Mastery Test 2 MCR 3U Friday September 30, 2011
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____1.Simplify x - 2y - 1 -2x +y -3
a. / -x - y - 4 / b. / -3x - 3y - 4 / c. / -2xy - 4 / d. / -x - y - 2____2.Which method would be easiest to solve the system:
a. / graphing / b. / substitution / c. / elimination____3.Factor completely:
a. / / b. / / c. / / d. /____4.If then
a. / x = 4 or x = -2 / b. / x = ±4 / c. / x = ±2 / d. / x = 2 or x = -4____5.The graph of x+y = 1 is
a. / Line 1 / b. / Line 2 / c. / Line 3 / d. / Line 4 / e. / Line 5____6.A line passes through A(-2,1) and has slope . What are the coordinates of another point on the line?
a. / (-3,-1) / b. / (0,0) / c. / (-3,3) / d. / (0,-1)____7.Determine the zeros (2 decimal places) of the relation with equation
a. / 1.72, -1.49 / b. / 0.72, -0.30 / c. / 0.88, -0.24 / d. / 0.94, -0.25____8.One solution to the system is approximately
a. / x=1.90, y=8.80 / b. / x=1.78, y=8.56 / c. / x=1.63, y=8.27 / d. / x=2.31, y=9.62____9.If f = {(2,4), (4,-3), (-3,2)} then f -1(-3) =
a. / 4 / b. / 2 / c. / -3 / d. / -2____10.If f = {(1,3), (-1,5), (3,-1)} then f(-1) =
a. / 5 / b. / -3 / c. / 3 / d. / -5____11.The equation of the parabola shown is
a. / / b. / / c. / / d. /____12.Use the diagram below to write the relation as a set of ordered pairs.
a. / {(0,2), (2,3), (-3,3)} / c. / {(-3,3),(0,2),(2,1)}b. / {(-3,3), (0,3), (2,1)} / d. / {(1,-3), (2,0), (3,2)}
____13.Which of the following arrow diagrams illustrate functions?
#1 #2
a. / neither / b. / #1 only / c. / #2 only / d. / both #1 and #2____14.If f(x) = , then f -1(x) =
a. / 2(x + 1) / b. / 2x + 1 / c. / / d. /____15.If g(x) = (x - 1)2 , then evaluate g(-3)
a. / 10 / b. / 4 / c. / 16 / d. / -16____16.A function is defined by . Its domain is
a. / {x | x 0, x } / c. / {x | x 3, x }b. / {x | x > 0, x } / d. / {x | x < 3, x }
____17.A function is defined by . Its range is
a. / {y | y 0, y } / c. / {y | y -3, y }b. / {y | y > 0, y } / d. / {y | y -3, y }
____18.The graphs of functions are...
#1 #2
a. / neither / b. / # 1 only / c. / # 2 only / d. / both____19.Which of the following diagrams correspond to the function defined by f(x) = –1 + 2x
a. / / c. /b. / / d. /
____20.A function f has I/O diagram
Which of the following diagrams correspond to ?
a. / / c. /b. / / d. /
____21.What is the next number in the sequence 2,5,9,14,...
a. / 20 / b. / 19 / c. / 18 / d. / 30____22.If then is equal to
a. / 18 / b. / 14 / c. / 20 / d. / 36____23.A function f is defined by
Which of the following tables of values correspond to ? (NOTE: the inverse of f)
a. / / b. / / c. / / d. /____24.A function g is defined in terms of another function f as shown in the diagram below.
What transformations must be applied to the graph of f to get the graph of g?
a. / A horizontal translation of 1 unit(s) to the right, and a vertical stretch of factor 2 / c. / A horizontal translation of 1 unit(s) to the left, and a vertical stretch of factor 2b. / A horizontal translation of 1 unit(s) to the right, and a vertical stretch of factor / d. / A horizontal translation of 1 unit(s) to the left, and a vertical stretch of factor
____25.The input/output diagram illustrates a number of transformations to y=f(x).
The correct sequence of horizontal transformations applied to f is...
a. / a horizontal stretch of factor followed by a hor. translation of 1 unit leftb. / a hor. translation of 1 unit right followed by a horizontal stretch of factor 2
c. / a hor. translation of 1 unit left followed by a horizontal stretch of factor
d. / a horizontal stretch of factor 2 followed by a hor. translation of 1 unit right
____26.The input/output diagram illustrates a number of transformations to y=f(x).
The correct sequence of vertical transformations applied to f is...
a. / a vertical stretch of factor 2 followed by a vertical translation of 3 units upb. / a vertical. translation of 3 units up followed by a vertical stretch of factor
c. / a vertical. translation of 3 units down followed by a vertical stretch of factor 2
d. / a vertical stretch of factor followed by a vertical. translation of 3 units down.
____27.The input/output diagram illustrates a number of transformations to y=f(x).
The equation for this new function is...
a. / / c. /b. / / d. /
____28.What is the next number in the sequence 12,6,3,1.5,...?
a. / -1.5 / b. / 0 / c. / 0.75 / d. / 0.5____29.If then is equal to
a. / 18 / b. / 8 / c. / 6 / d. / 9____30.A function is defined by . Its domain is
a. / {x | x 0, x } / c. / {x | x 3, x }b. / {x | x > 0, x } / d. / {x | x 3, x }
Mastery Test 2 MCR 3U Friday September 30, 2011
Answer Section
MULTIPLE CHOICE
1.ANS:A
The illustration of x - 2y - 1 -2x +y -3 is below. Note that x and y tiles are similar ... just different lengths: Simplifying, we get or -x - y - 4.
PTS:1BNK:3U 01 Algebra Review
2.ANS:B
Both equations are linear and one equation has a variable isolated already, so substitution would be easiest.
PTS:1BNK:3U 01 Algebra Review
3.ANS:C
Arrange the tiles into a rectangle:
So
PTS:1BNK:3U 01 Algebra Review
4.ANS:A
If then .
or
or
PTS:1BNK:3U 01 Algebra Review
5.ANS:D
We know the x-intercept is 1 (if we let y=0, we get x=1) and the y-intercept is 1 (if we let x=0, we get y=1), so it must be Line 4.
We also could have solved for y and used slope and y-intercept to identify the line.
PTS:1BNK:3U 02 Relationships Review
6.ANS:B
A slope of means that if we move 1 unit right then we move unit down to get to another point on the line. If we move 2 units right then we move 1 unit down to get to another point on the line, so from (-2,1), another point would be (0,0).
PTS:1BNK:3U 02 Relationships Review
7.ANS:A
Enter the equation into the calculator, select 6:ZStandard after pushing the Zoom button. Push 2nd Trace to select CALC. Select 2: zero. Use the scroll buttons to move the cursor to the left of one of the zeros and press ENTER. Scroll to the right of the zero, and then press ENTER twice. Round off the x-coord. of the zero to 2 decimal places. Repeat for the other zero.
PTS:1BNK:3U 03 Calculator Review
8.ANS:A
Enter the two equations into the calculator and clear any other out. Select 6:ZStandard after pushing the Zoom button. Push 2nd Trace to select CALC. Select 5: intersect, and press ENTER twice. Use the scroll buttons to move the cursor close to the point of intersection which has a positive x-coordinate and press ENTER. Round off the coordinates to 2 decimal places.
PTS:1BNK:3U 03 Calculator Review
9.ANS:A
The ordered pair (4,-3) shows us that f takes 4 and produces -3. Therefore, f-1 takes -3 and produces 4. I.e., f-1(-3) = 4.
PTS:1BNK:3U 04 Functions
10.ANS:A
The ordered pair (-1,5) shows us that f takes -1 and produces 5, so f(-1) = 5
PTS:1BNK:3U 04 Functions
11.ANS:A
The vertex is at (-3,-2), so the correct choice is .
PTS:1BNK:3U 02 Relationships Review
12.ANS:C
The numbers on the left are the possible values of x, and the numbers on the right are the values of y. The arrows show us that the ordered pairs are (-3,3),(0,2), and (2,1). It should be noted that these ordered pairs can be listed in any order - just don’t change the order inside each pair!
PTS:1BNK:3U 04 Functions
13.ANS:A
For a function, each element on the left must be mapped onto one element on the right.
Both diagrams show one element of the domain mapping onto 2 different elements of the range.
PTS:1BNK:3U 04 Functions
14.ANS:B
Function f takes the input (x) subtracts one and then divides it by 2 to get the output (y). The inverse function will multiply the input by 2, and then add1 so we have f-1(x) = 2x + 1
This problem can be solved algebraically as well (although it is a lot of unnecessary work) The equation for f is y =. The equation for the inverse is x = . If we multiply both sides by 2 we get 2x = y - 1. Ad 1 to both sides to get 2x + 1 = y or f-1(x) = 2x + 1.
PTS:1BNK:3U 04 Functions
15.ANS:C
g(-3) = ((-3) - 1)2
= (-4)2
= 16
PTS:1BNK:3U 04 Functions
16.ANS:C
In the real numbers, is defined only if the expression under the square root symbol is greater or equal to 0, so x - 3 0. The values of x that make x - 3 0 are all the numbers greater or equal to 3.
PTS:1BNK:3U 04 Functions
17.ANS:A
In the real numbers, will always give an answer which is greater or equal to 0 (the square root symbol actually means the ‘principal’ or positive square root). It is also possible for to produce any real number greater or equal to 0 (e.g., if x = 3, = 0)
Therefore 0
So ... y 0
PTS:1BNK:3U 04 Functions
18.ANS:A
Neither graph passes the vertical line test (one value of x ‘has’ more than one value of y).
PTS:1BNK:3U 04 Functions
19.ANS:A
In the expression, –1 + 2x , the first operation is to multiply by 2 and then add –1 (remember that –1 + 2x is really the same as 2x – 1)
The input/output diagram shows that we take the value of x and first multiply by 2 (the arrows show us the direction of the flow) and then take that answer and add –1.
PTS:1BNK:3U 04 Functions
20.ANS:D
In the original function, the first operation is to add 3, and then divide by 4 , so the inverse must perform the opposite operations in the opposite order (think ... putting your sock on and then your shoe - the first thing you must do to reverse the process is to remove the shoe and then remove the sock), so....
To undo “divide by 4 ”, we must multiply by 4. To undo “add 3“, we must subtract 3 or add –3 , so the I/O diagram for the inverse function is:
PTS:1BNK:3U 04 Functions
21.ANS:A
2 + 3 = 5
5 + 4 = 9
9 + 5 = 14
It appears that the differences between the terms is increasing by 1, so the next term should be 6 more than 14 or 20.
PTS:1BNK:3U 05 Sequences, Discrete Functions & Transformations
22.ANS:A
so
PTS:1BNK:3U 05 Sequences, Discrete Functions & Transformations
23.ANS:A
If the table of values is valid for the inverse, then if we reverse the values of x and y, the points must satisfy the definition for f.
is the correct table because...
and none of the other tables satisfy this. Another way to do this question is the enter the equation for f in the calculator and check the table of values it creates.
PTS:1BNK:3U 04 Functions
24.ANS:A
Probably the easiest way to do this question is to pick an ordered pair that might belong to f, say (2,1), and then ... see what happens when you work away from f in the I/O diagram.
So plug in 2 to the left of f, and 1 to its right as shown below.
...
The ‘flow’ is left to right so we note that if we add –1 to the new x coordinate, we get 2. This means that the new x coordinate is 1 more than the old one... so we must be moving all the points right 1 unit(s). When 2 is fed into f, out comes 1. Then, 1 is multiplied by 2 to get 2, so the new y-coordinates are 2 times bigger than the old y-coordinates ... so it is a vertical stretch of factor 2.
PTS:1BNK:3U 05 Sequences, Discrete Functions & Transformations
25.ANS:B
The original values for x would be input into f (just to its left), so if we trace back from there to the ‘new’ values of x, we see that the first horizontal transformation is a translation of 1 unit right (add 1 to the old x-values) to reverse the subtraction of 1. To invert the division by 2, the next operation would be a horizontal stretch of factor 2.
PTS:1BNK:3U 05 Sequences, Discrete Functions & Transformations
26.ANS:B
The original values for y would be output from f (just to its right), so if we trace from there to the ‘new’ values of y on the right, we see that the first vertical transformation is a vertical translation of 3 units up. The next operation would be to divide by 2 which corresponds to a vertical stretch of factor .
PTS:1BNK:3U 05 Sequences, Discrete Functions & Transformations
27.ANS:D
If we apply the sequences of operations listed in the input/output diagram to x, we get ...
PTS:1BNK:3U 05 Sequences, Discrete Functions & Transformations
28.ANS:C
Each term is one half of the previous term, so the next term is one half of 1.5 or 0.75.
PTS:1BNK:3U 05 Sequences, Discrete Functions & Transformations
29.ANS:B
so
PTS:1BNK:3U 05 Sequences, Discrete Functions & Transformations
30.ANS:B
In the real numbers, is defined only if the expression under the square root symbol is greater or equal to 0, so x 0. However, if x was equal to 0, we would get
which would be undefined, so x must be just greater than 0
PTS:1BNK:3U 04 Functions