Cracking the SAT Subject Test in Math 2, 2nd Edition 25 If the graph of y =f(x) is shown above, which of the following sets represents all the roots of f(x) ? (A) {x = −2, 0, 2} (B) {x = −4, −1, 0} (C[.]
25 If the graph of y =f(x) is shown above, which of the following sets represents all the roots of f(x) ? (A) {x = −2, 0, 2} (B) {x = −4, −1, 0} (C) {x = −1, 2} (D) {x = −4, −1, 2} (E) {x = −4, −1} SYMMETRY IN FUNCTIONS Symmetry Across the y-Axis (Even Functions) Some questions on the SAT Subject Test in Math 2 will ask about lines or points of symmetry of functions The most common line of symmetry to be asked about is the y-axis Imagine drawing a function symmetrically across the y-axis on a piece of paper If the paper were folded along the y-axis, the left and right halves of the graph would meet perfectly Functions with symmetry across the y-axis are sometimes called even functions This is because functions with only even exponents have this kind of symmetry, even though they are not the only even functions Look at the following graphs of even functions This is the algebraic definition of symmetry across the y-axis: A function is symmetrical across the y-axis when f(−x) = f(x) This means that the negative and positive versions of any x-value produce the same y-value Origin Symmetry (Odd Functions) A function has origin symmetry when one half of the graph is identical to the other half and reflected across the point (0, 0) Functions with origin symmetry are sometimes called odd functions, because functions with only odd exponents (as well as some other functions) have this kind of symmetry This is the algebraic definition of origin symmetry: A function has origin symmetry when f(−x) = −f(x) This means that the negative and positive versions of any x-value produce opposite y-values Symmetry Across the x-Axis Some equations will produce graphs that are symmetrical across the xaxis These equations can’t be functions, however, because each x-value would then have to have two corresponding y-values A graph that is symmetrical across the x-axis automatically fails the vertical-line test Questions asking about symmetry generally test basic comprehension of these definitions It’s also important to understand the connection between these algebraic definitions and the appearance of graphs with different kinds of symmetry DRILL 9: SYMMETRY IN FUNCTIONS Try these practice questions The answers can be found in Part IV Which of the following graphs is symmetrical with respect to the x-axis? (A) (B) (C) (D) (E) 17 If an even function is one for which f(x) and f(−x) are equal, then which of the following is an even function? (A) g(x) = 5x + 2 (B) g(x) = x (C) g(x) = (D) g(x) = x3 (E) g(x) = −|x| 30 If an odd function is one for which f(−x) = −f(x), and g(x) is an odd function, then which one of the following could be the graph of g(x)? (A) (B) (C) (D) (E) 34 Which of the following is true for the function ? I f(x) is even II f(x) is odd III f(x) is symmetrical across the x-axis (A) None of the above (B) I only (C) II only (D) III only (E) I, II, and III Periodic Functions A periodic function is a function that repeats a pattern of range values forever Always look for a pattern when you’re dealing with a periodic function Remember? We also talked about periodic functions in the Trigonometry section 40 Two cycles of periodic function f are shown in the graph of y = f(x) above What is the value of f(89) ? (A) −2 (B) −1 (C) (D) (E) Here’s How to Crack It In this question, we need to find the period of the function, that is, how many units along the x-axis the function covers before it repeats its range values Find the pattern From peak to peak, it goes from x = 1 to x = 8 This means that the function repeats itself every 7 units (the period is 7) Where does 89 fall in this pattern? Well, you want to take away multiples of from 89, to find out an equivalent range value on the graph above So, f(89) = f(82) = f(75)…and so on Since 89 ÷ 7 = 12 remainder 5, this means that f(89) = f(5) From the graph, f(5) = −1, and the answer is (B) Transformation of Function Graphs When giving you a function question, ETS may decide to fool around with the variable Sometimes you’ll be asked how this affects the graph of the function For example, ETS may show you f(x) and ask you about the graph of |f(x)| You can either Plug In points or know the following rules In relation to f(x): • f(x) + c is shifted upward c units in the plane • f(x) − c is shifted downward c units in the plane • f(x + c) is shifted to the left c units in the plane • f(x − c) is shifted to the right c units in the plane • −f(x) is flipped upside down over the x-axis • f(−x) is flipped left-right over the y-axis • |f(x)| is the result of flipping upward all of the parts of the graph that appear below the x-axis • Cf(x) is stretched vertically when C > 1 Positive y-values become bigger and negative y-values become smaller • or f(x) is stretched horizontally Of course, you may have to combine these rules If so, Plugging In some points may be the easiest way to go 31 The graph of y = f(x) is shown above Which of the following is the graph of y =−f(x + 1) ? (A) (B) (C) (D) (E) Mirror, Mirror on the Axis A function that seems to have a mirror image reflected in the y-axis is symmetrical across the y-axis Here’s How to Crack It To figure out what happens to the graph of f(x), just use the rules above The x + inside the parentheses shifts the graph one unit to the left If this were the final answer, the vertex would be at (−1, 1) Now you have to take care of the negative sign outside the function It reflects the entire function across the x-axis, so the vertex gets reflected to (−1, −1) and the parabola opens upward If you reflected first and then shifted to the left, you’d get the same result The answer is (A) Let’s look at one more: 47 The graph of f(x) is shown above Which of the following could be the graph of −3f(x) + 3? (A) (B) (C) ... To figure out what happens to the graph of f(x), just use the rules above The x + inside the parentheses shifts the graph one unit to the left If this were the final answer, the vertex would be at (−1, 1) Now you have to take care of the negative... about the graph of |f(x)| You can either Plug In points or know the following rules In relation to f(x): • f(x) + c is shifted upward c units in the plane • f(x) − c is shifted downward c units in the plane... Of course, you may have to combine these rules If so, Plugging In some points may be the easiest way to go 31 The graph of y = f(x) is shown above Which of the following is the graph of y =−f(x + 1) ?