The Music Project
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Show all of your work and fully explain your thinking process for each question. No credit will be awarded for answers with no work shown!

1. Mick wants to make a huge sideways lighted guitar for the top of his new “Harder Than Rock” Cafe. He models the top half of the guitar with the piecewise function:

	{		,
			,
f(x) =			,
			,

a. Graph f(x) on an appropriately sized graph.

b. Determine algebraically.

c. Determine algebraically.

d. Determine f(3.199) graphically. Be accurate to at least 2 decimal places.

e. Determine f(7.095) graphically. Be accurate to at least 2 decimal places.

f. Determine f(11.7) graphically. Be accurate to at least 2 decimal places.

g. Include the points (,), (3.199, f(3.199)), (7.095, f(7.095)), (11.7, f(11.7)) and (,) on your graph from part 1a.

h. What is the absolute maximum of f(x)?

i. Is f(x) a function? Why or why not?

j. Is f(x) a one-to-one function? Why or why not?

k. What are the domain and range of f(x)?

2. Musical notes may be written as follows:

Now, these notes may be thought of as being mapped to functions. For instance, these notes:

may be mapped on the a set of axes as follows:

Now, we could think of connecting these notes via some function, such as

f(x) = 2|x-3| for and that would look like:

The domain here would be and the range would be .

Note: Each tick (or line) in the vertical direction represents a shift of 2 as notes may appear in the spaces between the lines.

Note: Throughout question 2, we will be ignoring key, sharps and flats, so some of the key changes may not be exactly proper!

a. Consider the musical translation g(x) shown below. Use your knowledge of graphical transformations to write the equation g(x) for this graph.

b. What are the domain and range of g(x) from part a?

c. Graph h(x) = f(-x+5). Include the 4 music notes generated on your picture. What do you notice about the relationship between the four music notes on f(x) and h(x)? Musicians call this kind of a transformation a retrograde.

d. What are the domain and range of h(x) from part c?

e. Graph j(x) = -f(x). Include the 4 music notes generated on your picture. What do you notice about the relationship between the four music notes on f(x) and j(x)? Musicians call this kind of a transformation an inversion.

f. What are the domain and range of j(x) from part e?

g. If k(x) = -x, find k◦h(x). Graph k◦h(x). Include the 4 music notes generated on your picture. What do you notice about the relationship between the four music notes on f(x) and those on this graph? Musicians call this kind of a transformation a retrograde inversion.

h. What are the domain and range of k◦h(x) from part g?

3. The frequencies of musical notes are related by simple proportions. For instance, increasing the pitch by an octave (say from one A to the next higher A) doubles the frequency (so there is a 1:2 ratio from one note up to the same note in the next octave). For instance, on the keyboard below, if the C all the way on the left is middle C, then the first A above it has a frequency of 440 Hz. The A an octave above that (close to the right end of the keyboard shown below) would be at 880 Hz (in a ratio of 1:2 with the lower A).

Note: # is the symbol for sharp and b is the symbol for flat.

Note: A half-step is a step to the very next key (black or white) to the right. For instance, a half-step up from B is C, a half-step up from D# (or Eb) is E, and a half-step up from F is F# (or Gb). An octave is 12 half-steps up.

a. The ratio for an increase in pitch by a fifth (7 half-steps) is 2:3. So, we could figure out what the frequency of the E on the right half of the keyboard shown above by solving:

because E is a fifth above the A (which is at 440 Hz). Figure out what the frequency is of that E.

b. The ratio for an increase in pitch by a fourth (5 half-steps) is 3:4. What is the frequency of the D that is a fourth above the A at 440 Hz?

c. A third is an increase by four half-steps. For instance, when you go from A at 440 Hz to the C# above it at about 550Hz, you go up by a third. What is the ratio for an increase in pitch by a third?

d. Using you answer from part c, calculate the frequency of the F (or E#) that is a third above the C# at about 550 Hz.

Last Modified on Monday, January 26, 2004.