Sat - MC Grawhill part 43 ppt

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.. Multiply the number of negative terms per repeti

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 32, and the 30th term

is 32× 329= 327

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)  3 = 2x  3 The third term is

2(2x  3)  3 = 4x  6  3 = 4x  9 The fourth term

is 2(4x  9)  3 = 8x  18  3 = 8x  21 The fifth

term is 2(8x  21)  3 = 16x  42  3 = 16x  45.

2 C Each term in the sequence is the previous term

times 2 The first term, 1⁄8, is equal to 23 To find

the value of the 13th term, multiply the first term

by 2 twelve times or by 212to get your answer

23× 212= 23 + 12= 29

3 15 The first term is 400, after which each term is

20 less than 1⁄2the previous term The second term

is 1⁄2(400)  20 = 180 The third term is 1⁄2(180) 

20 = 70 The fourth term is 1⁄2(70)  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 = 831⁄3 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 1⁄3of 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 5 + (103)(6) = 613

7 B 31= 3; 32= 9; 33= 27; 34= 81; 35= 243; 36= 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 1⁄4of 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 ()(+) =  The

fourth term is (+)() =  The fifth term is ()() = + The sixth term is ()(+) =  The first six terms of the sequence are: , +, , , +,  The pattern , +,  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

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) = ax2 + bx + c, where c is the

y-intercept When a (the coefficient of x2) is

positive, the parabola is “open up,” and when a

is negative, it is “open down.”

y

x

1

f (x) = –x2 + 4x – 3

1

The graph above represents the function y = x2+

4x 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) 3 (B) 2 (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) = 3 (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, 3)

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

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?

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 3 and 3 are shown above Which of

the following describes the set of all x for

which g(x) ≥ f(x)?

(A) x≥3

(B) 3 ≤ x ≤ 1 or 2 ≤x≤3

(C) 1 ≤ 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) x3

(D) 2x+ 1

(E) 2x1

3. What is the least possible value of (x+ 2)2if

3 ≤x≤0?

(A) 3

(B) 2

(C) 1

(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, 2)

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( 1) = 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 3 4

8

f(x)

will always equal 16? Because the slope m of any linear function represents the amount that y in-creases (or dein-creases) whenever x inin-creases 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, (x1, y1)

and (x2, y2), the slope can be calculated with

(Also see Chapter 10, Lesson 4.)

y y

x x

2 1

2 1

−

−

5 Choose any two points on the graph and call their

coordinates (x1, y1) and (x2, y2) Then calculate the slope with

6 f(x) = ax2+ bx + c, where c is the y-intercept

7 It is a parabola that has a vertical line of symmetry through its vertex

y y

x x

2 1

2 1

−

−

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)2given 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)2is the square of a number What is the least

possible square of a real number? It must be 0,

because 02equals 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)2is

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

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 1/2less than h(−1), or 4 − 1/2 = 3.5

Lesson 3: Transformations

Functions with similar equations tend to have similar shapes For instance, functions of the quadratic form f(x) =

ax2+ 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 (3, 4) That is, the x coordinate is negated.

When it is reflected over the x-axis, it becomes (3, 4)

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 = f(x).

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 = x2 5 after it has been reflected over the x-axis?

3 How does the graph of y = 4f(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) = ax2+ 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?

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 = 1/2f(x)

(B) y = f(x 2)

(C) y = f(x + 2)

(D) y = f(x) + 2

(E) y = f(x)  2

2. The figure above shows the graph of the

func-tion y = g(x), which has a minimum value at the

point (1, 2) What is the maximum value of

the function h(x) = 3g(x)  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) (3, 4) (C) (3, 4) (D) (4, 3) (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) = x, and

h(x) = x2, then which of the following represents

the graph of y = g(f(h(x)))?