Page 72 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

GGrraade11,, College Preparrattiionn
3. Solving Problems Involving Exponential Relations
2. Connecting Graphs and Equations of Exponential Relations
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
1.9 solve problems, using an appropriate strategy (i.e., factoring, graphing), given equations of quadratic relations, including those that arise from real-world applications (e.g., break-even point)
Sample problem: On planet X, the height,
h metres, of an object fired upward from the ground at 48 m/s is described by the equation h = 48t – 16t2, where t seconds is the time since the object was fired upward. Deter- mine the maximum height of the object, the times at which the object is 32 m above the ground, and the time at which the object hits the ground.
By the end of this course, students will:
2.1 determine, through investigation using a variety of tools and strategies (e.g., graphing with technology; looking for patterns in tables of values), and describe the meaning of nega- tive exponents and of zero as an exponent
2.2 evaluate, with and without technology, numeric expressions containing integer
exponents and rational bases (e.g., 2–3, 63, 34560, 1.0310 )
2.3 determine, through investigation (e.g., by patterning with and without a calculator), the exponent rules for multiplying and dividing numerical expressions involving
exponents [e.g., 1 3x 1 2], and the (2 ) (2 )
exponent rule for simplifying numerical expressions involving a power of a power
3 2 [e.g., (5 ) ]
2.4 graph simple exponential relations, using paper and pencil, given their equations
[e.g., y = 2x, y = 10x, y = 1 x] (2 )
2.5 make and describe connections between representations of an exponential relation
(i.e., numeric in a table of values; graphical; algebraic)
2.6 distinguish exponential relations from linear and quadratic relations by making compar- isons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; inspecting graphs; comparing equations), within the same context when possible (e.g., simple interest and compound interest, population growth)
Sample problem: Explain in a variety of ways how you can distinguish exponential growth represented by y = 2x from quadratic growth represented by y = x2 and linear growth rep- resented by y = 2x.
By the end of this course, students will:
3.1 collect data that can be modelled as an exponential relation, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; meas- urement 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 describe some characteristics of exponential relations arising from real-world applications (e.g., bacterial growth, drug absorption) by using tables of values (e.g., to show a constant ratio, or multiplicative growth or decay) and graphs (e.g., to show, with technology, that there is no maximum or minimum value)
3.3 pose problems involving exponential relations arising from a variety of real-world applica- tions (e.g., population growth, radioactive decay, compound interest), and solve these and other such problems by using a given graph or a graph generated with technology from a given table of values or a given equation
Sample problem: Given a graph of the population of a bacterial colony versus time, determine the change in population in the first hour.
3.4 solve problems using given equations of exponential relations arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bounc- ing ball, compound interest) by substituting values for the exponent into the equations
Sample problem: The height, h metres, of a ball after n bounces is given by the equation h = 2(0.6)n. Determine the height of the ball after 3 bounces.
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