Maths Task-Functions and relations, cubic functions, quadratic functions

  

Maths Task-Functions and relations, cubic functions, quadratic functions, problem solving.docx
maths_task_functions_and_relations__cubic_functions__quadratic_functions__problem_solving.docx

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SECTION 1
SEND: Work for Submission
TECHNOLOGY FREE
1.
Change to interval notation, and sketch on a number line.
(a)
2.
3.
(b)
{x:-3?x<5} {x: x>-7}
Given ? = {1,2,3,4,5,6},A = {1,2, 3} and B = {3,4,5} find:
(a)
A|B
(b)
B?
(c)
A?B
(d)
A?B
Sketch the following function stating the (a) domain (b) co-domain and (c) the range.
{(x, y): y = x2 – 1, x?-1, y?R}
4.
What is the implied domain of
5.
Sketch:
y=
1
. Explain your answer.
2- x
? x2 , x ? 0
?
f ( x ) = ? x, 0 ? x ? 2
?2, x ? 2
?
6.
7.
(i)
What is the inverse of the function y = 2x – 3 ?
(ii)
Sketch the graph of y = 2x – 3 and its inverse.
(iii)
State the domain and range for both the function and its inverse.
Sketch the graph of
y=
1
– 1 , labelling all the important features.
x+2
What are the equations of the asymptotes?
State the domain and range.
8.
9.
Write down the equation of the rule obtained when the graph of y = x2is transformed by
(a)
a dilation of factor 5 from the y-axis. (i.e. parallel to the x-axis)
(b)
a dilation of factor
(c)
areflection in the x-axis.
(d)
areflection in the y-axis.
1
from the x-axis. (i.e. parallel to the y-axis)
4
A wire of length l is cut into two equal pieces, one of which is bent to form a circle and the other a square.
(a)
Find an expression for the radius of a circle in terms of l.
(b)
Find an expression for the side of a square in terms of l.
(c)
Construct the area function that expresses the total area in terms of l.
CAS Calculator
10.
Sketch the graph of (x- 1)2 + (y- 2)2 = 4, clearly showing the coordinates of the centre of the circle.
What is the radius of the circle?
Use your CAS calculator to find the coordinates of the x and y intercepts (if they exist.) Where
appropriate give your answers correct to 2 decimal places.
SECTION 2
You are encouraged to use your CAS calculator to assist you with problem solving tasks. Please use the
following checklist to gain full marks:
You have to:
? Includethorough explanations/workingoutfor each question;
? Include screen captures from your calculator using the appropriate
connectivity software or sketching them by hand.
? Give all answerscorrect to three decimal places.
? Sketch graphs for appropriate domains
? Label graph axes, showing correct variables and units
? Check that your answers make sense
Please note that you do not need to copy the questions.
The following rubric will be used to assess your work:
Problem 1 TheCricket Ball
Problem 2
(a) 5 marks
(a) 4 marks
(b) 1 mark
(b) 2 mark
(c) 1 mark
(c) 2 mark
(d) 1 mark
(d) 4 marks
(e) 2 marks
(e) 3 marks
(f) 2 marks
(f) 3 marks
Total 12 marks
Total 18 marks
Bike Tacks
Bonus marks:
Appropriate use of mathematical conventions, symbols and terminology
Poor 0 marks
Satisfactory 1 mark
Excellent 2 marks
Presentation of work and graphs
Poor 0 marks
Satisfactory 1 mark
Excellent 2 marks
Appropriate and effective use of CAS calculato3
Poor 0 mark
Satisfactory 1 mark
Good 2 marks
Total
Excellent 4 marks
37 marks
1.Cricket ball(5+1+1+1+2+2 = 12 marks)
(x ? 0 ) , represents the path of a cricket ball where
The equation d = 1 + x – 0.02 x
dis the vertical height and xthe horizontal distance inmetres.
2
(a)
Sketch the graph of this equation, clearly labelling key features such as axes, intercepts and
turning points. Give values correct to 3 decimal places.
(b)
Find the initial height of the ball when it is first struck.
(c)
Find the height of the ball after it has travelled horizontally for 15 metres.
(d)
Find the maximum height that the ball reaches.
(e)
(f)
Find the horizontal distance travelledwhile the ball is at the height greater
than 10 m.
Find how far the ball travels horizontally after being struck.
2. Bike Tracks
marks)
(4 + 2 + 2 + 2 + 4 + 3 + 3 = 18
Three bike tracks have been proposed in a small town. All distances have been measured in kilometres
from the centre of the town which is designated by the co-ordinate (0, 0).
(a)
The 1st biketrackfollows the path of a straight line and has to pass through the points (-10, 22)
and (10, 8).Find the equation of this bike track.
(b)
The 2nd bike track has to pass through the point (0,6) and must always make an angle of 60?
with the positive direction ofx-axis. Using algebra, find the equation of this bike track. Show all
steps of your working out.Give exact values of the coefficients.
(c)
Sketch 1st and 2nd tracks using the CAS calculator and find co-ordinates of their point of
intersection. Copy the CAS calculator screen output onto your page clearly showing the point of
intersection.
(d)
Two bike riders travel at the same speed along 1st and 2nd tracks one on each track. They both
start their rides from the y-axis and move towards the point of intersection of the two tracks.
Which of the riders would reach the point of intersection first?Include any relevant calculations
to justify your answer.
(e)
The 3rd bike track has a parabolic shape. It has to pass through the points
(0, 0),(4, 48), (16, 0). Find the equation of this track. Clearly show all your working out.
(f)
Find co-ordinates of the points of intersection of the 2 nd and 3rd bike tracks. Copy the CAS
calculator screen output onto your page clearly showing the points of intersection.
SECTION 3
SEND: Work for Submission
TECHNOLOGY FREE
1.
On the same set of axes , using different colours, sketch the graphs of
y = x3,
y = (x – 1)3 + 2,
y = (x + 2)3 – 1
and
y = -x3.
Clearly label the scale on your axes and clearly label each of your graphs.
Write the coordinates of the points of inflection for each of the graphs.
2.
Find, by polynomial division, the quotient and remainder when 2×3-3×2-3x- 2 is divided by 2x- 1.
3.
Use the remainder theorem to find the remainder when x3-5×2 + 7x- 6 is divided by x + 2.
4.
Without using a CAS calculator, factorise the following:
(a)
6×3 + 13×2 – 4
27×3 – 8
(b)
(c)
x3 + 3×2 – 4x
5.
Solve6x3 – 11×2 –4x + 4 = 0 algebraically (i.e. without using a CAS calculator.)
6.
When 3×3 + ax2 + bx + 7 and bx3-6×2 +11x + a are divided by x + 2, the remainders are 11 and -9
respectively. What are the values of a andb?
7.
CAS Calculator
Using a CAS calculator, graph
y=-
1
3
(
x+
3
2
) +5
3
.
Copy your graph from the screen, and on your drawing clearly show key features such as x and yintercepts and points of inflection. Give your values correct to two decimal places.
8.
Using Logger Pro, view one of the following sample movies:
•
•
•
•
Ball Toss
Bounce Back
Drop Zone
What Goes Up
Write a brief report about your chosen experiment giving an explanation of what is happening in the
video. Include in your report some of the important data points on the graph.
SECTION 4
SEND: Work for Submission
TECHNOLOGY FREE
1.
Without using your calculator, sketch the following. Find the x and y intercepts and explain fully how
you found these intercepts. i.e. clearly show all working out.
2.
(a)
y = (x – 3)(x – 1)(x + 2)
(b)
y = x3 + x2 – 5x + 3
(c)
y = (x-1)3 + 3
(d)
y = -(x – 2)4
(e)
y = -(x +1)2(x-2)(x-4)
Match each rule with the correct graph
shown on the right:
(a) y=(x– 6)2(x – 4)
(b) y=(x+ 6)2(x + 3)
(c) y= –(x+ 2)2(x – 3)
(d) y= x(x+ 5)2
3.
Find {x: x3 – 2×2 – 5x + 6 > 0}
4.
Find the equation of the quartic graph which passes through the origin and has just one turning point at
(-2, 8).
5.
CAS Calculator
Use your CAS calculator to find the maximum value of y for the function:
y = x3 + 4×2 + 2x- 1,
-4
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