Objective: The students will examine differences in the graphs of the function sin(x) when altered by constants. The students will sketch the graphs of these funtions correctly 95% of the time.
Materials: graphing calculators, overhead projector, transparecies and markers
Lesson: We know what the graph of sin x looks like. Let’s graph it on the overhead.
Where does the graph hit the y-axis? 0
Where does it touch the x-axis? 0,p,2p
What is the altitude of the function? 1
How about the lowest point? -1
graph on overhead
What is the frequency of this function? 2p
Now put this into Y1 on your calculator.
Now suppose we added a constant to the function. Someone give a number to add to the funtion. Now graph sin x + a in Y2.
What happened to the graph? It shifted up by a.
Graph more if needed until students see the change.
What do you think will happen if we subtract a constant? Graph sin x - a as Y3.
What is the change in the graph? It shifted down by a.
So what can we say happens when we add or sutract a constant?
sin x +- a =The graph moves up (add) or down (subtract) by the constant.
Now consider what happens if you add or subtract the constant to x instead of the function. Set Y1= sin(x =a) and Y2= sin(x-a) and find what the changes for each are.
Do this as much as you need to to find the changes.
Give students time to notice changes. Then as then ask for the changes and add them to the list on the overhead.
sin (x +- a) = The graph moves to the left (add) or right (subtract) by the constant.
Demonstrate by setting a = p/2.
Now continue to build the chart. In your groups, plot graphs for multiplying and dividing the function and just x. Write down what changes. Look for changes in postion, altitude, and frequency. Use p/2 sometimes to see the changes.
When students seem done, have them give the list of changes for each of the variations of the function. Add these to the list on the ovehead.
sin x * a = altitude changes by a multiple of a.
sin (x*a)= frequency changes by multiple of a inverse.
Notice that dividing is the same as multiplying by a fraction.
Graph examples if they seem needed.
Now that we have this chart, we can easily sketch any sine function.
For example, on your paper sketch 3/4 sin x +1/2.
What are the chan
ges? Altitude changes to 3/4 and graph shifts up 1/2.
3/4 sin x + 1/2
sin x
Now if you graph these in your graphing calculators, your graph should look like the calculators graph. This is a good way to check your graphs.
Now try the following functions in your groups. List the changes, then sketch on paper. Then check your graph on the calculator. If they are not the same try to find out why.
PROBLEMS:
sin(p/4 +x) graph shifts left by p/4
2sin(3/4 x) altitude changes to 2, frequency changes to 2p*4/3=(8p)/3
HOMEWORK:
Problems from book= list changes in graph, sketch, check on calculator if possible
Assesment: The students should be able to sketch in class with minimal problems. Homework will be done and 90% of it will be correct.