Teaching and Learning

Line segment graphs and review of coordinate geometry

Introduction

This learning activity is designed to use a line segment graph as a context for review of coordinate geometry. Parts 1 to 5 are intended to be completed without the use of technology. Parts 6 and 7 include the use of technology.

Part 1

Consider the set of points and coordinates given below:

Plot these points on a graph, and connect them to form the line segments AB, BC, CD, DE and EF.

Calculate the total area bounded by these line segments and the horizontal axis.

Part 2

For each line segment find:

1. the gradient
2. the coordinates of the midpoint
3. the length
4. the equation of the line that contains the line segment
5. the angle the line segment makes with respect to the horizontal, correct to the nearest degree

Part 3

Let P be the point with coordinates (1, 6)

1. find the equation of the line that passes through P and is parallel to the line segment AB
2. find the equation of the line that passes through P and is perpendicular to the line segment AB
3. draw both of these lines on the graph

Part 4

Let Q be the point with coordinates (11, 8)

1. find the equation of the line that passes through Q and is parallel to the line segment CD
2. find the equation of the line that passes through Q and is perpendicular to the line segment CD
3. draw both of these lines on the graph

Part 5

Find the coordinates of the point of intersection of the line passing through BC and the line with equation 3x - 2y = 24.

Part 6

Use technology to define a piecewise (hybrid) function with pieces for each of the line segments
AB, BC, CD and DE and plot its graph.

Clearly indicate each of the points A, B, C, D, E and F on the graph and include the line segment EF.

Part 7

Verify the results from Parts 1 to 5 of the learning activity, and plot corresponding graphs on the same graph as in Part 6.

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Example of learning activity: Exploring the graphs of power functions

Introduction

This learning activity explores power functions of the form f(x) = xⁿ by

• Using the graphing functionality of technology to explore power functions f(x) = xⁿ for varying values of n.
• After observing the patterns and characteristics for the varying sets of values for n, students should generalise and summarise their findings.

Part 1

Using technology, have students explore the graph of f: R → R, f(x) = xⁿ for the following sets of values of n. Have students sketch, without technology, their expected shape of the graphs for each set of  before verifying the correct form using their technology.

1. Consider n ∈ {0,2,4,6,8,10} by setting up a slider from 0 ≤ n ≤ 10 in steps of 2.

          Describe how the graph changes as n changes, including when n = 0.

          Explain how these changes relate to the rule of the function.

2. Consider n ∈ {1,3,5,7,9} by setting up a slider from 1 ≤ n ≤ 9 in steps of 2.

          Describe how this set of graphs differs to the one in a. and why.

          Explain why the graph for n = 1 looks different from the other graphs.

      c. Consider n ∈ {-10,- 8,-6,-4,-2} by setting up a slider from -10 ≤ n ≤ -2 in steps of 2.

          Describe how this set of graphs differs to the one in a. and why.

      d. Consider n ∈ {-9,7,-5,- 3,-1} by setting up a slider from -9 ≤ n ≤ -1 in steps of 2.

          Describe how this set of graphs differs to the one in b. and why.

Part 2

Summarise findings from a. to d. above and include a sketch of the generalised shape for each set of values of n, noting key characteristics and exceptions.

Part 3

Continue the explorations of the graph of f: R → R, f(x) = xⁿ for fractional values of n. Have students sketch their expected shape of the graphs for each set of n before verifying the correct form using their technology.

      e. Let  where w ∈ {0,1,2,3,4,…,10} with a slider from 1 ≤ w ≤ 10 in steps of 1.

          Describe how the general shape of the graph differs when w is odd or even.

          Explain how these changes relate to the rule of the function and why there is no graph for n = 0.

      f. Let   where w ∈ {-10,-9,-8,-7,…,-1} with a slider from -10 ≤ w ≤ -1 in steps of 1.

         Describe how the general shape of the graph differs when w is odd or even.

Part 4

Extend the summary to include findings from Part 3.

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Example of learning activity: How many different shapes of polynomial function graphs are there?

Introduction

This learning activity investigates the range of different shapes of polynomial graphs for some polynomial functions of low degree.

Part 1

There is only one shape for the graph of a linear functionf: R → R, f(x) = ax + b, which is a straight line.

1. Use a range of examples to illustrate how the graph of a linear function can be obtained from the graph of y = x by a combination of dilation, translation and possibly reflection in the horizontal axis.

          There is also only one shape for the graph of a quadratic function
          f: R → R, f(x) = ax² + bx + c, a ≠ 0, which is a parabola.


2. Use a range of examples to illustrate how the graph of a quadratic function can be obtained from the graph of y = x² by a combination of dilation, translation and possibly reflection in the horizontal axis.

Part 2

Consider the cubic polynomial function f: R → R, f(x) = ax³ + bx² + cx + d, a ≠ 0.

1. Systematically vary each of the coefficients in turn and summarise the different shapes of graphs that result.
2. Describe the shape of the graph in terms of the number of stationary points and/or points of inflection.
3. Is it possible to obtain the graph of any cubic polynomial function from the graph of y = x³ by a combination of dilation, translation and possibly reflection in the horizontal axis? Explain your reasoning, using examples as appropriate.

Part 3

Repeat the investigation in Part 2 for quartic polynomial functions

f: R → R, f(x) = ax⁴+ bx³ + cx² + dx + e, a ≠ 0

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Example of learning activity: Bisection for a cubic

Introduction

This learning activity applies the bisection method for finding an approximate solution to an irrational root of a cubic polynomial functionf: R → R, f(x) = x³ - 22x² + 163x + 404.

Part 1

1. Use the rational root theorem to show that the function f has no rational roots.
   
2. Plot the graph of f and use a table of values to identify the unit interval with integer endpoint that contains the real root of f.
   
3. Vary the graphing domain and use a systematic guess-check-refine approach to find an approximate value for the root correct to one decimal place.
   
4. Complete a table of values for the first four iterations of the bisection process.
   
5. Use a short program or technology application to apply the bisection process and determine the number of iterations required to obtain an approximate value correct to one decimal place, three decimal places and five decimal places.
   

Part 2

Repeat this analysis for a selection of other cubic polynomial functions.

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Example of learning activity: Are there rational roots?

Introduction

This learning activity explores when quadratic functions of the form f: R → R, f(x) = ax² + bx + c, a ≠ 0 have rational roots or not.

Part 1

Consider the family of quadratic functions defined by f: R → R, f(x) = x² - bx - 36

Identify values of b for which f has rational roots and illustrate these with their corresponding graphs.

Investigate for other integer values of c.

Part 2

Consider the family of quadratic functions defined by f: R → R, f(x) = x² - 8x + c .

Identify integer values of c for which f has rational roots and illustrate some of these with their corresponding graphs.

Investigate for other integer values of b.

Part 3

Use the rational root theorem to decide if the quadratic function f(x) = 2x² - 3x -7 has rational roots. Repeat this for several other choices of small integer values for the coefficients a, b and c.

The discriminant can also be used to determine whether a quadratic function has rational roots or not. Identify how this can be done by considering the discriminant of several quadratic functions that have rational roots.

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Example of learning activity: Filling vases with water

Introduction

There are many different shaped vases. If a vase is filled by pouring in water at a constant rate, the depth of water in the vase will be a function of how long the water has been pouring in from empty, and the shape of the vase.

Assuming that a vase has rotational symmetry about its central axis, its shape can be represented by a planar cross-section that contains this axis.

This task has two parts and involves qualitative analysis to tackle two problems and explain reasoning:

1. Given the cross-section of a vase, what do the graphs of the depth of water in the vase as a function of time, and the corresponding graph of the rate of change of the depth of water in the vase as a function of time look like?
2. Given a graph of the depth of water in the vase as a function of time, and the graph of the rate of change of the depth of water in the vase as a function of time, what is the shape of the vase?

Note that both functions will be strictly increasing functions as water is being poured in at a constant rate and the vases have a finite volume.

Part 1

Consider a collection of different shaped vases by means of their cross-section and construct a corresponding depth-time graph and rate of change in depth-time graph.

There are a number of related interactive demonstrations available online.

Students could be asked to draw an estimate of the rate of change in depth-time graph before each of the animations are shown.

Part 2

Consider a collection of different depth-time graphs and rate of change in depth-time graphs and construct corresponding cross-sections for vases.

There are various images of rate of change in depth-time graphs available online.

Students could be asked to draw an approximation of the shape of the container before its revealed.

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Example of learning activity: Experiments, simulation and probability

Introduction

This learning activity investigates probabilities obtained from experiments and simulations.

Part 1

1. Using two coins select one of the following outcomes: HH, HT, TT.
2. Toss the two coins simultaneously 120 times and graph the distribution of outcomes. How many times did your outcome occur.
3. Use technology to run a simulation of the experiment, repeat it 100 times. Discuss any similarities and differences in the shape of these distributions. For each run of the simulation, record the number of times the selected outcome occurs.
4. Plot the distribution of the number of times the selected outcome occurs across the 100 runs of the simulation, and describe this distribution.

Part 2

Consider two standard packs of 52 playing cards. Two people each have one complete pack. They randomise the order of the cards by thoroughly shuffling each pack.

The two people then turn up the top card of their pack and lay it on a table at the same time. The cards are compared, and it is noted if the pair of cards are an exact match or not.

This procedure is repeated until all pairs of cards have been similarly compared.

1. Students work in pairs to carry out this experiment 10 times, noting how many matching pairs of cards occur. The distribution of class results can be formed and discussed.
2. Use technology to implement a simulation of this experiment. Students run the simulation 100 times, 200 times, 300 times and so on until they have run the simulation 1000 times, in each case graphing the distribution of the proportion of times (zero, one, two, three …)  matching pairs of cards occur. In the long run, what is the most likely number of matching pairs of cards?
3. Form an estimate of the probability that no matching pairs of cards occur.

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Example of learning activity: Estimating radians, sine and cosine on a unit circle

Introduction

This learning activity is a quick visual activity based on estimating the location of points corresponding to radian measure on the circumference of a unit circle, and the (signed) length of line segments corresponding to sine and cosine. It also looks at finding rough visual estimates for solutions to equations of the form sin(x) = a or cos(x) = a in the interval [0,2π].

Part 1

1. Draw a unit circle using a scale of 10 cm = 1 unit.
2. Use the values  to place points corresponding approximately to 1, 2, 3, 4, 5 and 6 radians on the circumference of the circle.
   Use the appropriate vertical and horizontal line segments to find approximations for the corresponding values of sine and cosine.
   
3. Use the values  to approximately locate points on the circumference of the unit circle corresponding to 0.6, 2.3, 4.6 and 5.1 radians. Estimate the corresponding values for sine and cosine in each case.
   
4. Use technology to check these estimates.

Part 2

1. Draw a unit circle using a scale of 10 cm = 1 unit.
2. Draw in the horizontal line y = 0.3 and use this to estimate the solutions to sin(x) = 0.3 over the interval [0,2π]. Repeat this for various other horizontal lines y = k where -1 < k < 1.

3. Draw in the vertical line x = -0.8 and use this to estimate the solutions to cos⁡(x) = -0.8 over the interval [0,2π]. Repeat this for various other vertical lines x = k where -1 < k < 1.

4. Use technology to check these estimates.

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Example of learning activity: Data representations and conditional probability

Introduction

The same data is presented using three different representations, Venn diagram, two-way table and tree diagram, to strengthen students' understanding of the connections between them.

Students show and explain the conditional probability relation using each of the three representations.

Part 1

A travel survey of 100 Year 11 students found 16 per cent of students have visited Uluru but not Queensland and 74 per cent had visited a theme park in Queensland. Of the 74 per cent of students who have been to a theme park in Queensland, some have also visited Uluru. Ten per cent of the students surveyed have not been outside of Victoria.

Represent this data using each of these three representations:

• Venn diagram
• Two-way table
• Tree diagram.

Find how many students have been to Uluru and visited a theme park in Queensland, and the total number of students who have been to Uluru.

Part 2

Introduce students informally to the concept of conditional probability by posing some targeted questions, such as the following:

1. What is the probability that a given student has been to a theme park in Queensland?
2. What is the probability that a given student has been to a theme park in Queensland, if you know they've been to Uluru?

Show and discuss the difference between questions i. and ii. using the three representations.

Part 3

Using each of the three representations, have students show and explain the answer to the following question: What is the probability that a given student visited both Uluru and a theme park in Queensland, given that they've visited a theme park in Queensland?

Part 4

Have students relate each part of the conditional probability relation  to each of the three representations.

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Example of learning activity: Linear equations for cubic roots

Introduction

This learning activity uses linear functions that are tangents to a graph to determine an approximate value for an irrational root of a cubic polynomial function. That is, the horizontal axis intercept of the tangent to the graph of the function at a point close to the root approximates this root.

Part 1

Consider the cubic polynomial function f: R → R, f(x) = 2x³ - 34x² - 196x - 373 .

1. Plot the graph of the function and show that it only has one real root.
2. Find the unit interval [a, b] with integer endpoints that contains this root. By varying the plotting domain, find the approximate value of the root correct to one decimal place.
3. Use the rational root theorem to show that this root is irrational.

Part 2

1. Find the horizontal axis intercept, b₁ , of the tangent to the graph of f at (b, f(b)). Plot the graph of the tangent on the same set of axes as the graph of the function, with a suitable graphing domain to show the root and this approximation.
2. Find the horizontal axis intercept, b₂ , of the tangent to the graph of f at (b₁, f(b₁)). Plot the graph of the tangent on the same set of axes as the graph of the function, with a suitable graphing domain to show the root and this second approximation.
3. Find the horizontal axis intercept, b₃ , of the tangent to the graph of f at (b₂, f(b₂)). Plot the graph of the tangent on the same set of axes as the graph of the function, with a suitable graphing domain to show the root and this third approximation.

Part 3

1. Use technology to implement the algorithm for Newton's method, using both a and b as initial values.
2. Explore how quickly Newton's method works for other initial values.
3. Apply Newton's method to a cubic polynomial function with three real roots and explore the variation in sensitivity of the method based on the choice of initial value.

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Example of learning activity: Simulating rectilinear motion

Introduction

Straight line motion occurs in many contexts.

This learning activity looks at two approaches to representing rectilinear motion graphically, for a given position-time function, where the points of the graph represent the particle. The first approach is to construct a static time series plot of as set of points (t,x(t)) for integer values of t from 0 to some specified value, where t is in seconds and x(t) is in metres. The second approach is to construct a dynamic simulation (or animation) plot as a set of points (0,x(t)) for integer values of t from 0 to some specified value, where t is in seconds and x(t) is in metres per second. This will show the particle moving along the line x = 0: that is, along a vertical axis. These graphs can be used to describe the motion qualitatively. As the points are 'one second apart' the difference between consecutive positions gives the velocity of the particle in metres per second.

Task

For each of the following position-time functions:

1. Plot the set of points (t, x(t)) over the specified domain.
2. Plot the set of points(0, x(t)) over the specified domain and run the simulation.
3. Give a qualitative description of the motion of the particle

Position-time functions:

1. x(t) = 1.5t, 0 ≤ t ≤ 10
2. x(t) = 20 - 2t, 0 ≤ t ≤ 10
3. 
   
4. x(t) = 200 - 4t², 0 ≤ t ≤ 7
5. x(t) = 2t, 0 ≤ t ≤ 10 and 20-1.5t, 10 < t ≤ 20
6. x(t) = 25 + 6 cos(t), 0 ≤ t ≤ 12
7. x(t) = 2(t-1)(t-9), 0 ≤ t ≤ 10
8. x(t) = 40-2^3-t, 0 ≤ t ≤ 10

Part 2

1. Devise several simulations and show them to another student. Have the student devise a qualitative description of the motion and conjecture a possible position-time function as a model for the motion.
2. Identify a range of contexts that may be modelled by rectilinear motion over a given domain.

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Example of learning activity: Numerical approximations for derivatives

Introduction

This learning activity looks at numerical approximations to derivatives by left and right secants and central difference.

If the derivative of a function f is defined, then it can be evaluated from first principles by either of the two limits:

For small positive values of h, these correspond to the left secant (backward difference) and right secant (forward difference) approximations for the derivative, that is:

The central difference is the average of these, and is used by technology to calculate numerical values for derivatives:

In the following work let h = 0.0001.

Part 1

Consider the quadratic functionf: R → R, f(x) = x² - 3x .

1. Construct a table of values for the left secant, right secant and central difference approximations for this function for x from –2 to 5 in steps of 0.5.
2. Plot the corresponding points for the central difference approximation, and draw a straight line through them, stating its rule.
3. Repeat steps a. and b. for several other quadratic functions.

Part 2

Repeat Part 1 for several simple cubic polynomial functions, the square root function and the basic hyperbola.

Part 3

Carry out similar analysis for f(x) = sin⁡(x) over the interval [0,2π] in steps of 0.1. What does the graph of the approximate derivative function look like? Repeat this analysis for g(x) = cos⁡(x) over the interval [0,2π] in steps of 0.1. What does the graph of the approximate derivative function look like?

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Example of learning activity: Piecing things together with polynomials

Introduction

A modelling context is used to introduce piecewise functions. Students draw on their knowledge of polynomial functions, domain and range, and continuity to construct a piecewise function using a context of their choosing.

Example

The mountain outline in the image below has been used as a context for creating the piecewise function f, which is a combination of three linear functions and two quadratic functions.

Task

Choose a context of interest and use technology to construct and graph a suitable piecewise function that models a profile or curve. The solution should include the context image with the graph of the piecewise function superimposed and the rule of the piecewise function specified.

The piecewise function must include the following:

• a minimum of three sections
• at least two polynomials of degree 2 or higher
• be continuous across the domain of the function, with values calculated to a suitable level of accuracy at endpoints of connecting pieces.

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Example of learning activity: A product function

Introduction

This learning activity considers a family of product functions based on power and exponential functions, and explores key features and the behaviour of their graphs.

Part 1

Consider the function f: R → R, f(x) = x e^(-x).

1. Draw the graph of y = x and y = e^–x on the same set of axes, and show how the graph of f could be obtained from these graphs.
2. Describe the behaviour of the graph of the function as x → – ∞ and as x → ∞.
3. Find the coordinates of any points of intersection between the graph of y = e^-x and the graph of f.
4. Find the derivative of f and hence identify the location and nature of any stationary points and the location of any points of inflection.
5. Show that y = x is tangent to the graph of f at the origin.

Part 2

Consider the family of functions f_n: R → R, f(x) = xⁿ e^-x, where n is a positive integer. The function in step a. is the case for n = 1.

1. Draw graphs of f_n for several values of n, and identify the nature and location of any stationary points, and the location of any points of inflection.
2. Find the coordinates of any points of intersection between the graph of y = e^-x and the graph of  f_n.
3. Describe the behaviour of the graph of the function as x → – ∞ and as x → ∞.
4. State the intervals over which f_n is strictly increasing, and the intervals over which it is strictly decreasing.
5. Summarise the key features of the graphs of these functions in terms of n.
   

Part 3

Let g be a differentiable function with domain R. Carry out similar analysis for h(x) = g(x)e^-x.

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Example of learning activity: Antiderivatives from derivatives

Introduction

This learning activity looks at how differentiation of particular functions with rules of the form xⁿ  f(x), where n is a positive integer, leads to finding an antiderivative of a closely related function. This technique is sometimes known as integration by recognition.

Part 1

Let f(x) = e^x and consider derivatives of functions of the form xⁿ e^x.

1. Differentiate xe^x and hence find an antiderivative for xe^x.
   
2. Differentiate x²e^x and hence find an antiderivative for x²e^x.
3. Differentiate x³e^x and hence find an antiderivative for x³e^x.

Describe a general process for finding an antiderivative for xⁿe^x.

Part 2

Repeat this analysis for:

1. f(x) = sin⁡(x)

    

2. f(x) = log_e(x).
   

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