Sat - MC Grawhill part 44 pot

For in-stance, the statement “y varies directly as x and inversely as w” means that and “y varies inversely as the square of x” means that... D If the force varies inversely as the squar

2 C The function h(x) is equivalent to the function g(x) after it has been reflected over the x-axis,

ver-tically stretched by a factor of 3, and shifted down-ward one unit After these transformations, the vertex of the parabola will be at (1, 5), so the

maximum value is y = 5

3 D Call the original point (a, b) Its reflection over the line y = x is (b, a) (Draw a graph to see.) The reflection of this point over the x-axis is (b, a), and the reflection of this point over the y-axis is

(b, a) If the final point is (3, 4), then the

origi-nal point was (4, 3)

Concept Review 3

1 y = f(x 5) Although the 5 seems to suggest a

shift to the left (because when we subtract 5 from

a number, we move five units to the left on the

number line), this change actually shifts the graph

to the right To see why, look back at the first two

examples in Lesson 3, and pay particular attention

to how the changed equations produce the

individ-ual points on the graph and how these points

com-pare with the points on the original graph It also

may help to pick a simple function, such as y = x2,

graph it by hand (by choosing values for x,

calcu-lating the corresponding values for y, and plotting

the ordered pairs), and then graph y = (x 5)2in

the same way to see how the graphs compare

2 y = x2+ 5 Since the point (x, y) is the reflection

of (x, y) over the x-axis, reflecting any function

over the x-axis simply means multiplying y by 1

This “negates” every term in the function

3 It is the original graph after it has been “flipped”

vertically and “stretched” vertically The graph of

y = 4f(x) is a “vertically stretched” version of

y = f(x) that also has been reflected over the x-axis.

Every point on y = 4f(x) is four times farther from

the x-axis as its corresponding point on y = f(x) and

is also on the opposite side of the x-axis

4 Overall shape and maximum and minimum

val-ues The graph of y = f(x + 15) is simply the graph

of y = f(x) shifted to the left 15 units It maintains

the shape of the original graph and has the same

maximum and minimum values (That is, if the

greatest value of y on y = f(x) is 10, then the

great-est value of y on y = f(x + 15) is also 10.)

5 Zeroes, vertical lines of symmetry, and x

coordi-nates of maximum and minimum values The

graph of y = 6f(x) is simply the graph of y = f(x)

that has been “vertically stretched” by a factor of

6 Imagine drawing the graph of y = f(x) on a

rub-ber sheet and then attaching sticks across the top and bottom of the sheet and pulling the sheet until it’s six times as tall as it was originally The

stretched graph looks like y = 6f(x) Although most

of the points move because of this stretch, the

ones on the x-axis do not These points, called the zeroes because their y coordinates are 0, remain

the same, as do any vertical lines of symmetry and

the x coordinates of any maximum or minimum

points

6 The graph of h(x) = ax2+ bx + c, since it is qua-dratic, looks like a parabola If a is negative, the parabola is “open down.” (Remember that y = x2

is the graph of the “open up” parabola y = x2after

it has been “flipped” over the x-axis.) Also, notice that c is the “y-intercept” of the graph, since h(0) = a(0)2+ b(0) + c = c Therefore, the graph is an up-side-down parabola with a positive y-intercept.

The only choice that qualifies is (B)

7 The point on the d-axis represents the starting

depth of the water If the tank begins with twice as

much water, the starting point, or “d-intercept,”

must be twice that of the original graph Also, if the water drains out at twice the rate, the line must be twice as steep Since twice as much water drains out at twice the rate, the tank should empty

in the same amount of time it took the original tank to drain The only graph that depicts this sit-uation is (A)

Answer Key 3: Transformations

SAT Practice 3

1 D The transformation in (D) is a shift of the

orig-inal function upward two units This creates a

tri-angular region above the x-axis with a greater

height and base than those of the original graph,

and therefore creates a greater overall area The

transformation in (A) will create a triangle with

area 1/2A, the transformations in (B) and (C) are

horizontal shifts, and so will not change the area

The downward shift in (E) will reduce the height

and base, and therefore the total area

4 A A point and its reflection over a line are both

equidistant to that line Imagine that point Q has

a y coordinate of 4 (any value between 1 and 6 will

do) This implies that point Q is two units from the

line y = 6, and therefore, point P also must be two

units from the line y = 6 and must have a y

coordi-nate of 8 Point Q also must be three units from

the line y = 1, so point R also must be three units

from the line y = 1 and must have a y coordinate

of 2 Therefore, the length of PR is 8  (2) = 10.

5 E You can determine the equation defining the

function through substitution: y = g(f(h(x))) = g(f(x2)) = g(x2+ 1)) = x2 1, which describes an

“open-down” parabola with vertex at (0, 1) Notice that this sequence of transformations takes

the standard parabola y = x2and shifts it up one unit and then reflects the new graph over the

x-axis.

Lesson 4: Variation

Direct Variation

The statement “y varies directly as x” means

that the variables are related by the equation

y = kx, where k is a non-zero constant This

equation implies that x and y go up and down

proportionally For instance, whenever x is

multiplied by 3, y is also multiplied by 3

The table and graph above show three examples

of direct variation functions Notice that (1) every

graph passes through the origin, (2) as k increases, so

does the slope of the graph, and (3) for any given k,

whenever x is doubled (or tripled or halved), so is the

corresponding value of y.

Example:

If x varies directly as y and x = 20 when y = 60,

then what is the value of x when y = 150?

First find the value of the constant k by substituting

the values of x and y into the equation y = kx.

y = kx

Now we know that the equation is y = 3x.

Substitute y = 150: 150 = 3x

Inverse Variation

The statement “y varies inversely as x” means

that the variables are related by the equations

y = or xy = k, where k is a non-zero constant These equations imply that x and y go up and down inversely For instance, whenever x is multiplied by 3, y is divided by 3.

The table and graph above show an example of an inverse variation function Notice that (1) the graph

never touches the x-or y-axis, (2) as x increases, y

de-creases, and (3) for every point on the graph, the

product of x and y is always the constant k, in this case k = 1

Example:

If x varies inversely as y and x = 40 when y = 10, then what is the value of x when y = 25?

First find the value of the constant k by substituting the values of x and y into the equation xy = k.

xy = k

Now we know that the equation is xy = 400 Substitute y = 25: x(25) = 400

Joint and Power Variation

A variable can vary with more than one other variable or with powers of a variable For

in-stance, the statement “y varies directly as x and inversely as w” means that and “y varies inversely as the square of x” means that

x

w

k x

Concept Review 4: Variation

1 What equation is equivalent to the statement y varies directly as the square of x?

2 If y varies inversely as x, then the of x and y is a constant

3 If y varies directly as x, then the of x and y is a constant

4 Describe the features of the graph of a direct variation function

5 If w varies directly as v3and w = 16 when v = 2 , what is the value of w when v = 3?

6 If y varies inversely as the square of x, then what will be the effect on y if the value of x is doubled?

7 The variable a varies inversely as b If b = 0.5 when a = 32, then for how many ordered pairs (a , b) are both

a and b positive integers?

8 If x varies directly as the square root of y and directly as z, and if x = 16 when y = 64 and z = 2, what is the value of z when y = 36 and x = 60?

SAT Practice 4: Variation

1. If p varies inversely as q and p = 4 when q = 6,

then which of the following represents another

possible solution for p and q?

(A) p = 8 and q = 12

(B) p = 8 and q = 10

(C) p = 10 and q = 12

(D) p = 12 and q = 1

(E) p = 12 and q = 2

2. Which of the following describes one possible

relationship between the values of m and n

shown in the table above?

(A) n varies directly as m

(B) n varies inversely as m

(C) n varies directly as the square of m

(D) n varies inversely as the square of m

(E) n varies directly as the square root of m

3. If the function f is defined by the equation

f(x, y) = x2y3 and f(a, b) = 10, what is the value

of f(2a, 2b)?

(A) 50

(B) 100

(C) 160

(D) 320

(E) 640

4. At a fixed temperature, the volume of a sample

of gas varies inversely as the pressure of the

gas If the pressure of a sample of gas at a fixed

temperature is increased by 50%, by what

per-cent is the volume decreased?

(E) 100%

331

3

5. The force of gravity between two stars varies in-versely as the square of the distance between the stars If the force of gravity between two stars that are four light-years apart is 64 ex-anewtons (1 exanewton = 1018newtons), what would the force between these stars be if they were eight light-years apart?

(A) 256 exanewtons (B) 128 exanewtons (C) 32 exanewtons (D) 16 exanewtons (E) 8 exanewtons

6. If the variable a varies directly as b and in-versely as c, and if a = 10x + 5 when c = 2 and

b = 10, then what is the value of a when b = 4 and c = 2x + 1?

.

1 2 3 4 5

7 8 9 6

1 0

2 3 4 5

7 8 9 6

1 0

2 3 4 5

7 8 9 6

1 0

2 3 4 5

7 8 9 6

7. If the variable y varies inversely as the square

of x, and if x > 0, then which of the following

operations will double the value of y?

(A) multiplying x by 2

(B) dividing x by 2

(C) multiplying x by

(D) dividing x by

(E) dividing x by 4

2 2

8. If y = 1 when x = 8 and y = 4 when x = 2, which

of the following could express the relationship

between x and y?

I y varies inversely as x

II y varies directly as the square of x

III y varies directly as x

(A) none (B) I only (C) I and II only (D) I and III only (E) I, II, and III

4 B It’s probably easiest to set up the equation, then choose simple values for the volume and pressure, and then “experiment.” Since the vol-ume varies inversely as the pressure, the product

of the volume and the pressure is a constant: vp = k Now choose simple values for v and p, such as 2 and 4: vp = (2)(4) = 8 = k Therefore, in this case,

the product of the volume and the pressure is always 8 If the pressure is increased 50%, then

it grows to 1.5(4) = 6 Now solve for the

corre-sponding value of v:

v(6) = 8

Divide by 6:

Therefore, the volume has decreased from 2 to

To calculate the percent decrease, use the “percent change” formula from Chapter 7, Lesson 5: Percent change =

4

2 3

1

1 3

−

4

3

Concept Review 4

1 y = kx2

2 product

3 quotient or ratio

4 It is a straight line passing through the origin with

a slope equal to k, the constant of proportionality.

For every point on the line, the ratio of the y

coor-dinate to the x coorcoor-dinate is equal to k

5 Write the general variation equation: w = kv3

Substitute w = 16 and v = 2: (16) = k(2)3

Write the specific variation equation: w = 2v3

6 Write the variation equation: y = k/x2 or x2y = k.

Choose any values for x and y: x2y = (1)2(3) = 3 = k

Write the specific variation equation: x2y = 3

Double the original value of x: (2)2y = 3

Simplify: 4y = 3

So what was the effect on y when you doubled the value of x? It went from 3 to 3/4, therefore, it was

divided by 4 or multiplied by 1/4

7 If a varies inversely as b, then ab = k, where k is a constant If b = 0.5 when a = 32, then

k = (0.5)(32) = 16 Therefore, in any ordered pair solution (a, b), the product of a and b must be 16 The only solutions in which a and b are both

pos-itive integers are (1, 16), (2, 8), (4, 4), (8, 2), and (16, 1), for a total of five ordered pairs

8 If x varies directly as the square root of y and directly as z , then First, substitute the

values x = 16, y = 64, and z = 2 to find k:

Substitute y = 36 and x = 60:

60 1 36= z

16= k( )(2 64)

Answer Key 4: Variation

SAT Practice 4

1 E Recall from the lesson that whenever two

vari-ables vary inversely, they have a constant product

The product of 4 and 6 is 24, so every other correct

solution for p and q must have a product of 24

also Choice (E) is the only one that gives values

that have a product of 24

2 D It helps first to notice from the table that as m

increases, n decreases, so any variation

relation-ship must be an inverse variation Therefore, only

choices (B) and (D) are possibilities If n varied

in-versely as m, then the two variables would always

have the same product, but this is not the case:

1  4 = 4, 2  1 = 2, and 4  25 = 1 However, if n

varied inversely as the square of m, then n and m2

would always have the same product This is true:

12 4 = 4, 22 1 = 4, and 42 25 = 4 Therefore,

the correct answer is (D)

3 D You are given that f(a, b) = a2b3= 10 Using the

definition, f(2a, 2b) = (2a)2(2b)3 = (4a2)(8b3)

= 32a2b3 Substituting a2b3 = 10, you get 32a2b3

= 32(10) = 320

5 D If the force varies inversely as the square of

the distance, then the product of the force and

the square of the distance is a constant For these

particular stars, the force times the square of the

distance is (4)2(64) = 1024 If they were eight

light-years apart, then the force would satisfy the

equation (8)2(f) = 1024, so f = 16.

6 4 Set up the variation equation:

a = kb/c

Substitute: 10x + 5 = k(10)/2

Simplify: 10x + 5 = 5k

Divide by 5: 2x + 1 = k

Substitute new values: a = (2x + 1)(4)/(2x + 1)

7 D If y varies inversely as the square of x, then their product x2y is a constant To keep it simple, pick x and y to be 1, so the product (1)2(1) = 1 To

find the value of x that would double y, simply double y and solve for x.

If x2(2) = 1, then This is the original

value of x divided by

8 B Since y increases as x decreases, any variation must be an inverse variation Since the product of

x and y is a constant (1  8 = 4  2 = 8), y varies inversely as x.

2

x= 1 2

Lesson 5: Data Analysis

Scatterplots and Lines of Best Fit

A scatterplot is a collection of points plotted on a

graph that is used to visualize the relationship

be-tween two variables A line of best fit is a straight line

that best “hugs” the data of a scatterplot This line

usually divides the points roughly in half This line

can be used to make predictions about how one of

the variables will change when the other is changed

The SAT may ask you to describe the basic

features of a line of best fit for a set of data, but

it won’t ask you to find this equation exactly.

You can usually just eyeball it: draw a line that

fits the data and cuts the points roughly in

half, and then notice whether the slope of the

line is positive, negative, or 0, and then notice

roughly where its y-intercept is

Example:

If the Smiths own as many pieces of electronic equipment as the Carsons do, how many cell phones do the Smiths own?

Start with the first table The Carsons own 3 + 4 +

2 = 9 pieces of equipment If the Smiths own the same number of pieces, they must own 9  4 = 5 tele-phones Now go to the “Telephones” table Since the Smiths have five telephones and two are not cell phones, they must own 5  2 = 3 cell phones

As with tables, always carefully read the labels

of pie charts to understand what the data

rep-resent before tackling the question In a pie

chart, a sector containing x% of the data has a central angle of (x/100)(360°).

Example:

In the pie chart above, what is the angle measure

of the sector represented by the color purple? Purple accounts for 20% of the circle, and 20%

of 360° = (0.2)(360°) = 72°

Example:

What is the approximate slope of the line of best

fit in the scatterplot above?

To estimate the slope of a line of best fit, pick two

points on your line of best fit and use the slope equation

(from Chapter 10, Lesson 4) to solve for m It appears

that the line of best fit connects the points (30, 40)

Tables, Charts, and Graphs

A table is a set of data arranged in rows and

columns If an SAT question includes a table,

al-ways read the table carefully first, paying special

attention to the axis labels, and try to understand

what the table represents before tackling the

question If you are given two tables, make sure

you understand how the two tables are related

90 40

90 30

50 60 5 6

Concept Review 5: Data Analysis

1 What is a line of best fit?

2 How is a line of best fit created?

3 How do you estimate the slope of a best fit line?

4 When given percentages in a pie chart, how do you determine how many degrees each sector represents?

Questions 5 and 6 refer to the bar graph at right:

5 The largest percent increase in number of accidents occurred between

which two days of the week?

6 Approximately what percentage of the accidents occurred on Friday,

Saturday, or Sunday?

Questions 7–9 refer to the pie chart at right:

7 The pie chart shows the results of a survey that asked 4,000 kindergarten

students their favorite color How many more students said yellow was

their favorite color than said blue was their favorite color?

8 What is the degree measure of the sector of the circle that represents red?

9 Using the data in the pie chart, how many students would have to change

their answer to blue in order for blue to account for 50% of the data?

Questions 10–11 refer to the tables at right:

10 According to the tables, which school ordered the most

amusement park tickets?

11 If all the students who went to Coaster Heaven from New

Haven Public and Hamden High bought 5-day passes, how

much more money did Hamden High spend than New

Haven Public did at Coaster Heaven?

5 10 15

Car Accidents (in thousands)

Sundays Mondays T

W Thursdays

HIGHWAY ACCIDENT FREQUENCY 2003