Q) My Year 11 class is investigating transformations of the sine function.
Some have difficulty understanding why sin (x + 60x) moves the graph 60x in the negative direction and sin2x gives 1Z2 the wavelength. Combining the two lost all of them, as sin (2x + 60x) moves the graph back only 30x. Is there a nice way to clarify this so they can recall the facts without plotting points?
A) Fundamental to understanding the function is to refer back to the unit circle in the first instance. This also makes the link between physics and maths more accessible. Imagine a complete circle with a rotating rod (length = 1 unit) to which an elastic line is attached at the circumference; the line meets the horizontal axis at a right angle. As the rod rotates through different angles the vertical and horizontal sides of the right-angled triangle change. This diagram shows the rod at 60x to the x-axis. The horizontal length, 0.5, is the numeric value of cos 60x (read from the x-axis) and the vertical height 0.87 is the numeric of sin 60x, indicated by the dotted line (can be read from the y-axis). A full description of this exercise can be found at www.mathagonyaunt.co.uk, June 14 2002. When the vertical lengths for sin x are plotted against the angle x as the rod turns, we have the following curve: The highest point of the curve from the x-axis is the amplitude (a). For the unit circle, a = 1. If the radius is increased to 3 units, a would be 3. This is also the length of the hypotenuse of the triangle. The wavelength (l) is the distance between the highest point of two consecutive peaks - equivalent to how far the rod has travelled around the circle for 0 . LESS THAN LESS THAN x LESS THAN LESS THAN 360, or 0 LESS THAN LESS THAN x LESS THAN LESS THAN 2 LESS THAN F128
To discuss transformations of the sine curve we are looking at what is happening in terms of wavelength and amplitude. For the case where y = sin2x, we have the rod travelling twice as "fast" as in the unit-circle case. So when x = 30o, sin x = 1ZC3 and sin 2x = 2ZC3, which is equivalent to sin 60o. This means the rod goes through a complete cycle in half the "time" , and the amplitude stays the same as the size of the circle hasn't changed. We can see this more clearly on the graph.
Next, we look at y = sin (x +30o) - similar to y = sin (x + 60o) in your question - to relate this to the function of a function created in the final graph of y = sin (2x + 60o). For y = sin (x +30o), consider x = 20o; we then have sin 20o = 0.34 and sin (20o + 30o0.77. Plotting this, you can see that the sine curve is translated 30o in the negative direction.
To bring clarity to the function y = sin (2x + 60x) it may be helpful to factorise the bracket to give a unit x, resulting in sin (2(x + 30x)).
Introducing brackets within the function leads us to consider the priority of operations within this function of a function. The final function is therefore the effect of performing a 30x shift left of the function sin 2x.
Plotting points using calculated values and a table of data will reinforce this for pupils wishing to solidify these suppositions.
In support of this auditory enlightenment, your pupils will benefit from seeing this illustrated visually and kinaesthetically using a resource designed for large-screen classroom presentation, such as eStarters, created by Matt Dunbar, a teacher from Manchester, which perfectly demonstrates this in a highly visual way, allowing the teacher to intuitively and swiftly manipulate the coefficients.
Matt Dunbar has also created a topic file for graphs of trigonometric functions which readers can download free from www.estarters.com.
The most common question that pupils ask is "What is this used for?" Some examples are: an animation of the sine curve based on a bicycle wheel and modelling temperature in New York Central Park (http:people.hofstra.
edufacultyStefan_Wanertrigtrig1.html); graphical representations of sound waves (www.fi.edufellows fellow2apr99soundvib.html); sine curves of ocean waves (http:hyperphysics.phy-astr.gsu.edu hbasewaveswatwav2.html); a discussion of light refraction and how sine curves are linked to mirages (www.brantacan.co. ukinterference.htm)
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
Email your questions to Mathagony Aunt at firstname.lastname@example.org Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX