Mathematics
Grade 12
15 min
Identify linear and exponential functions
Identify linear and exponential functions
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1
Introduction & Learning Objectives
Learning Objectives
Differentiate between linear and exponential functions given their algebraic equations.
Identify linear versus exponential behavior from a table of values by analyzing rates of change.
Distinguish between the graphs of linear and exponential functions based on their shape, rate of change, and end behavior.
Use the concept of first differences (for linear) and constant ratios (for exponential) to classify data sets.
Apply the concept of the derivative to confirm the nature of the rate of change for each function type (constant vs. proportional).
Model real-world scenarios involving constant additive change or constant multiplicative change with the appropriate function type.
Imagine your savings account. Does it grow by the same dollar amount each year, or...
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Key Concepts & Vocabulary
TermDefinitionExample
Linear FunctionA function that has a constant rate of change. Its graph is a straight line.f(x) = 4x - 7. For every 1-unit increase in x, f(x) increases by a constant 4 units.
Exponential FunctionA function where the output is multiplied by a constant factor for each unit increase in the input. Its rate of change is proportional to its current value.g(x) = 3(2)^x. For every 1-unit increase in x, g(x) is multiplied by a factor of 2.
First DifferencesIn a table with evenly spaced x-values, the first differences are the differences between consecutive y-values. If they are constant, the function is linear.For the points (1, 5), (2, 8), (3, 11), the first differences are 8-5=3 and 11-8=3. The constant difference of 3 indicates a linear relationship.
Common RatioIn a tabl...
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Core Formulas
Linear Function General Form
f(x) = mx + b
This is the standard form of a linear function, where 'm' is the constant slope (rate of change) and 'b' is the y-intercept (the initial value when x=0).
Exponential Function General Form
f(x) = ab^x
This is the standard form of an exponential function, where 'a' is the initial value (when x=0), and 'b' is the constant growth/decay factor (b > 0, b ≠ 1).
Linear Rate of Change (Calculus)
\frac{d}{dx}(mx + b) = m
The derivative of a linear function is the constant 'm'. This proves that the rate of change is constant for all values of x.
Exponential Rate of Change (Calculus)
\frac{d}{dx}(a \cdot b^x) = a \cdot b^x \cdot \ln(b)
The derivative of an exponential function...
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Challenging
A function f(x) is defined such that its derivative is f'(x) = ln(5) * f(x), and it passes through the point (0, 2). What is the equation for f(x)?
A.f(x) = 2(5^x)
B.f(x) = 5(2^x)
C.f(x) = 2(e^{5x})
D.f(x) = ln(5)x + 2
Challenging
A linear function f(x) and an exponential function g(x) = 4(2^x) intersect at x=2. At this point of intersection, which statement correctly compares their instantaneous rates of change?
A.Their rates of change are equal.
B.The rate of change of g(x) is greater than the rate of change of f(x).
C.The rate of change of f(x) is greater than the rate of change of g(x).
D.More information is needed to determine the linear function's rate of change.
Challenging
The populations of two towns are modeled by A(t) = 500t + 10000 and B(t) = 10000(1.05)^t. Which statement correctly compares their rates of population growth, A'(t) and B'(t), for t > 0?
A.A'(t) is always greater than B'(t).
B.B'(t) is always greater than A'(t).
C.A'(t) is initially greater than B'(t), but B'(t) will eventually become greater.
D.B'(t) is initially greater than A'(t), but A'(t) will eventually become greater.
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