Page 51 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

2.2 determine, through investigation using tech- nology, the roles of the parameters a, k, d, and cinfunctionsoftheformy=af(k(x–d))+c, and describe these roles in terms of transfor- mationsonthegraphoff(x)=ax (a>0,a≠1) (i.e., translations; reflections in the axes; verti- cal and horizontal stretches and compressions to and from the x- and y-axes)
Sample problem: Investigate the graph of f(x)=3x–d –5forvariousvaluesofd, using technology, and describe the effects of changing d in terms of a transformation.
2.3 sketch graphs of y = af (k(x – d )) + c by applying one or more transformations tothegraphof f(x)=ax (a>0,a≠1), and state the domain and range of the transformed functions
Sample problem: Transform the graph of f(x)=3x tosketchg(x)=3–(x+1) –2,andstate
the domain and range of each function.
2.4 determine, through investigation using techno- logy, that the equation of a given exponential function can be expressed using different bases [e.g., f(x) = 9x can be expressed as f(x) = 32x ], and explain the connections between the equivalent forms in a variety of ways (e.g., comparing graphs; using transformations; using the exponent laws)
2.5 represent an exponential function with an equation, given its graph or its properties
Sample problem: Write two equations to rep- resent the same exponential function with a y-intercept of 5 and an asymptote at y = 3. Investigate whether other exponential func- tions have the same properties. Use transfor- mations to explain your observations.
By the end of this course, students will:
3.1 collect data that can be modelled as an expo- nential function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; measurement tools such as electronic probes), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph
the data
Sample problem: Collect data and graph the cooling curve representing the relationship between temperature and time for hot water cooling in a porcelain mug. Predict the shape of the cooling curve when hot water cools in an insulated mug. Test your prediction.
3.2 identify exponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range (e.g., ambient temperature limits the range for a cooling curve)
Sample problem: Using data from Statistics Canada, investigate to determine if there was a period of time over which the increase in Canada’s national debt could be modelled using an exponential function.
3.3 solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the equations
Sample problem: The temperature of a
cooling liquid over time can be modelled
by the exponential function
x
T(x) = 60(12 )30 + 20, where T(x) is the
temperature, in degrees Celsius, and x is the elapsed time, in minutes. Graph the function and determine how long it takes for the tem- perature to reach 28oC.
3. Solving Problems Involving Exponential Functions
EXPONENTIAL FUNCTIONS
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Functions
MCR3U