Cracking the SAT subject test in math 2, 2nd edition

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

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

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 x-axis 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

6 Which of the following graphs is symmetrical with

respect to the x-axis?

(A)

(B)

(C)

(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)

(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) 0 (D) 1 (E) 2

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 7 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)

(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 + 1 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)

47 The graph of f(x) is shown above Which of the following could be the graph of −3f(x) + 3?

(A)

(B)

(C)