RISERS

2009-2010

1. If x²+y² = 73 and xy = 24, what is the value of (x + y)² ?

(A) 73 (B) 97 (C) 100 (D) 121 (E) 144

2.

The figure above shows the graphs of y = x² and y = a – x² for some constant a. If the length of PQ is equal to 6, what is the value of a?

(A) 6 (B) 9 (C) 12 (D) 15 (E) 18

3. The quadratic function g is given by g(x) = ax², + bx + c, where a and c are negative constants. Which of the following could be the graph of g?

(A)

(B)

(C)

(D)

(E)

QUADRATIC EQUATIONS 

2009-2010 2009-2010

3.

Which of the following could be the equation of the graph of function g, shown above?

(A) g(x) = x² -3x +2 (B) g(x) = x² -2x +1 (C) g(x) = x² +x + 3 (D) g(x) = x² +2x -1 (E) g(x) = x² +3x -2

5. If x² − 64 = 0, which of the following could be a value of x?

(A) −8 (B) −4 (C) 0 (D) 16 (E) 32

6.

The quadratic function graphed above has equation f(x) = a − x². Which of the following must be true?

(A) a > 0 (B) a < 0 (C) a is even.

(D) a is odd.

(E) a is the square of an integer.

FUNCTIONS 

2009-2010

1. The table above shows some values for the function f. If f is a linear function, what is the value of a + b?

(A) 24 (B) 36

(C) 48 (D) 72

(E) It cannot be determined from the information given.

2. The shaded region in the figure above is bounded by the x–axis, the line x = 4, and the graph of y = f(x).If the point (a, b) lies in the shaded region, which of the following must be true?

I. a 4 II. b a III. b f(a)

(A) I only (B) III only (C) I and II only (D) I and III only (E) I, II, and III

3.

The figure above shows the graph of y = g(x). If the function h is defined by h(x) = g(2x) + 2, what is the value of h(1)?

4. A group of students washed cars to raise money. The net amount A, in dollars, raised by washing k cars is given by the function A(k) = 4k – 30. If the group washed 15 cars, what is the net amount they raised?

(A) $10 (B) $15 (C) $20 (D) $25 (E) $30

6. For which of the following functions is )

7. The figure above shows the graph of the function h. Which of the following is closest to h(5) ? producing x units of a certain product is given by the function c(x) =

x x 200

600 + k,

where k is a constant and x 100. If 20 units were produced yesterday for a

total cost of $640, what is the value of k?

FUNCTIONS 

2009-2010 2009-2010

11. The table above gives values of the quadratic function f for selected values of x.

Which of the following defines f?

(A) f(x) = x² + 1 (B) f(x) = x² + 2 (C) f(x) = 2x² – 2 (D) f(x) = 2x² – 1 (E) f(x) = 2x² + 1

12. The graph of y = f(x) is shown above.

Which of the following could be the graph of y = f(x + 2)?

(A)

(B)

(C)

(D)

(E)

FUNCTIONS 

2009-2010 2009-2010

13. In the figure above, ABCD is a

rectangle. Points A and C lie on the graph of y = px³, where p is a constant. If the area of ABCD is 4, what is the value of p?

14. The total daily profit p, in dollars, from producing and selling x units of a certain product is given by the function

p(x) = 17x – (10x + b), where b is a constant.

If 300 units were produced and sold

yesterday for a total profit of $1,900, what is the value of b?

(A) –200 (B) –100 (C) 0 (D) 100 (E) 200

15. The graph of y = g(x) is shown above.

If g (2) = k, which of the following could be the value of g (k)?

(A) 2 (B) 2.5 (C) 3 (D) 3.5 (E) 5

16. The figures above show the graphs of the functions f and g. The function f is defined by f(x) = x³ – 4x. The function g is defined by g(x) = f(x + h) + k, where h and k are constants. What is the value of hk?

(A) –6 (B) –3 (C) –2 (D) 3 (E) 6

constants, which of the following could be the graph of f?

18. The table above gives values of the linear function g for selected values of t. Which of the following defines g?

(A) g(t)=

Questions 19–20 refer to the following functions g and h.

20. Which of the following is equivalent to h (m + 1)?

FUNCTIONS 

2009-2010 2009-2010

21. Let the function f be defined by

f(x) = x² + 18. If m is a positive number such that f (2m) = 2f (m), what is the value of m?

22. Let the function h be defined by

h (t) = 2(t³ – 3). When h (t) = –60, what is the value of 2 – 3t?

(A) 35 (B) 11 (C) 7 (D) –7 (E) –11

23. The table above represents a relationship between x and y. Which of the following linear equations describes the relationship?

(A) y = x + 1 (B) y = x + 4 (C) y = 3x (D) y = 4x (E) y = 4x – 1

FUNCTIONS 

2009-2010

1. Let the function f be defined by f(x) = 2x – 1.

If 2

1 f ( t ) = 4, what is the value of t?

(A) 2 3 (B) 2

7 (C) 2 9 (D) 4

49 (E) 4

81

2. Based on the graph of the function f above, what are the values of x for which f(x) is negative?

(A) –6 < x < 0 (B) 0 < x < 6 (C) 6 < x < 7 (D) –6 < x < 6

(E) –6 < x < 0 and 6 < x < 7

3. Let the function f be defined by

f(x) = x² + 18. If m is a positive number such that f (2m) = 2f (m), what is the value of m?

4. The rate for a telephone call between City A and City B is 50 cents for the first minute and 30 cents for each additional minute or portion thereof. Which of the following functions describes the cost, in dollars, of a phone call between these two cities that lasts for n minutes, if n is a positive integer?

(A) f(n) = 0.80n

(B) f(n) = 0.50 + 0.30 n (C) f(n) = 0.50 + 0.30 (n + 1) (D) f(n) = 0.50 + 0.30 (n – 1) (E) f(n) = 0.50n + 0.30(n – 1)

5. For all numbers x, the function f is defined by f(x) =(x+4) (x+2). Which of the following has a negative value?

(A) f ( 5) (B) f ( 4) (C) f ( -3) (D) f ( -2) (E) f ( -1)

FUNCTIONS 

2009-2010 2009-2010

6.A weather-watch camera is set so that its shutter opens every 31 seconds. If g (h) represents the number of times the camera’s shutter opens in h hours, which of the

following defines g?

(A) g (h) = 31h

(B) g (h) = 31* 3600h (C) g (h) = 31h/3600 (D) g (h) =3600h/31 (E) g (h) =31*3600/h

7.Let the function s be defined so that s(x) is the area of a semicircle with diameter x. If s (6) +s (8) =s (b), what is the value of b?

8. The table above gives values of the linear function k for selected values of x. Which of the following defines k?

(A) k (x) = -2x +1 (B) k (x) = -x +1 (C) k (x) =

2 1 2 1x (D) k (x) = x -1 (E) k (x) =2x-1

9. For all x ≥ -1, let the function f be defined as f(x) = x+1. Which of the following is equal to f (15) f (8)?

(A) f ( 1) (B) f (0) (C) f (1) (D) f (7) (E) f ( 7 )

10. The table above shows some values for the functions f and g. If f and g are linear functions, what is the value of a b?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

FUNCTIONS 

2009-2010 2009-2010

11. Rita’s dog weighed 5 pounds when she bought it. Over the next several years, the dog’s weight increased by 10 percent per year. Which of the following functions gives the weight, w, in pounds, of the dog after n years of weight gain at this rate?

(A) w(n)=5+0.1n (B) w(n) = 5(0.1)ⁿ (C) w(n) = 5(0.9) ⁿ (D) w(n) = 5(1.1) ⁿ (E) w(n) = 5(n) ^1.1

f x =kg(x) +2

12. The function f above is defined in terms of another function g for all values of x, where k is a constant. If t is a number for which f (t) = 30 and g (t) = 8, what does k equal?

13. Which of the following could be the equation of the graph of function g, shown above?

(A) g(x) = x²-3x +2 (B) g(x) = x² -2x +1 (C) g(x) = x² +x + 3 (D) g(x) = x² +2x -1 (E) g(x) = x² +3x -2

14. Let the function f be defined by f(x) =2x-1.

If2

1 f (t) = 10 what is the value of t?

(A) 9.5 (B) 3 (C) 3 (D) 9.5 (E) 10.5

15.For which of the following functions is it true that f (-2)= f(2)?

(A) f (x) = (B) f (x) = 2 - x (C) f (x)=x+1 (D) f (x)= x²+1 (E) f (x)=2 x³

FUNCTIONS 

2009-2010 2009-2010

16. The graph of a quadratic function and the graph of a linear function in the xy -plane can intersect in at most how many points?

(A) One (B) Two (C) Three (D) Four

(E) More than four

17. Which of the following could be the graph in the xy -plane of the function f (x) = 2x + 3?

A.

B.

C.

D.

E.

FUNCTIONS 

2009-2010

1.

The function f graphed above is defined for

−3 ≤ x ≤ 6. For which of the following values of x is f (x) < f(x) ?

(A) −3 (B) −2 (C) −1 (D) 1 (E) 5

2. For all x, the function f is defined by f (x) = 3x, and the function g is defined by g(x) = f (x) + 2. What is the value of g (50)?

3.

c (n) = 35 + 12n

To hold a birthday party for n people, the amount a party planner charges, in dollars, is given by the function c(n) above. How much, in dollars, does the planner charge to hold a birthday party for 12 people?

(Disregard the $ sign when gridding your answer.)

4.

2 )

( )

( ²

x x t

x x r

The functions r and t are defined above. For how many values of a is it true that

r (a) = t (a)?

(A) None (B) One (C) Two (D) Three

(E) More than three

5.

The function y= f(x), defined for −5≤ x ≤6, is graphed above. Which of the following gives all values of x for which f(x) is positive?

(A) 0< x ≤6 (B) 5< x ≤6 (C) −5≤ x <2

(D) −4< x <1 and 5< x ≤6 (E) −5≤ x < −4 and 1< x <5

FUNCTIONS 

2009-2010 2009-2010

6. If the function f is defined by

3 ) 6

( x

x

f ,

for what value of x does f(x) = 20?

7. If the function g is defined by g(x) =x³ −3x, what is the value of g (−3)?

(A) −36 (B) −18 (C) −3 (D) 18 (E) 36

8.

The table above gives values of the function f for several values of t. If the graph of f is a line, which of the following defines f (t)?

(A) f (t)=t-1 (B) f (t)=t+1 (C) f (t)=2t+1 (D) f (t)=2t-1 (E) f(t)=1-2t

9.

Based on the graph of the function f above, what are the values of x for which f(x) is positive?

(A) -2 x <1 or 8<x< 10 (B) -2 x <1 or 4<x<8 (C) 1<x<4 or 8<x 10 (D) -2 x 10

(E) 1<x<8

10.Let the function f be defined by f(x) = 5x for all numbers x. Which of the following is equivalent to f(p + r)?

(A) 5

r p (B) 5p+r (C) 5p+5r (D) 10 (p + r) (E) 25pr

FUNCTIONS 

2009-2010 2009-2010

11.The graph of y= f(x) is shown above.

If -3 x 6, for how many values of x does f(x) = 2?

(A) None (B) One (C) Two (D) Three

(E) More than three

12.

The graph of y= f(x) is shown above.

If -3 x 6, for how many values of x does f(x) = 2?

(A) None (B) One (C) Two (D) Three

13. If the function f is defined by f(x)=3x + 4, then 2f(x) +4=?

(A) 5x +4 (B) 5x+ 8 (C) 6x +4 (D) 6x+ 8 (E) 6x +12

14.

The graphs of the functions f and g are lines, as shown above. What is the value of

f (3) +g (3)?

(A) 1.5 (B) 2 (C) 3 (D) 4 (E) 5.5

FUNCTIONS 

2009-2010 2009-2010

15.

Based on the portions of the graphs of the functions f and g shown above, what are all values of x between -6 and 6 for which g (x) >f (x)?

(A) 6< x< 3 only (B) 3 <x< 0 only (C) 0 <x< 3 only (D) 3 <x< 6 only

(E) 6 <x< 3 and 0< x< 3

16. Which of the following is the graph of a function f such that f(x) =0 for exactly two values of x between- 5 and 5?

A.

B.

C.

D.

E.

OPERATIONS 

2009-2010

1. Let the operations ∆ and □ be defined for all real numbers a and b as follows.

a ∆ b = a + 3b a □ b = a + 4b

If 4 ∆ (5y) = (5y) □ 4, what is the value of y?

2. Let the operation be defined by b

a b b a

a for all numbers a and b, where a b. If 1 2 = 2 x, what is the value of x?

(A) 4 (B) 3 (C) 2 (D) l (E) 0

3. For all the numbers x and y, let the operation

□

be defined by

x

□

y =xy – y. If a and b are positive integers, which of the following can be equal to zero?

I. a

□

b II. (a + b)

□

b III. a

□

(a + b) (A) I only

(B) II only (C) III only (D) I and II (E) I and III

4. Let be defined as = x² – x for all values of x. If what is the value of a?

(A) 1 (B) 2

1 (C) 2 3 (D) 5 6 (E) 3

OPERATIONS 

2009-2010 2009-2010

5. For all positive integers j and k, let j

®

k be defined as the whole number remainder when j is divided by k. If 13

®

k = 2, what is the value of k?

6. Let be defined as

= x^y z^yfor all positive integers x, y, and z. What is the value of ?

Questions 7-9 refer to the following definition.

Let # be defined by a # b = ab + a + b for all numbers a and b.

7. 2 # 5 = (A) 7 (B) 10 (C) 17 (D) 20 (E) 32

8. If 10 # h = 98, then h = (A) 8

(B) 9 (C) 10 (D) 11 (E) 12

9. For what value of x is the statement x

# y = x always true?

(A) - 2 (B) - 1 (C) 0 (D) 1 (E) 2

10. For all values of y, let y be defined by y = y²- 1.Which of the following is equal to (y ) ?

(A) y⁴ − 1 (B) y⁴ – y² − 1 (C) y⁴ + y²− 1 (D) y⁴ − 2 y² (E) y⁴ − 2 y²+ 1

OPERATIONS 

2009-2010 2009-2010

11. For all positive integers a and b, let ▲ be defined by a▲b = a ²+ b² − 2.

If c and d are positive integers, which of the following CANNOT be the value of c▲d ?

(A) 0 (B) 2 (C) 3 (D) 6 (E) 8

12. For all positive integers w and y, where w > y, let the operation be defined by w y= _w^w _y^y

2 2 .

For how many positive integers w is w 1 equal to 4 ?

(A) None (B) One (C) Two (D) Four

(E) More than four

13. For all numbers x and y, let x y be defined as x y = x² + xy + y². What is the value of (3 1) 1 ?

(A) 5 (B) 13 (C) 27 (D) 170 (E) 183

14. Let x be defined as x+

x 1 for all nonzero integers x. If

x = t, where t is an integer, which of the following is a possible value of t ? (A) 1

(B) 0 (C) −1 (D) −2 (E) −3

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