Cracking the SAT Subject Test in Math 2, 2nd Edition (D) (E) Here’s How to Crack It If the function is multiplied by −3, then it must be flipped across the x axis and stretched vertically Eliminate (A[.]
(D) (E) Here’s How to Crack It If the function is multiplied by −3, then it must be flipped across the xaxis and stretched vertically Eliminate (A) and (B), because they are not flipped across the x-axis If the function was stretched vertically, then the difference of the y values between the local minimum at approximately x = −1 and the local maximum at x = 0 must increase (i.e the points must be “stretched” apart) The graph of (D) has the same distance between the local minimums and maximums that the original has, so eliminate it Adding 3 outside of the parentheses would move the graph up 3 units and not displace the graph to the left; eliminate (E) because the graph was moved off the y-axis and choose (C) DEGREES OF FUNCTIONS The degree of a polynomial is the highest degree of any term in the polynomial The degree also determines at most how many distinct roots the polynomial will have For example, the function p(x) = x3 − 4x2 + 7x − 12 is a third-degree function This means that p(x) has at most three distinct roots These roots can be distinct or identical A sixth-degree function can have at most 6 distinct roots It can actually have anywhere from 0 to 6 distinct roots Let’s take a look at two sixth-degree functions: f(x) = x6 g(x) = (x − 1)(x − 2)(x − 3)(x − 4)(x − 5)(x − 6) The function f(x) has six roots, but they’re all the same: f(x) = 0 when x = 0, which makes the function equal • • • • • Basically, the function has six roots of zero—it has only one distinct root The function g(x) has six distinct roots: g(x) = 0 when x = 1, 2, 3, 4, 5, or 6 Another example is that a function might have four roots of 2, a root of 3, and a root of 4, for a total of three distinct roots The equation of this function would look like f(x) = (x − 2)4(x − 3)(x − 4) This is still a sixth-degree function, and it has six roots That’s the algebraic meaning of the degree of a function: It equals the maximum number of roots that the function has Math Vocab Remember that distinct means different The degree of a function tells you a great deal about the shape of the function’s graph Take a look at the graphs on the following pages This is the graph of a first-degree function All first-degree functions are linear functions, whose graphs are straight lines A first-degree function has no extreme values—that is, it has no point which is higher or lower than all of the others A second-degree function is usually a parabola The function graphed above must be at least a second-degree function A second-degree function has one extreme value, a maximum or minimum This function’s extreme value is a minimum A third-degree function can have as many as two local extreme values The function graphed on the previous page, which has a local maximum and a local minimum, must be at least a third-degree function A “local” maximum (or minimum) means the values of the function are bigger (or smaller) than all of the surrounding values, but that the function may be bigger (or smaller) at some distant values of x A fourth-degree function can have as many as three local extreme values The function above has three extreme values, two local maxima and a local minimum between them It must be at least a fourth-degree function An Important Distinction There is an important distinction to make The shape of the graph of f(x) = x6 will be very different from the shape of the graph of g(x) = x3 Even though the root of each is 0, it’s the degree that determines the shape So these graphs would be quite different Take a look at them on your graphing calculator By now, you should see the pattern A fourth-degree function can have a maximum of three extreme values in its graph; a fifth-degree function can have a maximum of four extreme values in its graph This pattern goes on forever An nth-degree function has a maximum of n distinct roots and a maximum of (n − 1) extreme values in its graph These two rules are the basis of a number of SAT Subject Test in Math 2 questions Take a look at the following practice questions DRILL 10: DEGREES OF FUNCTIONS The answers can be found in Part IV 8 If the graph above shows f(x), then f(x) has how many distinct real roots? (A) Three (B) Four (C) Five (D) Six (E) It cannot be determined from the information given 17 If the graph above is a portion of the graph of y = f(x), then which of the following could be f(x) ? (A) ax + b (B) ax2 + bx + c (C) ax3 + bx2 + cx + d (D) ax4 + bx3 + cx2 + dx + e (E) ax5 + bx4 + cx3 + dx2 + ex +f 20 If g(x) is a fourth-degree function, then which of the following could be the definition of g(x) ? (A) g(x) = (x − 3)(x + 5) (B) g(x) = x(x + 1)2 (C) g(x) = (x − 6)(x + 1)(x − 5) (D) g(x) = x(x + 8)(x − 1)2 (E) g(x) = (x − 2)3(x + 4)(x − 3) Fun with Functions As noted at the beginning of the chapter, functions are perhaps the most heavily-tested concept on the SAT Subject Test in Math 2 The techniques discussed in this chapter will help you crack the questions you will find on the test ETS likes to mix concepts and push limits on these questions, so be flexible in your approach, and if you’re stuck, look for ways to Plug In or Plug In the Answers! ... As noted at the beginning of the chapter, functions are perhaps the most heavily-tested concept on the SAT Subject Test in Math 2 The techniques discussed in this chapter will help you crack the questions... basis of a number of SAT Subject Test in Math 2 questions Take a look at the following practice questions DRILL 10: DEGREES OF FUNCTIONS The answers can be found in Part IV 8 If the graph above shows f(x), then f(x) has how many... There is an important distinction to make The shape of the graph of f(x) = x6 will be very different from the shape of the graph of g(x) = x3 Even though the root of each is 0, it’s the degree that determines the shape