Rencia LourensRADMASTE Centre

Using the CASIO fx-82ZA PLUS for functions in the FET band

Some remarksA calculator is a tool.Learners should

Know where answers come from.Understand mathematics.

Teachers shouldTeach the mathematics.Explain the reasoning behind why the

calculator methods work.BUT the calculator can (and should)

become a tool to assist.

CAPSFunctions form 35% of the Grade 12 paper

1, 45% in Grade 11 and 30% in Grade 10 (CAPS). The calculator can be used to support the calculations needed to draw and interpret the graphs of the functions.

Intersection of two graphsFind the points of intersection of the

straight line f(x) = x – 3 and the parabola g(x) = x2 – x – 6 if .

We need to beBe in TABLE mode.Have the DUAL table SETUP.

Who is NOT sure?

So how can I as a teacher use this to enhance understanding?

Some thoughtsThe meaning of simultaneous equations.The meaning of a plotted graph.

Next exampleFind the point(s) of intersections of the

graphs and This example is different from the previous

one becauseThe domain is not givenIt is not a convenient example where the

answer(s) are “in your face”.So we choose our own domain and start

with

x f(x) g(x)

-5 36 6.25

-4 24 5.25

-3 14 4.25

-2 6 3.25

-1 0 2.25

0 -4 1.25

1 -6 0.25

2 -6 -0.75

3 -4 -1.75

4 0 -2.75

5 6 -3.75

f(-2) > g(-2)f(-1) < g(-1)

f(3) < g(3)f(4) > g(4)

Hence somewhere between x = -2 and x = -1 we will have f(x) = g(x). (We will look at the other value later on).

We keep the table as is, but change our domain to

We also change the steps and make that 0,25.

x f(x) g(x)

-2 6 3.25

-1.75 4.3125 3

-1.5 2.75 2.75

-1.25 1.3125 2.5

-1 0 2.25

Also somewhere between x = 3 and x = 4 we will have f(x) = g(x).

We keep the table as is, but change our domain to

We also change the steps and make that 0,25.

x f(x) g(x)

3 -4 -1.75

3.25 -3.1875 -2

3.5 -2.25 -2.25

3.75 -1.1875 -2.5

4 0 -2.75

The graphs and intersect at(-1,5; 2,75)(3,5; 2,25)

Turning point of a parabolaFind the turning point of We do not know the range so we will start

with We do not need the second function, so we

CAN disable the second function.

x f(x)

-5 44

-4 31

-3 20

-2 11

-1 4

0 -1

1 -4

2 -5

3 -4

4 -1

5 4

So how can I use this as a teacher to enhance understanding?

Some thoughtsThe meaning of symmetryThe minimum valueThe meaning of a plotted graphThe shape of a quadratic function

Just checking – the turning point is (2; -5)

New exampleFind the turning point of We do not have a domain so we start with .

x f(x)

-5 118

-4 78

-3 46

-2 22

-1 6

0 -2

1 -2

2 6

3 22

4 46

5 78

The turning point should be

somewhere between x = 0

and x = 1

So……..We keep the table and change the domain

to….

And we make the steps…..

x f(x)

0 -2

0.25 -2.75

0.5 -3

0.75 -2.75

1 -2

The turning point is

Next exampleFind the turning point of .We do not know the domain hence…

x f(x)

-5 96.5

-4 70

-3 47.5

-2 29

-1 14.5

0 4

1 -2.5

2 -5

3 -3.5

4 2

5 11.5

The turning point should be

somewhere between x = 1

and x = 3

We will change the domain to .The steps should be .

x f(x)

1 -2.5

1.25 -3.5

1.5 -4.25

1.75 -4.75

2 -5

2.25 -5

2.5 -4.75

2.75 -4.25

3 -3.5

The turning point should be

somewhere between x = 2 and x = 2.25

We will change the domain to .The steps should be .

x f(x)

2 -5

2.0625 -5.0234375

2.125 -5.03125

2.1875 -5.0234375

2.25 -5

The turning point is

Finding the intercepts with the axesFind the intercepts with both the axes of

the graph of .Domain .Steps of 1

x f(x)

-5 56

-4 42

-3 30

-2 20

-1 12

0 6

1 2

2 0

3 0

4 2

5 6

y intercept

x intercept

x intercept

Just checking……. Where will the

turning point be?

So how can I as a teacher use this to enhance understanding?

Some thoughtsThe meaning of vs the meaning of .The meaning of a plotted graphSolving of quadratic equation

Next exampleFind the intercepts with both the axes of .Domain .Steps of 1.

x f(x)

-5 -43

-4 -31

-3 -21

-2 -13

-1 -7

0 -3

1 -1

2 -1

3 -3

4 -7

5 -13

y-intercept

Turning point should be

here

No x-intercept?

Seems as there are no x-intercepts.

Focus on turning point first.

Will be between x=1 and x=2.

The turning point is below the x-axis.

All the graph values are below the x-axis.

So no x-intercepts.

x

1 -1

1.25 -0.8125

1.5 -0.75

1.75 -0.8125

2 -1

Next exampleFind the intercepts with both the axes of .Domain .Steps of 1.

x f(x)

-5 -119

-4 -75

-3 -39

-2 -11

-1 9

0 21

1 25

2 21

3 9

4 -11

5 -38

y-interceptTurning point should be

here

x-intercept should be

here

x-intercept should be

here

Somewhere between x = -2 and x = -1 the one x-intercept should lie and somewhere between x = 3 and x = 4 the other x-intercept should lie.

So we are going to look at smaller domains and smaller steps.

x f(x)

-2 -11

-1.75 -5.25

-1.5 0

-1.25 4.75

-1 9

x-intercept

x f(x)

3 9

3.25 4.75

3.5 0

3.75 -5.25

4 -11

x-intercept

Looking at the reciprocal function

Work with domain

x f(x)

-5 1.333333

-4 1.2

-3 1

-2 0.6666666

-1 0

0 -2

1 ERROR

2 21

3 9

4 -11

5 --38

y-intercept

Asymptote

x-intercept

Finding equations of graphsWe now need to move to the STATS modeLet us have a look at the MenuIs everybody sure how to get into STATS

mode?

Example – linear functionWe are going to work with the linear

regression.Find the equation of the straight line

through (-1; -1) and (2; 5).Type in the two points in the table (data). Find the coefficients remembering that in

stats the linear equation is . The equation is .Or as we know it .

Example – Quadratic function with intercepts givenTyping error on page 10 (first bullet please

change to Quadratic and not linear).Enter the three points in the table (data).Find the coefficients remembering that in

stats the linear equation is . The equation is .Or as we know it .

Example – Quadratic function with any three points.

Enter the three points in the table (data).Find the coefficients remembering that in

stats the linear equation is . The equation is .Or as we know it .

Example – Exponential function* with any two points.

Enter the two points in the table (data).Find the coefficients remembering that in

stats the linear equation is . The equation is .*The CASIO fx-82ZA PLUS calculator can

only do the exponential graph of the form .

Example – Quadratic function with the turning point and another point.

We need to find ANOTHER point. From the turning point we know the axis of

symmetry is at x = -1The point symmetrical to (0;5) will be (-2; 5).Enter the three points.Find the coefficients remembering that in

stats the linear equation is . The equation is .Or as we know it .