The Sine Function

• graph the sine function y=Asin(Bx-C)+D and, through experiments,
• determine the graphical significance of the values of A, B, C, and D in the equation.

1. Launch GrafEq by double clicking its icon. Click on the title screen to remove it.
2. Type in the equation y=Asin(Bx-C)+D.
3. Press the Tab key, and assign a value to A by typing “A=1”.
4. Press the Tab key again, and assign a value to B by typing “B=1”.
5. Press the Tab key again, and assign a value to C by typing “C=0”.
6. Press the Tab key again, and assign a value to D by typing “D=0”. Your input window is now complete.
7. Press the Return key to get the Create View window. The default mode (Cartesian) is correct. For the y-axis, the default bounds (-10 to 10) are fine. For the x-axis,
   • click on the upper bound box and change it from 10 to 5, then
   • click on the lower bound box and change it from -10 to -5.
   
8. Click on the Create button. GrafEq will plot the graph of y=sinx.
9. The side-bar is currently showing general information. To display ticked axes:
   • click on the General button and drag the highlight bar to Ticks;
   • under the Simple submenu, click on the box which has dense ticks on both axes;
   • click the Active checkbox to display the ticked axes.
   
10. Note that this graph
    • has an amplitude of 1,
    • has a period a bit greater than 6, and
    • passes through the origin (0,0).

1. For this section of the lesson, we will keep the values of B, C, and D constant at 1, 0, and 0 respectively, so our reference curve, with A at 1, is y=sinx. If you had started this lesson from the beginning, these values are already set in the previous section.
2. Change the value of A from 1 to 2:
   • click on the Ticks button and drag the highlight bar to General;
   • click on the Relation #1 button to display the relation input window;
   • click on the “A=1” constraint, and change 1 to 2.
3. Press the Return key to view the sine curve y=2sinx. Note that the period of the curve is unchanged and no shift occurs. Q: What is amplitude of this curve? Enter this value on line 2 of the table.
4. Repeat the necessary steps, as described above, to graph y=-3sinx. Note that the period of the curve remains unchanged, while the graph is flipped horizontally. Q: What is amplitude of this curve? Enter this value on line 3 of the table.
5. Anticipate and enter on lines 4 and 5 of the table the effects of changing the value of A to -4 and 5 respectively. Confirm your answer with GrafEq by graphing y=-4sinx and y=5sinx (by setting A=-4 and A=5 respectively).
6. Q: What general conclusion can you draw about the graphical significance of the value of A in our sine function y=Asin(Bx-C)+D? Put your answer on the conclusion line of the table.

1. For this section of the lesson, set the value of A to 3, and keep the value of D and C at 0, so our reference sine curve, with B at 1, is y=3sinx.
2. To clearly show the graphical significance of the values of B and C in our sine function, we want to introduce an extra set of custom ticks on our graph. To do this,
   • choose New Custom Ticks from the Graph menu,
   • enter “x=k(π/2)”,
   • click on the Labels/Upper box, and choose Labels/Alt#1,
   • click on the lines/solid box, and choose lines/dotted,
   • click on the Tick Font/Moderate box, and choose Tick Font/Smaller, and then
   • click on the Active box.
   
3. Study the reference sine curve with its new custom ticks added. Note that
   • the amplitude of the curve is 3,
   • the curve passes through (0,0), and
   • the period is 2π.
   
4. Change the value of B from 1 to -2, to view the sine curve y=3sin(-2x). Note that the amplitude is unchanged, while the graph is flipped horizontally. The period is reduced by 1/2, from 2π to π. With these observations, complete line 7 of the table.
5. Anticipate and enter on lines 8, 9, 10, and 11 of the table the effects of changing the value of B to 2, 1/2, -1/2, and 4 respectively. Confirm your answer with GrafEq.
6. Q: What general conclusion can you draw about the graphical significance of the value of B in our sine function y=Asin(Bx-C)+D? Put your answer on the conclusion line of the table.

1. For this section of the lesson, reset the value of B to 1, and keep the values of A and D at 3 and 0 respectively, so our reference sine curve, with C at 0, is y=3sinx. The amplitude of the curve is 3, and the period is 2π. The curve passes through the origin (0,0); its phase shift is 0.
2. Change the value of C from 0 to (-π/2), to view the sine curve y=3sin(x+π/2). Note that the amplitude and the period of the curve are unchanged. The curve is shifted by π/2 to the left, from the origin (0,0),. Enter these observations on line 13 of the table.
3. Change the value of C to π/2, to view the sine curve y=3sin(x-π/2). Complete line 14 of the table with your observation.
4. Keeping the value of C at π/2, change the value of B from 1 to 2, to view the sine curve y=3sin(2x-π/2). Note that the amplitude is unchanged while the period is reduced by 1/2, from 2π to π. The curve is shifted by π/4 to the right, from the origin (0,0). Enter these observations on line 15 of the table.
5. Anticipate and enter on lines 16, 17, and 18 of the table the effect of changing the values of B and C as given on the table. Confirm your answer with GrafEq.
6. Q: What general conclusion can you draw about the graphical significance of the value of C in our sine function y=Asin(Bx-C)+D? Put your answer on the conclusion line of the table.

1. For this section of the lesson, set the value of A to 2, B to 1, and C to 0, so our reference sine curve, with D at 0, is y=2sinx. The amplitude of the curve is 2, and the period is 2π. The curve passes through the origin (0,0); both its phase shift and vertical shift are 0.
2. Change the value of D from 0 to 1, to view the sine curve y=2sinx+1. Note that the amplitude, the phase, and the period of the curve are all unchanged. The entire graph is shifted upwards by 1. Complete line 20 of the table.
3. Anticipate and enter on lines 21 and 22 of the table the effects of changing the value of D to -2, and 3 respectiely. Confirm your answer with GrafEq.
4. Q: What general conclusion can you draw about the graphical significance of the value of D in our sine function y=Asin(Bx-C)+D? Put your answer on the conclusion line of the table.