Introduction
A context such as the following can be used to investigate drug absorption, using a product function model involving circular functions and exponential functions.
For each of the following functions the behaviour and variety of shapes of their graphs is to be investigated. The modelling domain and corresponding range should be identified, as well as key features such as axis intercepts, stationary points and points of inflection, symmetry, asymptotes, and the shape of the graph over its natural domain, using the derivative function for analysis as applicable.
The task will begin with an investigation of a graph that might model the concentration of a certain drug in a patient's system over time. The use of parameters in the family of the function gives students the opportunity to explore the effect the size of parameters has on the graph and hence on the magnitude of the drug in a patient's system over time. Students then explore a similar function that may model the situation more closely.
Component 1
Consider the function with rule
f(x) = e-x sin(x).
- Graph the function identifying its key features and explain how the shape of its graph can be deduced from its component functions.
The graph of
d(t) = Ae-k tsin(kt), where
A and
k are positive real constants, can be used to describe drug absorption in a patient's bloodstream, using units mg/litre per unit of time in minutes.
- Consider the special case where
A = 1 and k = 1, and discuss this with respect to a dose of a drug taken at t = 0.
- Select several pairs of values of
A and
k where
1 ≤ A ≤ 10 and
0.1 ≤ k ≤ 1, and explore and interpret features of the graph of
d(t).
- Discuss the role of the sine function, the exponential function, and constants
A and
k where in determining the shape of the graph of d(t).
Component 2
Consider the function
d:[0,4π] → R,d(t) = Ae-kt sin(kt), where
d(t) measures units mg/l per unit of time in minutes.
- Let
A = 10 and
k = 0.2. Graph this function, identifying its key features, and construct a corresponding table of values.
- Identify and interpret the maximum rates of increase and decrease, and when the concentration is half of its maximum value.
- Investigate what happens to the graph when
A and
k are systematically varied, and discuss any patterns.
Jordan is in hospital and needs a particular drug to manage pain.
- Let
dj :[0,10] → R,dj (t) = 20e-0.5t sin(0.5t) where the particular drug in Jordan's bloodstream is measured in mg/l and time is measured in minutes. Draw the corresponding graph and compare this with the investigations above.
Component 3
Investigate any points of intersection between graphs of
f:[0,4π] → R,f(t) = Ae-kt sin(kt) and
g:[0,4π] → R,g(t) = Ae-kt.
Discuss where these points of intersection exist in relation to the stationary point(s) of the graph of
f(t).
Areas of study
The following content from the areas of study is addressed through this task.
|
Area of study |
Content dot points |
| Functions, relations and graphs | 2, 3, 4, 5, 6 |
| Algebra, number and structure | 4, 5, 6 |
| Calculus | 3, 4, 5 |
| Data analysis, probability and statistics | –
|
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
|
Outcome |
Key knowledge dot points |
Key skills dot points |
|
1 | 1, 2, 4, 6, 7, 8, 9, 10, 12 | 1, 2, 8, 9, 10, 11, 12 |
|
2 | 1, 2, 3, 5 | 1, 2, 3, 4, 5, 6, 7 |
|
3 | 1, 2, 3, 4, 5, 8 | 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12 |
Introduction
A context such as the following could be used to investigate key features of the graphs of some polynomial functions of a real variable formed by products of other polynomial functions.
For each of the following functions the behaviour and variety of shapes of their graphs is to be investigated. The maximal domain and corresponding range should be identified, as well as key features such as axis intercepts, stationary points, points of inflection and symmetry, and the shape of the graph over its natural domain, using the derivative function for analysis as applicable.
The number, location and nature of key features should be determined with respect to different combinations of the parameters that define the product functions, and the different types of graphs identified and classified.
Part 1
Investigate the nature of graphs of polynomial product functions of the form
f:R → R,f(x) = xn (a - x)m
where
n and
m are positive integers and
a ∈ R.
Part 2
Investigate the nature of graphs of polynomial product functions of the form
f:R → R,f(x) = xn (a - x)m
where
n and
m are positive integers and
a ∈ R.
Part 3
Investigate the nature of graphs of polynomial product functions of the form
f:R → R,f(x) = (a - x)n (b - x)m
where
n and
m are positive integers and
a
and
bare real numbers such that
a ≠ b.
Areas of study
The following content from the areas of study is addressed through this task.
|
Area of study |
Content dot points |
| Functions, relations and graphs | 1, 4, 5 |
| Algebra, number and structure | 1, 2, 5 |
| Calculus | 1, 3, 4, 5 |
| Data analysis, probability and statistics | –
|
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
|
Outcome |
Key knowledge dot points |
Key skills dot points |
|
1 | 1, 2, 7, 9, 10, 11 | 1, 2, 5, 6, 7, 9, 10, 11, 12 |
|
2 | 1, 2, 3, 5 | 1, 2, 4, 5, 6, 7 |
|
3 | 1, 2, 3, 4, 5, 6, 8 | 1, 2, 3, 4, 5, 6, 7, 9, 11, 12
|
The application task is to be of 4–6 hours' duration over a period of 1–2 weeks.
Introduction
A context such as the following could be used to develop an application task that investigates how a product function of an exponential decay function and a circular function can be used to model the motion of the
pendulum of a clock after the driving force has stopped.
Component 1
Introduction of the context through specific cases or examples
- Draw the graphs of
f:R → R,f(t) = 5e-kt for several (at least five) values of 0 < k < 1 on the same set of axes.
- Draw the corresponding graphs of
f(-t) and
- f(t) on the same set of axes, and comment on the similarities and differences between these graphs.
- Draw the graphs of
g:[0,2π] → R,g(x) = sin(ax) for several (at least five) values of
a together on the same set of axes. State the period for each function, and comment briefly on the similarities and differences between these graphs.
Component 2
Consideration of general features of the context
- For
k = 0.2;a = 1, sketch the graphs of
ƒ1(t) = 5e–kt ,
ƒ2(t) = –5e–kt and
s(t) = 5e–kt sin(at), where
t ∈ [0, 4π].
- Find the derivative of
s(t), in terms of
t,
k and
a, and hence for
k = 0.2;
a = 1, find the coordinates of the first two maximum/minimum points for
s(t) with
x coordinates closest to the y-axis.
- Find the coordinates of any points of contact between the graphs of s(t) and
ƒ1(t) and between the graphs of s(t) and
ƒ2(t). Briefly comment on the relationship between these points of contact and the graph of
sin(t). Hence, state the exact
x coordinate for the point of intersection closest to the
y-axis.
- State the coordinates of intersection between the graphs of
s(t) and
sin(t), over the given domain. Comment on these findings. Hence, give the exact coordinates of these intersection points.
Component 3
Variation or further specification of assumption or conditions involved in the context to focus on a particular feature or aspect related to the context
Brian has recently purchased a grandfather clock. The rate at which the hands of the clock move is controlled by a pendulum, which is kept in regular motion by slowly descending weighted chains.
When the weights reach their lower point and stop moving, the pendulum swing begins to change, causing the hands of the clock to slow down and gradually stop. From the time when the swing begins to change, the horizontal displacement,
s cm, of the point,
P, at the end of the pendulum, from the vertical, as shown in the following diagram, can be modelled by functions with the rule
s(t) = 5e–ktsin(at), where
t > 0 is the time in seconds after the pendulum swing begins to change and
k and
a are real constants. For Brian's clock,
k = 0.2 and
a = 1.
- Find the horizontal displacement of the pendulum for several seconds after the weights stop descending, and draw a series of diagrams corresponding to the position of the pendulum at these times.
- Draw a series of diagrams of the position of the pendulum the first several maxima. If the pendulum is deemed to have come to rest when the swing is less than 0.01 cm, find how long the pendulum takes to come to rest.
- Brian has a friend, Jana, who also bought a similar grandfather clock. Jana's clock is modelled by the same rule for the horizontal displacement when the weights stop descending, where
k = 0.4 and
a = 1. Draw the graph of the two pendulums' horizontal displacement for
t ≥ 22. Compare the behaviour of the two pendulums and discuss how the different values for
k affect the motion of the point
P after the swing of the pendulum begins to change.
Areas of study
The following content from the areas of study is addressed through this task.
|
Area of study |
Content dot points |
| Functions, relations and graphs | 2, 3, 4, 5, 6 |
| Algebra, number and structure | 4, 5 |
| Calculus | 2, 3, 4, 5 |
| Data analysis, probability and statistics | –
|
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
|
Outcome |
Key knowledge dot points |
Key skills dot points |
|
1 | 1, 2, 3,4, 6, 7, 9, 10, 11 | 1, 2, 6, 7, 8, 9, 10, 11, 12 |
|
2 | 1, 2, 5 | 1, 2, 4, 5, 6, 7 |
|
3 | 1, 2, 3, 4, 5, 6, 8 | 1, 2, 3, 4, 5, 6, 7, 9, 11, 12 |
The application task is to be of 4–6 hours' duration over a period of 1–2 weeks.
Introduction
A context such as the following could be used to develop an application task that investigates how a variety of functions, and piecewise (hybrid) functions constructed from these, could be used to model sections of pathway, such as parts of a bicycle track adjacent to a river, creek or wetland: for example, the Yarra Bend public park in Melbourne.
The process of constructing such a function is called
splining.
Component 1
Introduction of the context through specific cases or examples. Students should
Consider the problem of determining a quadratic function
f:R → R,f(x) = ax2 + bx + c, the graph of which passes through three specified points. Suppose two of these points,
A and
B, have coordinates (1, 4) and (2, 2) respectively. The third point,
C, has an
x-coordinate of 4 and is given as (4, k) where
k is an arbitrary real constant.
Explore the effect of varying
k on the graph of the function.
- Suppose that
C is determined to be (4, 1.5). Investigate cubic functions of the form
f:R → R,f(x) = ax3 + bx2 + cx + d with graphs that pass through the points
A,
B and
C.
Explore the effect of
d on the behaviour of the graphs of these cubic functions. Identify a value of
d that gives a cubic function closely matching the quadratic function that passes through the same three points.
- A fourth point, D, has coordinates (0,
m). For different values of
m find pairs of quadratic functions, the first pair containing points D,
A and
B and the second containing the points
B and
C. These two curves must be smoothly joined at B. Determine the effect of
m on the behaviour of the graphs produced.
Component 2
Consideration of general features of the context. Students should
Consider the various sections of the river using different combinations of specified coordinates and dimensions. The following provides a sample.
A new bicycle track is to be constructed along the Yarra River in Kew between two pedestrian bridges labelled
A and
B on the map shown below.
The track cannot be constructed on the western side of the river due to the presence of the golf course.
The track is to follow the curves of the river on the eastern side. That is, it will go from A to
B by the boathouse kiosk, passing between the river and Smith Oval.
- Explore how a model can be developed between the pedestrian bridges
A and
B using a series of smoothly joined quadratic functions.
- Design a measure for how well the pathway matches the curve of the river and apply it to the model.
Component 3
Variation or further specification of assumption or conditions involved in the context to focus on a particular feature or aspect related to the context. Students should
Improve the fit of your bicycle track, according to the measure you have designed, by using a combination of different types of functions.
Alternatively, identify an outline, curve or path in some other context and suitably model this by a piecewise function, which may include function types other than polynomial functions.
Areas of study
The following content from the areas of study is addressed through this task.
|
Area of study |
Content dot points |
| Functions, relations and graphs | 1, 2, 5, 6 |
| Algebra, number and structure | 1, 4, 5, 6 |
| Calculus | 3, 4 |
| Data analysis, probability and statistics | –
|
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
|
Outcome |
Key knowledge dot points |
Key skills dot points |
|
1 | 1, 4, 6, 7, 9, 10, | 1, 6, 9, 10, 12 |
|
2 | 1, 2, 3, 5 | 1, 2, 3, 5, 7 |
|
3 | 1, 2, 3, 4, 5, 8 | 1, 2, 3, 4, 5, 6, 7, 9, 11, 12
|
The application task is to be of 4–6 hours' duration over a period of 1–2 weeks.
Introduction
A context such as the following could be used to develop an application task that investigates graphs of polynomial functions of the form
f:R → R,f(x) = m(x - a)s(x - b), the key features of these graphs, and the number of solutions to equations of the form
f(x) = p, where
p ∈ R.
Component 1
Introduction of the context through specific cases or examples. Students should
- Consider the function
f:R → R,f(x) = (x - 1)2 (x - 2). Sketch the graph of
y = f(x), and clearly indicate all key features. Find the values of
x for which
f(x) = p has one, two or three solutions, where
p is a real number.
- State the transformations required to map the graph of
y = f(x) onto the graph of
. If there is a turning point at (2, 3), find all possible values of
n and
k. Sketch the corresponding graphs.
- State the transformations required to map the graph of
y = f(x) onto the graph of
y = Af(x - h). If there is a turning point at
(–1, 4), find all possible values of
A and
h. Sketch the corresponding graphs.
- The graph of
y = f(x) is mapped onto graph of
y = Af(n(x - h)) + k. Discuss how the values of
A,
n,
h and
k change the graph of the original function under various transformations.
Component 2
Consideration of general features of the context. Students should
- Now consider the function
f:R → R, f(x) = m(x - a)2(x - b) where
m, a, b ∈ R.
Investigate the graphs of
y = f(x) for combinations and ranges of values of the parameters
a,
b and
m.
- In each of the cases in step a., find the values of
p ∈ R for which
f(x) = p has one, two or three solutions.
- State the transformations required to map the graph of
y = f(x) onto the graph of
y = Af(nx) + k, where
A,n and
k ∈ R. Investigate how
A,n and
k, and a, b and m relate to the location and nature of the stationary points of the graph of
y = Af(nx) + k.
Component 3
Variation or further specification of assumption or conditions involved in the context to focus on a particular feature or aspect related to the context. Students should
- Consider the function
f: R → R, f(x) = m(x - a)s (x - b), where
m,a,b ∈ R a and
s ∈ N. Investigate the graphs of
y = f(x) for cases where
a < b,m > 0 and
s ∈ N. What generalisations can be made?
- Let
f: R → R, f(x) = (x - a)s (x - b) where
a,b ∈ R, a < b and
s ∈ N. Find the values of
p for which
f(x) = p has zero, one, two or three solutions when
s = 1, 2, 3, 4 and 5. What generalisations can be made?
Areas of study
The following content from the areas of study is addressed through this task.
|
Area of study |
Content dot points |
| Functions, relations and graphs | 1, 3, 4, 5 |
| Algebra, number and structure | 1, 4, 5 |
| Calculus | 3, 4, 5 |
| Data analysis, probability and statistics | –
|
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
|
Outcome |
Key knowledge dot points |
Key skills dot points |
| 1 | 1, 2, 3, 9, 10, 11 | 1, 2, 6, 9, 10, 11, 12 |
| 2 | 1, 2, 3, 5 | 1, 2, 4, 5, 6, 7 |
| 3 | 1, 2, 3, 4, 5, 8 | 1, 2, 3, 4, 5, 6, 7, 9, 11, 12
|
The application task is to be of 4–6 hours' duration over a period of 1–2 weeks.
Introduction
A context such as the following could be used to develop an application task that investigates the use of a special kind of cubic polynomial function, called Bernstein polynomials, whose graphs form what are called Bezier curves. These are named after the French automobile engineer Pierre Bezier who developed their application to a new computer-aided design tool for the Renault car manufacturing corporation in the 1960s. Today, Bezier curves are a key component of graphic design applications.
Bezier curves use polynomial functions of low degree, such as cubic polynomials over a restricted domain, to specify the coordinates of the points that make up these curves. These functions, called Bernstein polynomials, provide local control of shape, based on a small set of points called
control points, and have
graphs that are continuous and smooth curves for which the derivative can be found at any point on the curve. Shapes constructed using drawing packages and the outlines of letters produced by printers are typically based on a set of routines that use these curves.
A cubic Bezier curve drawn over the interval
0 ≤ t ≤ 1 is produced by graphing a relation that has its
x and
y coordinates specified respectively by the cubic polynomial functions:
x = a(1 - t)3 + 3ct(1 - t)2 + 3et2(1 - t) + gt3
y = b(1 - t)3 + 3dt(1 - t)2 + 3ft2(1 - t) + ht3
Where the coefficients
a,
b,
c,
d,
e,
f,
g and
h are obtained from the coordinates of the four control points (a,
b), (c,
d), (e,
f) and (g,
h).
The gradient of the tangent to the curve for a particular value of
t can be determined, using the chain rule for differentiation, by the relationship:
The shape of the Bezier curve produced depends on the selection of coordinate values for the control points. Where more than one Bezier curve is used to produce a required shape, these curves will need to be joined smoothly to produce a good image. The three components for an application task could be developed as follows.
Component 1
Introduction of the context: through specific cases or examples students should
- Consider the simpler case of quadratic Bezier curves.
- Select four distinct points as control points and determine the equations for the
x and
y coordinates as functions of
t to specify a particular cubic Bezier curve. Represent the corresponding Bezier curve using a table of values and the graph of the relation.
- Consider the gradient of the tangent to the curve at various points on the curve, including the relation between the tangents to the first and last control points and the location of the second and third control points.
Component 2
Consideration of general features of the context students should
- Vary the control points and consider the Bezier curves produced, including cases that lead to, for example, straight lines and loops.
- Investigate the selection of control points to produce a reasonable representation of a particular shape: for example, how many letters of the alphabet can be reasonably approximated by a single Bezier curve?
- Consider how well a cubic Bezier curve matches the required shape and any limitations on possible shapes that can be represented using these curves.
Component 3
Variation or further specification of assumption or conditions involved in the context to focus on a particular feature or aspect related to the context. Students should
- Represent more complicated shapes formed by piecing together several Bezier curves.
- Identify the location of control points for a pair of cubic Bezier curves that are smoothly joined to represent other letters, or for several Bezier curves that are smoothly joined to represent a shape such as the outline or cross section of a car.
Areas of study
The following content from the areas of study is addressed through this task.
|
Area of study |
Content dot points |
| Functions, relations and graphs | 1, 5, 6 |
| Algebra, number and structure | 1, 5 |
| Calculus | 3, 4 |
| Data analysis, probability and statistics | –
|
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
|
Outcome |
Key knowledge dot points |
Key skills dot points |
| 1 | 1, 3, 4, 7, 10, 11 | 1, 2, 7, 9, 11, 12, 13 |
| 2 | 1, 2, 3, 5 | 1, 2, 3, 4, 6, 7 |
| 3 | 1, 2, 3, 4, 5, 8 | 1, 2, 3, 4, 5, 6, 7, 9, 11, 12
|
The modelling or problem-solving task is to be of 2–3 hours' duration over a period of 1 week.
Introduction
Polls provide a topical and regular insight into the relative popularity of political parties over time, in particular as events occur and are reported in the media and trends change.
Popularity on a two-party preferred basis as indicated by polls is a context for inference about proportions with respect to a population based on sampling. Consider a country that has a population of around 15 million voters on an electoral roll. Polls inform public consideration and debate on various matters of policy.
Part 1
- Plot graphs of the distribution of sample proportions for sample sizes of 50, 100 and 200 for
p = 0.43, 0.52, 0.61
- Explain what these graphs indicate.
Part 2
- Randomly select an integer in the range [30, 60] and use this to generate a population of 1000 voters, with that value as the percentage of the population who would vote for a given party on a two-party preferred basis.
- Generate 50 random samples of size
n = 60 from this population and use each of these to find a point estimate for the true population proportion. Graph the distribution of the sample proportions and state its mean and standard deviation.
- Use each point estimate to construct a confidence interval for p at a 90% level of confidence. Graph all of these intervals together as a set of horizontal line segments, one under the other, and use them to explain the relationship between the true value of the population proportion,
p, and this set of confidence intervals for a 90% level of confidence.
Part 3
The
Margin of Error Table relates sample size, sample proportion and margin of error at a 95% level of confidence.
- Show how the figures for the row corresponding to a sample size of 2000 are obtained.
- Draw graphs of several functions to illustrate how the maximum margin of error varies for different sample sizes and levels of confidence.
- Suppose that it costs $50 per individual response gathered as part of a survey. Discuss what you think might be a reasonable combination of sample size, level of confidence, margin of error and total cost.
Areas of study
The following content from the areas of study is addressed through this task.
|
Area of study |
Content dot points |
| Functions, relation and graphs | – |
| Algebra, number and structure | – |
| Calculus | – |
| Data analysis, probability and statistics | 1, 2, 4
|
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
|
Outcome |
Key knowledge dot points |
Key skills dot points |
|
1 | 1, 14, 16, 17 | 1, 16, 18, 19, 20 |
|
2 | 1, 2, 4, 5 | 1, 2, 3, 4, 6, 7 |
|
3 | 1, 2, 3, 4, 6, 7, 8 | 1, 3, 4, 5, 8, 9, 10, 11 |
The modelling or problem-solving task is to be of 2–3 hours' duration over a period of 1 week.
Introduction
A context such as the following could be used to develop a two-part problem-solving task that involves a composition of functions leading to the standard normal distribution, and the use of simulations to investigate the distribution of proportions in probability experiments.
Part 1
Let
f: R → R,f(x) = ex and
g: R → R,g(x) = -x2.
- Plot the graphs of
f(x) and
g(x) and explain how the shape of the graph of
f(g(x)) can be deduced from these.
- Plot the graph of
f(g(x)) and clearly identify its key features.
- Use sets of trapeziums to form a sequence of under-estimates and over-estimates for the area bounded by the graph of
f(g(x)) and the horizontal axis between
x = –10 and
x = 10.
- Use the results from step c. to find approximate values for a and
b such that
h: R → R,h(x) = ae-bx2 forms a probability density function, with mean 0 and standard deviation 1.
- Plot the graph from step d. on the same set of axes as the standard normal distribution and comment on similarities and differences.
Part 2
- A pair of standard dice are rolled simultaneously. Use technology to simulate this experiment for 60 rolls of the dice and record the set of outcomes. What is the proportion of rolls for which the two dice had the same value?
- Run the simulation 100 times and plot the distribution of this proportion. Describe the distribution.
- Now consider two identical packs of 10 cards numbered 1 to 10. Both packs are shuffled thoroughly. The first card is turned over from each pack and the result is recorded. This is then repeated for the second card from each pack, the third card from each pack and so on, through to the 10th and final card of each pack. Use technology to simulate this experiment and record the set of outcomes. How many pairs of cards in the experiment had the same value?
- Run this simulation 100 times and plot the distribution of proportions for the number of times when the pair of cards had the same value. Describe this distribution.
- Give an estimate for the probability that there is at least one pair of matching cards and explain how this estimate was obtained.
Areas of study
The following content from the areas of study is addressed through this task.
|
Area of study |
Content dot points |
| Functions, relations and graphs | 2, 3, 4, 5 |
| Algebra, number and structure | 3, 5 |
| Calculus | 3, 4, 7, 9 |
| Data analysis, probability and statistics | 1, 2, 3
|
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
|
Outcome |
Key knowledge dot points |
Key skills dot points |
|
1 | 1, 2, 4, 6, 7, 11, 12, 13, 14, 17 | 1, 2, 4, 5, 12, 14, 15, 16, 17, 18, 19, 20 |
|
2 | 1, 2, 4, 5 | 1, 2, 3, 4, 5, 6, 7 |
|
3 | 1, 2, 3, 4, 7, 8 | 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13 |
The modelling or problem-solving task is to be of 2–3 hours' duration over a period of 1 week.
Introduction
A context such as the following could be used to develop a modelling or problem-solving task that involves modelling travel over different terrains at different average speeds for each terrain, and using this information to optimise the time of travel. Bushwalkers travel over different types of terrain, from cleared to dense bush. The denseness of the bush and the ruggedness of the terrain influence the average speed of travel. By planning a route to take such factors into consideration, the total time taken to travel from one point to another can be optimised. In calculating estimates of the time for a particular route, a walker uses his or her average speed for each different type of terrain they are likely to encounter.
For a walk through a particular type of terrain, the distance travelled,
d km, can be calculated as the product of the average speed,
v km/h, and the time travelled at this speed, t hours. On a typical two-day walk a bushwalker might cover a distance of up to 30 km with walking speeds of up to 5 km/h over cleared terrain.
Part 1
- For a typical two-day walk, choose several representative values for average speed and draw a graph of the relationship between
t and
d for each of these values.
- Similarly, choose several representative values for the distance to be travelled and draw a graph of the relationship between t and
v for each of these values.
- Discuss the key features of each of the two families of graphs and the differences between them.
Part 2
A bushwalk is planned from Ardale to Brushwood. As shown in the diagram below.
The direct route, a distance of 14 km, goes entirely through rugged bush country. However, there is a large square clearing situated as shown. This clearing has one diagonal along the perpendicular bisector of the direct route and one corner,
C, at the midpoint of the direct route.
One of the bushwalkers believes that time will be saved if they travel from Ardale to Brushwood on a route similar to the one shown passing through
P and
Q, where the section
PQ is parallel to the direct route. The side length of the square clearing is 7 km, and the part of this route that goes across the square clearing is parallel to the direct route.
- Choose a suitable variable, and hence determine a mathematical relationship that can be used to determine the total time for a route of this type. Draw the graph of this relationship and discuss its key features.
- Find and describe the route for which the travelling time will be least and compare it with the direct route.
Areas of study
The following content from the areas of study is addressed through this task.
|
Area of study |
Content dot point(s) |
| Functions, relations and graphs | 6 |
| Algebra, number and structure | 5 |
| Calculus | 1, 4, 5 |
| Data analysis, probability and statistics | – |
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
|
Outcome |
Key knowledge dot points |
Key skills dot points |
|
1 | 1, 4, 6, 7, 9, 10 | 1, 6, 9, 12 |
|
2 | 1, 2, 3, 5 | 1, 2, 3, 4, 5, 6, 7 |
|
3 | 1, 2, 3, 4, 5, 6, 8 | 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13 |
The modelling or problem-solving task is to be of 2 - 3 hours duration over a period of 1 week.
Introduction
A context such as the following could be used to develop a modelling or problem-solving task which involves modelling the shape of a designer two-part window feature for a section of wall, and the dimensions and area of the design.
The section of the wall is 4 m wide and 3.5 m high. The window is a symmetrical design which fits in the middle of the wall horizontally. The base of the window forms a straight line 0.5 m above and parallel to the floor and is 2 m in length. The lower part of the window has two straight line slant edges, and these are 1.5 m apart at the height of 1.5 m from the floor. The window designer is considering a range of possibilities for the upper part of the window, the highest point of which is to be at most 3 m from the floor.
The designer constructs a graph of the window design using a set of axes with the origin on the floor at the middle of the wall.
Part 1
Initially the designer considers and upper part consisting of two-line segments which join onto the top of the lower straight edges, at an angle of 45° to the horizontal, and extend to the point where they meet.
- Use functions to define the sections of the window's edges and draw a graph showing the wall and the window, labelling all key points with their coordinates.
- Find the area of the window.
- Consider a family of related designs where the angle the edges of the upper part make with the horizontal is varied. Show several examples and calculate the area of the window in each case. What is the largest possible value for this angle, and the corresponding area of the window?
Part 2
As an alternative, the designer considers using an arch for the top part of the window.
- Draw the graph where the arch is a semi-circle. Calculate the area of the window. Do the lower and upper parts of the window join smoothly?
- Draw several graphs for the upper part defined by the family of functions with rule of the form
f(x) = ax2 + b, where a and b are non-zero real constants.
- For the case where the lower and upper parts of the window join smoothly, calculate the corresponding area of the window.
- What happens when a function with rule of the form
g(x) = a sin(bx) + c is used to model the upper arch, if the two parts are to be smoothly joined?
Part 3
The designer decides that while a smooth join of the two parts of the window is a critical requirement, it is not necessary for the arch to be smooth at its top point, so a symmetrical hybrid function based on part of the graph of some other function could be used to represent the arc.
Consider some other possible modelling functions and identify which of these gives the maximum window area for different choices of functions and defining parameters.
The following images show some arches from buildings along North Terrace in Adelaide.
Areas of study
The following content from the areas of study is addressed through this task.
|
Area of study |
Content dot point |
| Functions, relations and graphs | 2, 3, 6 |
| Algebra, number and structure | 5, 6, |
| Calculus | 3, 4, 5, 6, 10 |
| Data analysis, probability and statistics | -
|
Outcomes
The following outcomes, key knowledge and key skills are addressed through this task.
|
Outcome |
Key knowledge dot point |
Key skill dot point |
|
1 | 1, 2, 4, 7, 10, 12 | 1, 2, 6, 10, 12, 13, 14, 15 |
|
2 | 1, 2, 3, 5 | 1, 2, 3, 5, 7 |
|
3 | 2, 3, 4, 5, 8 | 2, 3, 4, 5, 6, 7, 9, 11, 12, 13
|