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91028 4 credits Investigate relationships between tables, equations or graphs

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Assessment specifications Things to take notice of

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“Candidates may be required to understand the difference between graphs representing situations involving continuous data from graphs representing situations involving discrete data.” Always ask yourself if you can have half of the value e.g. you can’t buy half a cell phone- this is discrete data Always ask yourself if you can have negative values e.g. you can’t pay someone for negative time on a graph

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What to expect Writing a rule for a linear (straight line) pattern This means that you are adding (or subtracting) the same number

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Using table of values

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Example 1

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When we substitute in ‘1’, the value of m must be ‘3’ i.e. the coefficients must add up to ‘3’

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We can only have values from ‘1’ onwards as we don’t have a zero pattern or a negative pattern.

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We can’t have pattern number 1 1 / 2 so we have discrete data

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Draw the graph starting from n = 1

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The pattern number is the input so is on the x-axis

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The number of matches is the output so is on the y-axis

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Do not talk about this graph having a gradient. Say instead what the increase (or decrease) is.

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Example 2 Timbuktu and Casablanca are linked by a camel route which is 1710 km long. One camel caravan leaves Timbuktu for Casablanca and travels 60 km on the first day, 58 km on the second, 56 km on the next day and so on. Another caravan leaves Casablanca for Timbuktu and travels 75 km on the first day, 72 km on the second, 69 on the next day, and so on. A. How many days does it take before the two caravans meet? B. How far from Timbuktu do the two caravans meet?

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The distance travelled for each camel caravan

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Do the two camel caravans meet after 10 days? What is wrong with this graph?

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The information is just how far they travel on each of the days and does not show the total distance from each camel caravan is from Timbuktu. We need to construct another table.

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This is no longer linear and the distance travelled each day becomes our first difference.

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The total distance travelled from Timbuktu to Casablanca is a quadratic function of the form D = at 2 + bt + c

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D = at 2 + bt + c a = -1, c = 0, a + b + c = 60 gives b = 61

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D = -t 2 + 61t TEST the formula for t = 4 D = -16 + 244 = 228

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Similarly for the second camel caravan

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D = at 2 + bt + c a = 1.5, c = 1710, a + b + c = 1635 gives b = -76.5

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D = 1.5t 2 – 76.5t +1710 TEST the formula for t = 4 D = 24 – 306 + 1710 = 1428

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They meet after 19 days at a distance of 798 km from Timbuktu

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What is wrong with the graph?

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It shows negative values which do not apply.

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Solving using the calculator Either use EQUA- POLY Or SOLV

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Example 3

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Finding differences is not working

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Look for ratio

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This means it is an exponential

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It starts at 80 and increases by 1.5 times for each successive value

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Recognising linear data

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An equal increase in a regular interval i.e. for every increase of ‘10’ in x, there is a 0.28 increase in ‘y’.

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The equation is y = 0.028x + 0.27

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Unequal intervals

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Equation is y = 0.95x + 1.44

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Interpreting linear graphs Think: Start values Gradient (increases/decreases) Intersection points Relative positions

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Writing equations for parabolas Think Form of the equation Points on the curve that you know Start values

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Example Zane is throwing screwed up pieces of paper into a rubbish bin. The graph below shows the height of a piece of paper above the floor. The height of the piece of paper above the floor is y, in metres, and the horizontal distance of the piece of paper from Zane is x, in metres. The graph has the equation y = 0.2(x + 2)(4 – x).

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y = 0.2(x + 2)(4 – x)

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y = 0.2(x + 2)(4 – x) x-intercepts are x = -2, x = 4 Midpoint is when x = 1

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y = 0.2(x + 2)(4 – x) when x = 1 y = 1.8 m is the highest point

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y = 0.2(x + 2)(4 – x) when x = 0 y = 1.6 m is the initial height

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(b) Zane decides to increase the difficulty by moving further from the rubbish bin. He stands 5.3 m from the nearest side of a rubbish bin that is 30 cm in height.

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This time he releases the piece of paper 2.0 metres above the floor. It reaches a maximum height of 3.0 m above the floor when it has travelled 2 metres from Zane.

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Form an equation to model the path of the piece of paper and use it to determine whether Zane gets the piece of paper into the rubbish bin.

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You must justify your answer with clear mathematical reasoning, including showing exactly how you used the equation to obtain your answer.

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There are two forms of the equation that you should consider:

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As you are not given any intercepts, it is better to choose the first equation.

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You have one unknown (k) so you need 1 point. In this case, when x = 0, y = 2

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Substitute in x = 5.3

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This is below the level of the rubbish bin, so Zane fails to get it in. (Diagram is misleading)

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Example

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A rugby player places the ball on the ground 46 metres from the goal posts. He steps back, runs towards the ball and kicks it towards the goal posts. For the goal to count, the ball must pass over a bar 3 metres above the ground between the goal posts – in other words, the ball must be at least 3 metres high after it has travelled 46 metres from where it was kicked. The ball ’ s path through the air can be modelled with a parabola.

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24 m 46 m Maximum height 36 m Goal 3 m high Diagram is NOT drawn to scale.

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The highest point the ball reaches is 36 metres. It reaches this point when it has travelled 24 metres horizontally. Write the equation of the ball ’ s path. Use it to determine if the goal is scored by stating how high the ball is when it reaches the bar. You must show the equation you have formed and the calculations used to support your answer.

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Consider both forms of the equation. 24 m 46 m Maximum height 36 m Goal 3 m high Diagram is NOT drawn to scale.

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Either form will get you the answer. 24 m 46 m Maximum height 36 m Goal 3 m high Diagram is NOT drawn to scale.

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For the first equation use the point (0, 36) For the second equation use the point (24,0) 24 m 46 m Maximum height 36 m Goal 3 m high Diagram is NOT drawn to scale.

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24 m 46 m Maximum height 36 m Goal 3 m high Diagram is NOT drawn to scale. You could also have if your axis was taken as shown

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When x = 46 all equations will give the same answer

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Example Hamish has made a small water rocket as a science project. This is a soft-drink bottle that is pressurised with, say, a bike pump. It then flies through the air. The graph shows the height of the rocket for one of Hamish ’ s launches. The rocket is launched from the top of a fence post at H.

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d h H Fence post Diagram is NOT drawn to scale.

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The graph has the equation h = 0.04(d + 3)(15 – d) where h is the height of the rocket above the ground, and d is the horizontal distance from the rocket ’ s starting point. Both h and d are measured in metres. How high does the rocket get at the top of the curve?

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The graph has the equation h = 0.04(d + 3)(15 – d) where h is the height of the rocket above the ground, and d is the horizontal distance from the rocket ’ s starting point. Both h and d are measured in metres. How high does the rocket get at the top of the curve? The midpoint of d = -3 and d = 15 is d = 6 Substitute this into the equation.

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Here is the diagram of a suspension bridge over a river. The bridge has been modelled with straight lines and a parabola. Diagram is NOT drawn to scale. R P A B river S Q

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Diagram is NOT drawn to scale. R P A B river S Q

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