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

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 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 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 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 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

P = 35h = 25h +60 h = 6 P = 35h = 25h h = 12 P = 35h = 25h +180 h = 18

We are looking for the intersection points P = 35h = 25h 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

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=x 2 –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)

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)

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)

(axes placed at LH end) h= −x(x−10)/ 10 OR h=–0.1x 2 +x OR h=–0.1(x–5)

(centrally placed axes) h= -(x-5)(x+5) /10 OR h = –0.1(x – 5) 2 OR h = –0.1x

(axes placed at RH end) h= -x(x-10)/ 10 OR h=–0.1x 2 –x OR h=–0.1(x+5)