Mathematics
Grade 9
15 min
Match exponential functions and graphs
Match exponential functions and graphs
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1
Introduction & Learning Objectives
Learning Objectives
Identify the key features of an exponential graph, including the y-intercept, asymptote, and direction of growth or decay.
Determine if an exponential function of the form y = a * b^x represents growth or decay by analyzing its base 'b'.
Calculate the y-intercept of an exponential function directly from its equation.
Match a given exponential function to its corresponding graph using its key features.
Match a given exponential graph to its corresponding function from a list of options.
Explain how the values of 'a' and 'b' in the equation y = a * b^x affect the graph's starting point and shape.
Ever seen a post go viral and spread to millions in just a few hours? 📈 That explosive pattern is a real-life picture of an exp...
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Key Concepts & Vocabulary
TermDefinitionExample
Exponential FunctionA function where the variable is in the exponent, written in the form y = a * b^x. Here, 'a' is the initial value (and can't be zero), and 'b' is the constant base (must be positive and not equal to 1).y = 3 * (2)^x is an exponential function where the initial value is 3 and the base is 2.
Exponential GrowthThis occurs when the base 'b' in an exponential function is greater than 1 (b > 1). The graph starts slow and then increases very rapidly from left to right.The function y = 10 * (1.5)^x shows exponential growth because the base, 1.5, is greater than 1.
Exponential DecayThis occurs when the base 'b' is between 0 and 1 (0 < b < 1). The graph decreases rapidly at first and then flattens out as...
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Core Formulas
Standard Form of an Exponential Function
y = a \cdot b^x
This is the general equation. 'a' is the initial value (the y-intercept), 'b' is the base that determines growth or decay, and 'x' is the exponent.
Condition for Exponential Growth
b > 1
If the base 'b' is greater than 1, the function models growth. The graph will curve upwards, increasing from left to right.
Condition for Exponential Decay
0 < b < 1
If the base 'b' is a fraction or decimal between 0 and 1, the function models decay. The graph will curve downwards, decreasing from left to right.
The Y-Intercept Rule
\text{y-intercept} = (0, a)
In the equation y = a * b^x, the value of 'a' is always the y-coordinate of the y-intercept....
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Challenging
An exponential graph passes through the points (1, 6) and (2, 18). Which of the following is the equation for this graph?
A.y = 2 * (3)^x
B.y = 3 * (2)^x
C.y = 6 * (3)^x
D.y = 1 * (6)^x
Challenging
The graph of y = a * b^x passes through (-1, 10) and (0, 5). What is the equation?
A.y = 10 * (0.5)^x
B.y = 5 * (0.5)^x
C.y = 5 * (2)^x
D.y = 10 * (2)^x
Challenging
Given f(x) = 4*(2)^x and g(x) = 2*(4)^x. Which statement correctly describes the relationship between their graphs?
A.f(x) has a higher y-intercept and is always steeper than g(x).
B.g(x) has a lower y-intercept and is always less steep than f(x).
C.g(x) has a lower y-intercept but becomes steeper than f(x) as x increases.
D.f(x) and g(x) are the same graph.
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