GEOGEBRA WITH GRAPHIC

MATH 6700Independent Study Using the GeoGebra software to plot the graphics of functions in College Algebra The Minh Tran Outline 1 Plot the graphics of quadratic and absolute function

MATH 6700

Independent Study

Using the GeoGebra software to plot the graphics of functions in

College Algebra The Minh Tran

Outline

1) Plot the graphics of quadratic and absolute function

a) Quadratic function

b) Absolute function

2) Plot the graphics of rational functions with vertical, horizontal, and slant asymptotes a) Rational functions with vertical and horizontal asymptotes

b) Rational functions with vertical and slant asymptotes

3) Plot exponential and logarithmic functions

a) Exponential function

b) Logarithmic function

c) The inverse characteristics of exponential and logarithmic functions

CONTENT

1) Plot the graphics of quadratic and absolute function

a) Quadratic function

First, we would plot the basic function f(x) x= 2 by the following steps:

Step 1:

● Input the function f(x) = x^2 directly at the last row of your worksheet in GeoGebra

We would see the figure follows as:

● Type “Enter”, we have the graph of the basic function f(x) x= 2:

Step 2:

By the applying of the formula: ( ) ( )2

F x = a x h− + k for the basic functionf(x) x= 2

We would plot a new graphic of the function ( ) ( )2

F x = x 3 3− + , which is compared

to the basic function f(x) x= 2

Where

a = 1>0: The graphic of f(x) will open up

h = 3 > 0: The graphic of f(x) will be translated 3 units to the right

k = 3 > 0: The graphic of f(x) will be translated 3 units up

Hence, the vertex A(0,0) of the graphic of f(x) would be translated to the vertex B(3,3)

● Input the function F(x) = (x-3)^2 + 3 at the last row of your worksheet and press

“Enter”

,and then we can change the colors for the graphics by

● Click on each graph with your mouse directly, and

● Choose the colors at the tool as the figure below:

Now, we continue to get the two vertices A(0,0) and B(3,3) of the graph f(x) and F(x) by

● Choose the tool:

● Set the points at the vertex of each graphic

We have the two graphics of f(x) and F(x) with two vertices in the Cartesian coordinate system, which is shown as the figure:

Finally, it is easy to see that the graphic of f(x) translated from the vertex A(0,0) to the vertex B(3,3) of the graphic of F(x)

Step 3:

As you know, we would use an effect of visual aid in the learning and teaching process with GeoGebra software That is using of the slip of graphics by setting two parameters

a and b Generally, instead of the two vertices A(0,0) and B(3,3) by using the parameter

a for abscissa x and the parameter b for ordinates y

To do this, we choose the slider tool

● Choose the tool on the toolbar

● Set the two values a and b on the empty space of the coordinate system

● Double click on the symbol and continue to choose the values of min and max for a and b from 0 to 3 with the step is 0.5 at the Slider window

● At the Basic window, we choose the tools such as Show Object, Show Label, Animation On as the figure:

The figure below displayed the slip of the graphic of f(x) to the graphic of F(x)

b) Absolute function

Similarly, we also can plot the graphs of absolute function follows as:

From the graphic of the basic function f(x) = |x|, it will be translated to the graph of G(x)

=|x+4|- 4 and F(x)=|x-2|+4 with the translations from the vertex A(0,0) to the vertices B(2,4) and C(-4,4) Thus, we have the following figure

As the graph of quadratic function above, we would show the slip of the graphic of f(x)

to the graphic of G(x) and the slip of the point B from A(0,0) to C(-4,4)

2) Plot the graphics of rational functions with vertical, horizontal, and slant asymptotes

a) Rational functions with vertical and horizontal asymptotes

We would continue to plot the rational function f x ( ) 2x 4

x 3

+

=

− with the vertical

asymptote x = 3 and the horizontal asymptote y = 2 by the following steps:

Step 1:

● Input the function f(x) = (2x +4)/(x-3)

● Input the vertical asymptote x = 3,

● Input the horizontal asymptote g(x) = 2 directly at the last row of your worksheet

● Change the colors of the graphics, vertical and horizontal asymptote

We would see the figure follows as:

Step 2:

● Input the vertical asymptote x = m at the last row of your worksheet

● Double click on the letter m

● Choose Basic mode

● Choose the Animation On mode by setting a parameter m which runs from -5 to 5

● Input f(x) = (2x + m) / (x - m) at the last row of your worksheet

Hence, the vertical asymptote x= m would slip from -5 to 5 on the x- axis in the figure:

b) Rational functions with vertical and slant asymptotes

We can plot the rational function f x ( ) x² 2x 1

x 1

− + +

=

− by the following steps:

Step 1:

By the algebraic analysis, from the rational function

− + +

= = − + +

We have found the vertical asymptote is x = 1 since

x 1

x² 2x 1

x 1

lim

→

− + + = −∞

−

and the slant asymptote is g(x) = -x +1 since

x

2 0 m

x 1

li

−

● Input the function f(x) = (-x² + 2x + 1) / (x - 1)

● Input the slant asymptote g(x) = -x +1

● Input x=1

● Change the colors of the graphics, vertical and slant asymptote

We see the figure:

Step 2:

Input the vertical asymptote x = m and the slant asymptote g(x)= - x + m and then using the Amination On tool by setting a parameter m which runs from -5 to 5 so the vertical x

= m and the slant asymptote would slip from -5 to 5 on the x- axis in the figure:

● Choose the value m with the tool

● Double click on the letter m

● Choose Basic mode

● Choose the Animation On mode by setting a parameter m which runs from -5 to 5 with the step is 0.1

● Input f(x) = f(x) = (-x² + 2m x + 1) / (x - m)

● Input the vertical asymptote x = m

● Input the slant asymptote g(x) = - x + m

Hence, the vertical asymptote x= m would slip from -5 to 5 on the x- axis in the figure:

3) Plot exponential and logarithmic functions

a) Exponential function

We would plot the exponential functions f(x) 2 ; g(x) 4 ; h(x) 8= x = x = x and

F(x) ; G(x) ; H(x)

     

= ÷ = ÷ = ÷

     

Step 1:

● Input the exponential functions f(x) = 2^x and press “Enter”

● Input the exponential functions g(x) = 4^x and press “Enter”

● Input the exponential functions h(x) = 8^x and press “Enter”

● Input the exponential functions f(x) = (1/2)^x and press “Enter”

● Input the exponential functions g(x) = (1/4)^x and press “Enter”

● Input the exponential functions h(x) = (1/8)^x directly at the last row of your worksheet and press “Enter”

● Change the colors of the graphics

We have the figure:

Step 2:

● Input the exponential function f(x)=2^(mx) and press “Enter”

● Using the Animation On tool by setting a parameter m which runs from 0 to 5 with the step is 1 so we have the graphics of the exponential functions including

f(x) 2 ; g(x) 4 ; h(x) 8 ; k(x) 16 ;l(x) 32= = = = =

b) Logarithmic function

We would plot the logarithmic functions f(x) log x; g(x) log x; h(x) log x= 2 = 4 = 8

Step 1:

● Type f(x) = log(2,x) and press “Enter” button directly at the last row of your worksheet Similarly, input g(x) = log(4,x); k(x) = log(8,x) The graphics is shown by the figure

Step 2:

● Input the exponential function f(x) = log(2m, x), and then using the Animation On tool

by setting a parameter m which runs from 0.1 to 10 with the step is 0.1

c) The inverse characteristics of exponential and logarithmic functions

We would plot the exponential and logarithmic functions f(x) 2 ; g(x) x; k(x) log x= x = = 2

● Input the exponential functions above by typing f(x) = 2^x and press “Enter” button

● Type g(x) = x and press “Enter” button

● Type k(x) = log(2,x) and press “Enter” button directly at the last row of your worksheet and change the colors of the graphics

We set the points A, B, C, D, E, F are the inverse characteristics of exponential and logarithmic functions

For f(x) = 2^x:

x = 0 => f(x) = 1: A (0, 1)

x = 1 => f(x) = 1: C (1, 2)

x = 3 => f(x) = 8: E (3, 8)

For g(x) = log(2,x):

g(x) = 1 => x = 2: A (1, 0)

g(x) = 2 => x = 4: C (2, 1)

g(x) = 3 => x = 8: E (8, 3)

● Input these points on the graphic as in the part 1a)

We have the figure: