Page 127 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

A. EXPONENTIAL FUNCTIONS OVERALL EXPECTATIONS
By the end of this course, students will:
SPECIFIC EXPECTATIONS
By the end of this course, students will:
1.1 determine, through investigation with tech- nology, and describe the impact of changing the base and changing the sign of the expo- nent on the graph of an exponential function
1.2 solve simple exponential equations numeric- ally and graphically, with technology (e.g., use systematic trial with a scientific calculator to determine the solution to the equation 1.05x = 1,276), and recognize that the solu- tions may not be exact
Sample problem: Use the graph of y = 3x to solve the equation 3x = 5.
1.3 determine, through investigation using graph- ing technology, the point of intersection of
the graphs of two exponential functions
(e.g., y = 4− x and y = 8x + 3), recognize the x-coordinate of this point to be the solution
to the corresponding exponential equation (e.g., 4−x = 8x + 3), and solve exponential equations graphically (e.g., solve
2x + 2 = 2x + 12 by using the intersection
of the graphs of y = 2x + 2 and y = 2x + 12) Sample problem: Solve 0.5x = 3x + 3
graphically.
1.4 pose problems based on real-world applica- tions (e.g., compound interest, population growth) that can be modelled with exponen- tial equations, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation
Sample problem: A tire with a slow puncture loses pressure at the rate of 4%/min. If the tire’s pressure is 300 kPa to begin with, what is its pressure after 1 min? After 2 min? After 10 min? Use graphing technology to determine when the tire’s pressure will be 200 kPa.
By the end of this course, students will:
2.1 simplify algebraic expressions containing
integer and rational exponents using the laws
1
of exponents (e.g., x3 ÷ x2 , √x6y12)
a3b2c3
Sample problem: Simplify √a2b4 and then evaluate for a = 4, b = 9, and c = – 3. Verify your answer by evaluating the expression with- out simplifying first. Which method for eval-
uating the expression do you prefer? Explain.
2.2 solve exponential equations in one variable
by determining a common base (e.g., 2x = 32, 45x −1 = 22(x + 11), 35x + 8 = 27x)
x
5x + 8
Sample problem: Solve 3
= 27
by determining a common base, verify by sub- stitution, and investigate connections to the
intersection of y = 35x + 8 and y = 27x using graphing technology.
2.3 recognize the logarithm of a number to a given base as the exponent to which the base must be raised to get the number, recognize the operation of finding the logarithm to be the inverse operation (i.e., the undoing or reversing) of exponentiation, and evaluate simple logarithmic expressions
EXPONENTIAL FUNCTIONS
1. solve problems involving exponential equations graphically, including problems arising from real-world applications;
2. solve problems involving exponential equations algebraically using common bases and logarithms, including problems arising from real-world applications.
1. Solving Exponential Equations Graphically
2. Solving Exponential Equations Algebraically
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Mathematics for College Technology
MCT4C