Applications in Physics - Wave Interference

MHF4U Adding and Subtracting Functions

Functions can be added or subtracted from each other to form new functions. To do this you can either add both functions together algebraically, graphically or through a table of values.

Superposition principle – the sum of 2 or more functions can be found by adding the -coordinate of the functions for each -coordinate.

Destructive Interference Constructive Interference

Example 1 Given and , determine a) b) c)

Example 2 Use the graph of and to sketch the graph of and .

Example 3 Investigating the Domain

Graph and . Then graph .

What is the domain of ?

In general, for two functions and , what is the domain of ?

Example 4 Given and a) Find the equation of

b) Find the equation of

c) Graph all 4 graphs.

Example 5 Given and , sketch the graphs of a) b) c) d) . ( and -intercepts accurate, but maximums and minimums not accurate) Example 6 Given and determine and graph: a)

Example 7 Given and , Sketch a graph of a) b) c) d) State the domain of all 3 Example 8 The Profit Function The Athletic Council is selling T-shirts to raise money for new equipment. There is a fixed cost of $200 for producing the T-shirts, plus a variable cost of $5 per T-shirt made. Council has decided to sell the T- shirts for $8 each. a) Write an equation to represent i) The total cost, , as a function of the number, , of T-shirts produced

ii) The revenue, , as a function of the number, , of T-shirts produced.

b) Then, graph these functions on the same set of axes. Identify the point of intersection and explain the meaning of its coordinates.

c) Profit, , is the difference between revenue and expenses. Develop an algebraic and a graphical model for the profit function and graph it on the same grid as above.

d) Under what circumstances will the Athletic Council lose money? Make money?

e) Identify the domain and range of the cost, revenue, and profit functions in the context of this problem.