϫ3 ≤ x ≤ 0? (A) Ϫ3 (B) Ϫ2 (C) Ϫ1 (D) 0 (E) 1 4. The table above gives the value of the linear func- tion f for several values of x. What is the value of a + b? (A) 8 (B) 12 (C) 16 (D) 24 (E) It cannot be determined from the information given. 5. The graph on the xy-plane of the quadratic function g is a parabola with vertex at (3, Ϫ2). If g(0) = 0, then which of the following must also equal 0? (A) g(2) (B) g(3) (C) g(4) (D) g(6) (E) g(7) 6. In the xy-plane, the graph of the function h is a line. If h(Ϫ1) = 4 and h(5) = 1, what is the value of h(0)? (A) 2.0 (B) 2.2 (C) 3.3 (D) 3.5 (E) 3.7 x 2 34 8a bf(x) will always equal 16? Because the slope m of any linear function represents the amount that y in- creases (or decreases) whenever x increases by 1. Since the table shows x values that increase by 1, a must equal 8 − m, and b must equal 8 + m. There- fore a + b = (8 − m) + (8 + m) = 16. 5. D Don’t worry about actually finding the equation for g(x). Since g is a quadratic function, it has a vertical line of symmetry through its vertex, the line x = 3. Since g(0) = 0, the graph also passes through the origin. Draw a quick sketch of a parabola that passes through the origin and (3, −2) and has an axis of symmetry at x = 3: Concept Review 2 1. A set of instructions for turning an input number (usually called x) into an output number (usually called y). 2. As an equation (as in f(x) = 2x), as a table of input and output values, and as a graph in the xy-plane. 3. f(x) = mx + b, where m is the slope of the line and b is its y-intercept. 4. If the table provides two ordered pairs, (x 1 , y 1 ) and (x 2 , y 2 ), the slope can be calculated with . (Also see Chapter 10, Lesson 4.) yy xx 21 21 − − 5. Choose any two points on the graph and call their coordinates (x 1 , y 1 ) and (x 2 , y 2 ). Then calculate the slope with . 6. f(x) = ax 2 + bx + c, where c is the y-intercept. 7. It is a parabola that has a vertical line of symmetry through its vertex. yy xx 21 21 − − Answer Key 2: Functions SAT Practice 2 1. C In this graph, saying that g(x) ≥f(x) is the same as saying that the g function “meets or is above” the f function. This is true between the points where they meet, at x =−1 and x = 2. 2. B Since f(x) = x + 2, f(g(1)) must equal g(1) + 2. Therefore g(1) + 2 = 6 and g(1) = 4. So g(x) must be a function that gives an output of 4 when its input is 1. The only expression among the choices that equals 4 when x = 1 is (B) x + 3. 3. D This question asks you to analyze the “outputs” to the function y = (x + 2) 2 given a set of “inputs.” Don’t just assume that the least input, −3, gives the least output, (−3 + 2) 2 = 1. In fact, that’s not the least output. Just think about the arithmetic: (x + 2) 2 is the square of a number. What is the least possible square of a real number? It must be 0, because 0 2 equals 0, but the square of any other real number is positive. Can x + 2 in this problem equal 0? Certainly, if x =−2, which is in fact one of the allowed values of x. Another way to solve the problem is to notice that the function y = (x + 2) 2 is quadratic, so its graph is a parabola. Choose values of x between −3 and 0 to make a quick sketch of this function to see that its vertex is at (−2, 0). 4. C Since f is a linear function, it has the form f(x) = mx + b. The table shows that an input of 3 gives an output of 8, so 3m + b = 8. Now, if you want, you can just “guess and check” values for m and b that work, for instance, m = 2 and b = 2. This gives the equation f(x) = 2x + 2. To find the missing outputs in the table, just substitute x = 2 and then x = 4: f(2) = 2(2) + 2 = 6 and f(4) = 2(4) + 2 = 10. Therefore, a + b = 6 + 10 = 16. But how do we know that a + b CHAPTER 11 / ESSENTIAL ALGEBRA 2 SKILLS 415 y x y = g(x) O The graph shows that the point (0, 0), when reflected over the line x = 3, gives the point (6, 0). Therefore g(6) is also equal to 0. 6. D The problem provides two ordered pairs that lie on the line: (−1, 4) and (5, 1). Therefore, the slope of this line is (4 − 1)/(−1 − 5) =−3/6 =−1/2. Therefore, for every one step that the line takes to the right (the x direction), the y value decreases by 1 / 2 . Since 0 is one unit to the right of −1 on the x-axis, h(0) must be 1 / 2 less than h(−1), or 4 − 1/2 = 3.5. 416 McGRAW-HILL’S SAT Lesson 3: Transformations Functions with similar equations tend to have similar shapes. For instance, functions of the quadratic form f(x) = ax 2 + bx + c have graphs that look like parabolas. You also should know how specific changes to the function equation produce specific changes to the graph. Learn how to recognize basic transformations of functions: shifts and reflections. To learn how changes in function equations produce changes in their graphs, study the graphs below until you understand how graphs change with changes to their equations. Horizontal Shifts The graph of y = f(x + k) is simply the graph of y = f(x) shifted k units to the left. Similarly, the graph of y = f(x – k) is the graph of y = f(x) shifted k units to the right. The graphs below show why. Vertical Shifts The graph of y = f(x) + k is simply the graph of y = f(x) shifted k units up. Similarly, the graph of y = f(x) – k is the graph of y = f(x) shifted k units downward. The graphs below show why. Reflections When the point (3, 4) is reflected over the y-axis, it be- comes (Ϫ3, 4). That is, the x coordinate is negated. When it is reflected over the x-axis, it becomes (3, Ϫ4). That is, the y coordinate is negated. (Graph it and see.) Likewise, if the graph of y = f(x) is reflected over the x-axis, it becomes y = Ϫf(x). CHAPTER 11 / ESSENTIAL ALGEBRA 2 SKILLS 417 Concept Review 3:Transformations 1. What equation describes the function y = f(x) after it has been shifted to the right five units? 2. What equation describes the function y = x 2 Ϫ 5 after it has been reflected over the x-axis? 3. How does the graph of y = Ϫ4f(x) compare with the graph of y = f(x)? 4. What specific features do the graphs of y = f(x) and y = f(x + 15) have in common? 5. What specific features do the graphs of y = f(x) and y = 6f(x) have in common? 6. The quadratic function h is given by h(x) = ax 2 + bx + c, where a is a negative constant and c is a positive constant. Which of the following could be the graph of y = h(x)? 7. The figure above is a graph showing the depth of water in a rectangular tank that is being drained at a con- stant rate over time. Which of the following represents the graph of the situation in which the tank starts with twice as much water, and the water drains out at twice the rate? 418 McGRAW-HILL’S SAT SAT Practice 3:Transformations 1. The shaded region above, with area A, indicates the area between the x-axis and the portion of y = f(x) that lies above the x-axis. For which of the following functions will the area between the x-axis and the portion of the function that lies above the x-axis be greater than A? (A) y = 1 / 2 f(x) (B) y = f(x Ϫ 2) (C) y = f(x + 2) (D) y = f(x) + 2 (E) y = f(x) Ϫ 2 2. The figure above shows the graph of the func- tion y = g(x), which has a minimum value at the point (1, Ϫ2). What is the maximum value of the function h(x) = Ϫ3g(x) Ϫ 1? (A) 7 (B) 6 (C) 5 (D) 4 (E) It cannot be determined from the infor- mation given. 3. A point is reflected first over the line y = x, then over the x-axis, and then over the y-axis. The resulting point has the coordinates (3, 4). What were the coordinates of the original point? (A) (3, 4) (B) (Ϫ3, Ϫ4) (C) (3, Ϫ4) (D) (Ϫ4, Ϫ3) (E) (4, 3) Note: Figure not drawn to scale. 4. In the figure above, point Q is the reflection of point P over the line y = 6, and point R is the re- flection of point Q over the line y = 1. What is the length of line segment PR? (A) 10 (B) 11 (C) 12 (D) 13 (E) 14 5. If the functions f(x), g(x), and h(x) are defined by the equations f(x) = x + 1, g(x) = Ϫx, and h(x) = x 2 , then which of the following represents the graph of y = g(f(h(x)))? CHAPTER 11 / ESSENTIAL ALGEBRA 2 SKILLS 419 . 418 McGRAW-HILL’S SAT SAT Practice 3:Transformations 1. The shaded region above, with area A, indicates the area between the x-axis and the portion of y = f(x) that lies above the x-axis. For. 1 / 2 . Since 0 is one unit to the right of −1 on the x-axis, h(0) must be 1 / 2 less than h(−1), or 4 − 1/2 = 3.5. 416 McGRAW-HILL’S SAT Lesson 3: Transformations Functions with similar equations. quadratic function? 7. What kind of symmetry does the graph of a quadratic function have? 414 McGRAW-HILL’S SAT SAT Practice 2: Functions y x 1 1 y = f(x) y = g(x) 1. The graphs of functions f and g