1. Shawna Haider Associate Professor Mathematics
Making Connections: Tools and Techniques for Online, Hybrid, and Live Courses
Shawna Haider Associate Professor Mathematics
shawnahaider.com

2. The Courses Intermediate Algebra College Algebra Trigonometry Calculus
Differential Equations

3. The Format
Traditional Classroom
Hybrid
Online
Flexible Entry

4. Using Technology to Make Connections

6. Symmetric about the y axis
FUNCTIONS
Symmetric about the origin

7. Even functions have y-axis Symmetry4 3 2 1
For an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

8. Odd functions have origin Symmetry2 1 -2 -3
For an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.

9. x-axis Symmetry
We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function.

10. INCREASING
CONSTANT
DECREASING
Functions

11. It is not increasing OR decreasing but remaining constant
What is this function doing on the interval (-7, -2)? 8 7 6 5 4
What is this function doing on the interval (-2, 2)? 3 2
It is INCREASING 1
It is DECREASING 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 -2 -3
What is this function doing on the interval (2, 7)? -4 -5 -6 -7
x = -7
x = -2
x = 2
x = 7

12. These are functions that are defined differently on different parts of the domain.
WISE
FUNCTIONS

13. This then is the graph for the piecewise function given above.
This means for x’s less than 0, use f(x) = -x but for x’s greater than or equal to 0, use f(x) = x2
What does the graph of f(x) = -x look like?
What does the graph of f(x) = x2 look like?
Remember y = f(x) so let’s graph y = - x which is a line of slope –1 and y-intercept 0.
Remember y = f(x) so lets graph y = x2 which is a square function (parabola)
Since we are only supposed to graph this for x< 0, we’ll stop the graph at x = 0.
Since we are only supposed to graph this for x  0, we’ll only keep the right half of the graph.
This then is the graph for the piecewise function given above.

14. and

15. Above is the graph of
As you can see, a number added or subtracted from a function will cause a vertical shift in the function.
VERTICAL SHIFTS
What would f(x) + 1 look like? (This would mean taking all the function values and adding 1 to them).
What would f(x) - 3 look like? (This would mean taking all the function values and subtracting 3 from them).

16. HORIZONTAL SHIFTS
Above is the graph of
As you can see, a number added or subtracted from the x will cause a horizontal shift in the function but opposite way of the sign of the number.
What would f(x-1) look like? (This would mean taking all the x values and subtracting 1 from them before putting them in the function).
What would f(x+2) look like? (This would mean taking all the x values and adding 2 to them before putting them in the function).

17. We could have a function that is transformed both vertically AND horizontally.
up 3
left 2
Above is the graph of
What would the graph of look like?

18. reflects about the x -axis
From our library of functions we know the graph of
moves up 1
Graph using transformations
reflects about the x -axis
moves right 2
Winplot

19. y = 2
x = 4

20. y = 2
y = 1
x = 4

21. Multiplication of Matrices
If A = [aij ] is an m × n matrix, B = [bij ] is an n × p matrix and C = AB, then
Commuter's Beware!
Find AB
Find BA

22. Tablet PC for examples quickly and easily

23. Outlook

24. Some Favorite Java Applets
Interactive Differential Equations
Function Grapher
Graphs of Function, Derivative and Tangent
MML
Derivative graphs puzzle
Plotting Parametric Equations and Visualizing Linearization
Mean Value Theorem
Riemann Sum

25. Some Maple Examples
Slope Fields
Euler’s Method/Linearization

26. Learning units with applets – coordinate systems, variables, equations, trigonometry, analytical geometry, lines functions, calculus, probability & stats, power & Fourier series
Interactive Math Tests – Sets, functions, algebra, calculus
Dynamic Calculus
Great Calculus Applets
Course in a Box – Flash – Precalculus
MERLOT – Multimedia Educational Resource for Learning and Online Teaching
National Library of Virtual Manipulatives

27. Using transformations, graph the following: