Graphs
2012
Question One
(a) The table below gives the adult single train fares for
travel from the centre of a city.
(i) On the grid below, sketch the graph of the adult train
fares against the number of stations from the centre of
the city.
• A child’s fare is $1.50 for the first stage.
Each additional stage, for a child, increases the
fare by 75 cents.
• If a graph was drawn for the child’s fares,
describe the similarities and differences
between the graphs of the child’s fare and the
adult’s fare.
describe the similarities and
differences
describe the similarities and
differences
• Child’s fare increases are
smaller.
• Child’s fare starts at a
lower value.
• Increase for children’s
fares 3/5 of adult.
•
Both have constant
increases.
• Blake receives a copy of his bank statement
and finds he is overdrawn (he has a negative
amount in the bank).
• He starts a saving plan.
The graph below shows the amount of money
Blake hopes to have in his bank account, $S, if
he follows his savings plan for n weeks.
(i) How much does Blake plan to bank
each week?
(i) He puts in $400 in 10 weeks so $40
per week
(i) Give the equation for the graph of Blake’s saving plan in terms of $S, the
amount in Blake’s account, and n, the number of weeks after the start of his
saving plan.
(ii) S = 40n – 200
• Blake’s grandmother thinks he should be saving more.
• At the end of 4 weeks she tells him that if the amount in his bank
account at the end of 9 weeks is $300, she will give him $50.
• He increases the fixed amount he saves each week from the end of
week 4.
• He reaches his grandmother’s target of $300 in his account and
banks the $50 from his grandmother.
• He continues saving at the increased rate after banking the $50
from his grandmother. Describe how the graph changes from week
4 onwards.
• Hints: you can do this by giving equations for some parts of the
graph. You may find it helpful to sketch the graph using the grid on
page 4.
Weeks 4 - 9
• After 4 weeks, he has a
balance of -$40 and he
must reach $300 by 9
weeks.
• Gradient = 340/5=$68
per week
• Equation is now
• S = 68n + c
Weeks 4 - 9
• Equation is now
• S = 68n + c
• The point (9, 300) must
lie on this line
• i.e.
• 300=68 x 9 + c
• C = -312
• Equation is
• S = 68n - 312
Weeks 4 - 9
• The graph is steeper
than before because he
now saves $68 per
week
Week 9
• Grandma gives him $50
and so we get a vertical
jump.
From 9 weeks
• He still saves at the
same rate of $68, so the
gradient stays the same
but the line moves up
by $50 so the new
equation is
• S = 68n - 262
From 9 weeks
• The graph is parallel to
the last graph.
Question Two
• Emma is employing Ian to build a deck at her
house.
She provides all the building material.
She pays Ian $P for the number of hours, h, that
he works.
She also pays for Ian’s travel to her home each
day.
Ian works for 8 hours each day.
He knows the deck will take more than 4 hours to
build.
To help Emma know how much she can expect to
pay, Ian provides the following table:
On the grid below, plot a graph showing the payment
required for the number of hours worked.
Payment ($), P
(b) How much does Ian charge for his
travel each day?
• $60
• (c) Explain why the graph rises more steeply
after 8 hours.
• The gradient remains the same so the graph
does not rise more steeply but there is a jump
of $60 as there is another payment for travel
to the job.
• What would Emma expect to pay if the work
took 30 hours?
• Explain your calculation.
• She would have to pay for 30 hours at a rate of
$25 plus 4 days of travel at $60 per day.
• S=25h + 60d
• S=25 x 30 + 60 x 4 = $990
• Zarko lives next door to Emma and says he
could build the deck for her. He does not need
to be paid for his travel, but he charges $35 an
hour.
• How long would the work take if the payments
to Zarko and Ian were the same?
• Explain how you calculated your answer. Hint:
there may be more than one solution.
This question is badly worded!!!
• Zarko lives next door to Emma and says he
could build the deck for her. He does not need
to be paid for his travel, but he charges $35 an
hour.
• How long would the work take if the payments
to Zarko and Ian were the same?
• Explain how you calculated your answer. Hint:
there may be more than one solution.
We are looking for the intersection
points
We are looking for the intersection
points
• P = 35h = 25h +60
• h=6
• P = 35h = 25h + 120
h = 12
P = 35h = 25h +180
h = 18
We are looking for the intersection
points
• P = 35h = 25h + 240
• H = 24
• This is not an intersection
as after 24 hours, the
payment is the lower
value i.e. $780 and not
$840
• There are no more
intersection points
Another builder gives Emma a graph, showing
the amount she would charge for 8 hours work.
Give the rule for the payment that this builder would
receive for the first 8 hours that she worked.
The fixed fee is $60 and
the gradient is 120/3 =40
P = 40h + 60
Question Three
Zara plots a graph of the number of people, p, against
the number of tables.
Explain how the equation and graph relate to the
number of people at the tables.
There is room for another 4 people for every extra table so
the gradient is ‘4’ and ‘4’ is the number in front of ‘n’ in the
equation.
The ‘2’ represents the 2 people sitting on the ends of the
tables. ‘2’ is the number added to the 4n in the equation and
‘2’ is where the graph would intersect the ‘y’ axis but is not
meaningful here.
Graph has been drawn as discrete points as you cannot have part
people or tables OR a comment that n and p are whole numbers.
On the grid below, sketch the graph of
y = –(x – 2)(x + 4)
give the x and y intercepts
X-intercepts:(-3, 0), (3, 0),
y-intercept(0, 9)
the equation of the graph.
y = (x +3)(x – 3) OR y=x2 –9 .
• The parabola is moved 1 unit to the right and
2 units up.
Give the equation of the parabola in simplified
form in its new position
• AND give the y-intercept.
• The parabola is moved 1 unit to the right and
2 units up.
Give the equation of the parabola in simplified
form in its new position
• y-intercept (0, -6)
y - 2 = ( x - 1- 3) ( x - 1+ 3)
y = ( x - 4 )( x + 2) + 2
y = x - 2x - 6
2
• The parabola is moved 1 unit to the right and
2 units up.
Give the equation of the parabola in simplified
form in its new position
• y-intercept (0, -6)
y - 2 = ( x - 1) - 9
2
y = ( x - 1) - 7
2
y = x - 2x - 6
2
• In a children’s play park, a ball is kicked so that
its flight path can be modelled by the
equation
• h = – ax(x – 6)
where h metres is the height of the ball when
it is x metres from the point from where it is
kicked.
If the maximum height of the ball is 2 m, what
is the value of a?
The maximum of ‘2’ happens when x = 3
• h = – ax(x – 6)
2 = -3a ´ -3
2
a=
9
•
•
•
•
•
•
(axes placed at LH end)
h= −x(x−10)/ 10
OR
h=–0.1x2 +x
OR
h=–0.1(x–5)2 +2.5
•
•
•
•
•
•
(centrally placed axes)
h= -(x-5)(x+5) /10
OR
h = –0.1(x – 5)2
OR
h = –0.1x2 + 2.5
•
•
•
•
•
(axes placed at RH end)
h= -x(x-10)/ 10
OR
h=–0.1x2 –x
OR
h=–0.1(x+5)2 +2.5