Algebra.Com's Answer #841594 by math_tutor2020(3817)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
\n" );
document.write( "Many teachers and textbooks decided that sin x is fine when others prefer sin(x) to make it more clear what the input argument is.
\n" );
document.write( "The second version leads to more parenthesis but it's worth it I think.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Something like sin B(x + C) is the same as sin( B(x + C) ) to tell the reader \"The B(x+C) part is inside the sine function\".
\n" );
document.write( "It's a bit confusing I know.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "No, y = A sin B(x + C) + D is NOT the same as y = A*sin(Bx - C) + D; however, the two are somewhat similar at least.
\n" );
document.write( "The A, B, and D play the exact same role for either template. The C is a bit different. \r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Here's an example graph showing that y = A sin B(x + C) + D is NOT the same as y = A*sin(Bx - C) + D
\n" );
document.write( "https://www.desmos.com/calculator/sbgketg7yy\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Check out this page
\n" );
document.write( "https://math.libretexts.org/Courses/Rio_Hondo/Math_175%3A_Plane_Trigonometry/02%3A_Graphing_Trigonometric_Functions/2.04%3A_Transformations_Sine_and_Cosine_Functions
\n" );
document.write( "to refresh your memory of what each parameter does.
\n" );
document.write( "The template that link uses is y = A*sin(Bx - C) + D\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "However, since your teacher is using the template
\n" );
document.write( "y = A*sin(B(x + C)) + D
\n" );
document.write( "it's best to stick to it\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "|A| = amplitude
\n" );
document.write( "B = used to determine period
\n" );
document.write( "C = used to determine phase shift
\n" );
document.write( "D = vertical shift and midline\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "The highest and lowest points on this sinusoidal curve is when y = 1 and y = -3 respectively.
\n" );
document.write( "The midpoint is (max+min)/2 = (1+(-3))/2 = -1 which is the midline. Therefore D = -1\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "The vertical gap between those y values is 4 units. Count out the spaces. Or subtract and use absolute value. Half of this vertical distance is the amplitude. So either A = 2 or A = -2. Both will work. I'll go with A = 2 for simplicity sake, and to match what your answer key shows (note: if you picked A = -2, then the C value would be different from C = -pi/3 while everything else is the same).\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "The lowest point occurs when y = -3
\n" );
document.write( "Circle two neighboring valley points and record their x values. Two such x values are x = pi/6 and x = 5pi/6
\n" );
document.write( "The horizontal gap between them is 5pi/6 - pi/6 = 4pi/6 = 2pi/3
\n" );
document.write( "This is the period. It's the length of each cycle. The curve repeats itself every 2pi/3 horizontal x units.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "T = period
\n" );
document.write( "T = 2pi/B
\n" );
document.write( "B = 2pi/T
\n" );
document.write( "B = 2pi/( 2pi/3 )
\n" );
document.write( "B = 2pi * (3/(2pi))
\n" );
document.write( "B = 3
\n" );
document.write( "Note how,
\n" );
document.write( "T = 2pi/B
\n" );
document.write( "T = 2pi/3
\n" );
document.write( "So we confirm we have the correct B value.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "The last thing to find is the value of C.
\n" );
document.write( "So far we have:
\n" );
document.write( "A = 2
\n" );
document.write( "B = 3
\n" );
document.write( "D = -1
\n" );
document.write( "Plug those into the template y = A*sin( B(x + C) ) + D and we get y = 2*sin( 3(x + C) ) - 1
\n" );
document.write( "Then plug in one of the min or max points on the curve. Let's say we picked (x,y) = (pi/2, 1)\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Let's solve for C.
\n" );
document.write( "y = 2*sin( 3(x + C) ) - 1
\n" );
document.write( "1 = 2*sin( 3(pi/2 + C) ) - 1
\n" );
document.write( "1 + 1 = 2*sin( 3(pi/2 + C) )
\n" );
document.write( "2 = 2*sin( 3(pi/2 + C) )
\n" );
document.write( "2/2 = sin( 3(pi/2 + C) )
\n" );
document.write( "1 = sin( 3(pi/2 + C) )
\n" );
document.write( "3(pi/2 + C) = arcsin(1)
\n" );
document.write( "3(pi/2 + C) = pi/2
\n" );
document.write( "3pi/2 + 3C = pi/2
\n" );
document.write( "3C = pi/2 - 3pi/2
\n" );
document.write( "3C = -2pi/2
\n" );
document.write( "3C = -pi
\n" );
document.write( "C = -pi/3\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Here's a Desmos graph to confirm our answers
\n" );
document.write( "https://www.desmos.com/calculator/iosov8wrk1
\n" );
document.write( "GeoGebra is another graphing tool I use all the time.
\n" );
document.write( " \n" );
document.write( " |
![](\"https://i.ibb.co/3YYvXqR/x.png\")
\r