Square Root Function

Transcription

1 Square Root Function D Eample of an appropriate solution Equation of square root function R() = a h k 5 = a = 3a = 3a 3 = a R() = 3 4 Zero of the function R() = = = 3 4 = 3

2 6 = 9 = Answer : The company began to make a profit after.8 months. Accept an answer in [.7; 3]. 3 C 4 The graph of the real function is f() (9, ) (8, ) 0 (5, 0) (0, -)

3 5 C 6 B 7 C 8 C 9 C 0 The solution set of the inequality is [7, 3[ ]43, +[.

4 Eample of an appropriate method Finding the rules of the form y a b h k Function : (h, k) = (0, 3) (, y) = (4, ) Function : (h, k) = (, 0) (, y) = (6, ) Let b = Let b = a 4 0 = 3 a 4 = - a = - y = = a 0 a 4 = a = y = Finding the time at the intersection - 3 = - 3 = 6 9 = - 6 = - = 6

5 3.6 Time wanted 3.36 =.36 Answer:.36 seconds after it has been launched, the nd projectile will be higher than the st projectile. A

6 3 Eample of an appropriate solution Equation of the square root function Verte (0,,4) Point (6, 0) f A b h 0 A a a a A k.4 b So, f() = Coordinates of the point where she stops (0, h) f h h Answer: Carol stops at a height of 0.5 m.

7 4 Eample of an appropriate method Rule of the function : time in seconds f() : altitude in metres f() = a h k f() = a 0 0 f() = a 0 f(5) = 0 therefore 0 = a 5 0 f() = = 5a - = a Time at which Caroline entered the tunnel f() = 6 therefore 6 = = - = 4 = After 4 seconds, Caroline entered the tunnel. Time at which Caroline eited the tunnel f() = 4 therefore 4 = - 0

8 -6 = - 3 = 9 = After 9 seconds, Caroline eited the tunnel. Time during which Caroline was in the tunnel 9 4 = 5 seconds Answer: Caroline was in the tunnel for 5 seconds. Note: Students who used an appropriate method in order to determine the rule of the function have shown that they have a partial understanding of the problem. 5 B

9 6 Eample of an appropriate solution f cos 5 3 Maimum 3 + = 5 The verte of the square root function is (5, 4), the base of the cup is at a height of.59 cm in the Cartesian plane (5. 4). Calculation of the value of when g() =.59 (where h() =.59). y (5, 4) g The equation of the ais of symmetry is = 5. Therefore the width of the foot is (6.49 5) =.98 Answer: The width of the foot of the cup is 3 cm.

10 7 A

11 8 Eample of an appropriate solution y-ais (4, 3) (, ) (0, 5) -ais The square root function must be in the form: y = a + k Substituting (0, 5) we get: 5 = a 0 + k So k = 5 Substituting (4, 3) we get: 3 = a So a = 4 So the function is: y = At the ring, y = = So =.5 But.5 represents the radius of the gold ring in centimetres.

12 So the circumference is: C = (.5) 4.3 cm Therefore the gold ring will cost: 4.3 = 8.6 cents. Answer Rounded to the nearest cent, the cost of the gold ring is 8 cents. Note Do not deduct any marks if the student wrote 9 cents.

13 9 Eample of an appropriate method Step : Find the height of the rocket at 5 seconds 500 (50, 500) H(5) = 00 5 = 00(5) = 000 m Height (m) (36, y) Step : Find the equation of the nd stage (5, 000) H (t) = a h k Time since launch (sec.) Using (h, k) = (5, 000), we get H (t) = a Substituting (50, 500), we get 500 = a = a = 5a a = 500 a = 300 Therefore, H (t) = seconds after the firing of the second stage (at 5 sec.) is 36 sec.

14 Step 3: Find the image of 36 in H H (36) = m Answer Rounded to the nearest metre, the height of the rocket seconds after firing of the nd stage was 995 m. Do not penalize students for not rounding off correctly.

15 0 Eample of an appropriate solution Find the zero of the rational function 4 f() = = 0 4 = 0 ( 0) = 4 0 = = Verte of square root function (, 0) y = a h + k y = a 8 = a 6 8 = a 4 8 = a 4 = a y = 4 9 seconds later y = 4

16 y = 4 9 y = 4(3) y = Answer: Nine seconds after it catches its prey, the hawk will be metres above the ground. Note: Students who have found the verte of the square root function have shown they have a partial understanding of the problem.

17 Eample of an appropriate solution Find the equation of the absolute value function Verte (, 0) point (0, ) y a 0 a a y Find the value when y = y or or Find the equation of the square root function Starting point (.5, ) point (3.5, 3) y a 3 a a a So y.5

18 Find the time when the height is 5 m y Answer: The ball hits the wall 6.5 seconds after it was hit by the racket. Note: Students who use an appropriate method in order to determine the starting point of the square root function have shown they have a partial understanding of the problem. Do not penalize students who rounded their final answer. The range of f () is -, -5]

19 3 Eample of an appropriate solution f 3 - using verte and y intercept. Draw Flip (, y) of verte and y intercept y Find rule of inverse f f 0 a h a a - 9a - 9 a k (-, ) (0, ) (, 0) V(, -) f - 9, - Answer: The rule of correspondence for the path of the second missile is - - f, - or f, -,. 9 9 Alternate solution Domain: ]-, ] Range: [-, [

20 [ [-,, : : y y y y y y y f y f Answer: The rule of correspondence for the path of the second missile is [ [-,, 9 - y.

21 Name : Group : Date : Mathematics Question Booklet The members of a school finance committee decided to print and sell souvenir picture albums as a fundraising project. They established that the profit P(), in dollars, from the sale of these $5 albums could be determined by the following equation : P where represents the number of albums to be printed and sold. What is the minimum number of albums the committee must sell to realize a profit? A) 00 albums C) 35 albums B) 50 albums D) 395 albums

22 After one month in operation, a company's revenue has grown according to the formula R a h k where R() is the profit in millions of dollars after a period of months. R() Revenue in millions ($) (0, 5) (, -4) months Losses amounted to $400 the first month. After ten months, $5000 in profits was recorded. After how many months in operation did the company begin to make a profit? Show your work. 3 f 4 3. Given the function defined by the equation 6 What is the domain of this function? A) + B) [-6, C) [-, D) [4,

23 4 A real function is defined in the interval [0, 9] by f 9. Draw the graph of this function.

24 5 The standard equation of the function graphed below is of the form: g a b h k. g() (h, k) Which of the following is true? A) a > 0 and b > 0 C) a < 0 and b < 0 B) a > 0 and b < 0 D) a < 0 and b > 0

25 6 The graph on the right represents function f y The rule of this function is of the form a b h k f. Which of the following statements is FALSE? A) a ]-, 0[ C) h ]0, + [ B) b ]0, + [ D) k ]0, + [ 7 Which of the following functions has an inverse that is itself a function? A) Cosine function C) Square root function B) Step function D) Absolute value function

26 8 The function f() = represents the function of the outside temperature, in degrees Celsius, in relation to the number of hours elapsed,, since the beginning of observations. Which of the following graphs represents g(), the inverse of the function f()? A) g () C) g () B) D) g () g ()

27 9 A missile is picked up by an airplane s radar. The path of the missile across the radar screen is represented by the following rule of correspondence: A( s ) -0 6( s 3) 5000 where A(s) represents the altitude in metres and s, the time in seconds. Which statement is true? A) The domain of the function is [-3, +[. B) On the radar screen, the path of the missile is directed left. C) According to the rule of correspondence, the function is decreasing. D) According to the rule of correspondence, the function is increasing between 0 and 3. 0 What is the solution set of the following inequality?

28 Two missiles are launched seconds apart. The paths they follow over a span of 8 seconds can be represented by two different square root functions, as illustrated below: y Height (m) 4 (0, 3) (6, ) (, 0) (4, ) Time (s) How many seconds after the nd projectile has been launched, will it be higher than the st projectile? Show all your work.

29 The rule of function f represented by the following graph is f() =. y (, -) What is the rule of its inverse f -? A) f - () = where - B) f - () = where C) f - () = where - D) f - () = where

30 3 Carol's father made a slide during the winter. Placed on a Cartesian plane scaled in metres, the slide follows a square root function, as shown below. The top of the slide,.4 m above the ground, coincides with the verte of the square root function. The foot of the slide is 6 m horizontally from the top. Carol starts going down the slide. However, her scarf becomes jammed in the sled causing her to stop at a horizontal distance of 0 m. y.4 m 4 0 m h (6, 0) At what height did she stop? Show all your work.

31 4 Caroline slides down a waterslide at an aquatic park. The following graph represents Caroline's altitude in relation to her sliding time. This curve represents a square root function whose verte is point P(0, 0). y Altitude (metres) P (0, 0) 5 Time (seconds) A section of the slide is covered by a tarpaulin, forming a tunnel. Caroline enters the tunnel when she is at an altitude of 6 m. She eits the tunnel at an altitude of 4 m. How long was Caroline in the tunnel? Show all your work.

32 5 a b h. The equation of function f, shown in the Cartesian plane below, is k f The sign of parameter b is changed and the value of parameter k is decreased. j i f g h Which of the following functions represents these changes? A) Function g C) Function i B) Function h D) Function j

33 6 A cup of sorbet is designed in the Cartesian plane on the right. y The top part of the cup varies according to the trigonometric function: f cos cm The foot of the cup corresponds to two square root functions. The equation of one of the square root g functions is 4? The height of the cup is.4 cm. What is the width of the foot of the cup? Show all your work.

34 7 Which of the following is the inverse of: f() = 3 + 8, 0? A) f - () = 3 8, 8 C) f - () = 8, 8 3 B) f - () = ( 8) 3, 8 D) f - () = 8, 8 9

35 8 A new glass has been designed by rotating part of a graph of a square root function about the ais containing the stem of the glass. (Assume the width of the stem to be zero.) As illustrated in the diagram, the diameter of the lip of the bowl is 8 centimetres. The glass stands 3 centimetres in height and the top of the stem of the glass is 5 centimetres high. A decorative gold ring is to be painted around the bowl centimetres from the bottom of the glass, at a cost of cents per centimetre. 8 cm Lip Gold Ring Bowl 3 cm cm Stem 5 cm Base How much will it cost to paint the gold ring around the bowl? Round your answer to the nearest cent. Show all your work.

36 9 A two-stage model rocket is launched from ground level. Its st stage (engine) powers the rocket vertically, according to the rule H(t) = 00 t, where H(t) is height, in metres, and t is time, in seconds, after launch. At 5 seconds, the ehausted st stage is ejected, and the nd stage fires. The height of the rocket after the first y a h. 5 seconds can be epressed according to a new square root function of the form k 50 seconds after the initial launch the rocket reaches a height of 500 m. 500 Height (m) nd Stage st Stage Time since launch (sec.) Rounded to the nearest metre, what is the height of the rocket seconds after the firing of the nd stage?

37 0 After flying for 0 seconds, a hawk swoops down, catches a mouse, and immediately takes off with its prey. The path of the hawk's descent has been determined to be in the shape of a rational function and the path of its ascent is in the shape of a square root function. The point of ascent corresponds to the verte of the square root function. Four seconds after the hawk catches the mouse, it is 8 m above the ground. 4 The rational function is f() =. 0 y 0 How many metres above the ground will the hawk be 9 seconds after it catches its prey? Show all your work.

38 A tennis ball is hit by a racket from a height of metres and follows the path of an absolute value function. One second later the ball hits the ceiling, which has a height of 0 metres. On its way down, the ball bounces off a table that is metre high. After the bounce, its path is a semi parabola. One second after the ball hits the table, it reaches a height of 3 metres before hitting a wall at a height of 5 metres. Height (m) 0 5 m? Time (s) How many seconds after the ball was hit by the racket did it hit the wall? Show all your work.

39 A rocket is shot into the air by a submarine located at the verte of the following square root function: f 3-5 The rocket follows the path of the square root function. What is the range of f ()? 3 Two missiles are being tested at a military base. One missile followed the path described by 3 - f before self-destructing. The second missile followed the inverse path of the first, before it self-destructed. What is the rule of correspondence for the path of the second missile?

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