Shawna Haider Associate Professor Mathematics Making Connections: Tools and Techniques for Online, Hybrid, and Live Courses Shawna Haider Associate Professor Mathematics shawna.haider@slcc.edu shawnahaider.com

The Courses Intermediate Algebra College Algebra Trigonometry Calculus Differential Equations

The Format Traditional Classroom Hybrid Online Flexible Entry

Using Technology to Make Connections

Symmetric about the y axis FUNCTIONS Symmetric about the origin

Even functions have y-axis Symmetry 4 3 2 1 For an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

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

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

INCREASING CONSTANT DECREASING Functions

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

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

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.

and

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).

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).

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?

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

y = 2 x = 4

y = 2 y = 1 x = 4

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

Tablet PC for examples quickly and easily

Outlook

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

Some Maple Examples Slope Fields Euler’s Method/Linearization

Learning units with applets – coordinate systems, variables, equations, trigonometry, analytical geometry, lines functions, calculus, probability & stats, power & Fourier series http://www.univie.ac.at/future.media/moe/galerie.html Interactive Math Tests – Sets, functions, algebra, calculus http://www.univie.ac.at/future.media/moe/tests.html Dynamic Calculus http://web.monroecc.edu/pseeburger/ Great Calculus Applets http://higheredbcs.wiley.com/legacy/college/anton/0471472441/explorations/antonapplets2.htm Course in a Box – Flash – Precalculus http://distance-ed.math.tamu.edu/Precalculus_home/index.htm MERLOT – Multimedia Educational Resource for Learning and Online Teaching http://www.merlot.org/merlot/index.htm National Library of Virtual Manipulatives http://nlvm.usu.edu/

Using transformations, graph the following: