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Questions and Answers
Which graph shows exponential growth?
Which graph shows exponential growth?
Which equation represents an exponential function that passes through the point (2, 36)?
Which equation represents an exponential function that passes through the point (2, 36)?
f(x) = 4(3)^x
Which best describes the graph of the function f(x) = 4(1.5)^x?
Which best describes the graph of the function f(x) = 4(1.5)^x?
The graph passes through the point (0, 4), and for each increase of 1 in the x-values, the y-values increase by a factor of 1.5.
What is the multiplicative rate of change of the function f(x) = 2(5)^x?
What is the multiplicative rate of change of the function f(x) = 2(5)^x?
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What are the domain and range of the function on the graph?
What are the domain and range of the function on the graph?
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Which equation represents an exponential function with an initial value of 500?
Which equation represents an exponential function with an initial value of 500?
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What is the multiplicative rate of change of the function described in the table?
What is the multiplicative rate of change of the function described in the table?
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Which equation can be used to find the value, y, of a limited-edition poster after x years?
Which equation can be used to find the value, y, of a limited-edition poster after x years?
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Which equation can be used to predict the population y after x years, given the population at year 1 and year 2?
Which equation can be used to predict the population y after x years, given the population at year 1 and year 2?
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Which equation represents an exponential function that passes through the point (2, 80)?
Which equation represents an exponential function that passes through the point (2, 80)?
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Which is the graph of f(x) = 1/4(4)^x?
Which is the graph of f(x) = 1/4(4)^x?
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Which equation represents the value y of a collector's item after x years if it increases exponentially?
Which equation represents the value y of a collector's item after x years if it increases exponentially?
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Which table represents exponential growth?
Which table represents exponential growth?
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Which is the graph of f(x) = 5(2)^x?
Which is the graph of f(x) = 5(2)^x?
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What is the multiplicative rate of change of the function?
What is the multiplicative rate of change of the function?
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Which is the graph of f(x) = 100(0.7)^x?
Which is the graph of f(x) = 100(0.7)^x?
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Which best describes a population of bacteria after each day of treatment with an antibiotic?
Which best describes a population of bacteria after each day of treatment with an antibiotic?
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What is the multiplicative rate of change of the function?
What is the multiplicative rate of change of the function?
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What is the multiplicative rate of change of the function?
What is the multiplicative rate of change of the function?
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Which is the graph of f(x) = 3(2/3)^x?
Which is the graph of f(x) = 3(2/3)^x?
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Which conclusion about f(x) and g(x) can be drawn from the table?
Which conclusion about f(x) and g(x) can be drawn from the table?
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What is the common ratio of the sequence -2, 6, -18, 54?
What is the common ratio of the sequence -2, 6, -18, 54?
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What value, written as a decimal, should Lena use as the common ratio for the graphed geometric sequence?
What value, written as a decimal, should Lena use as the common ratio for the graphed geometric sequence?
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Which explicit function represents the geometric sequence of heights of a bouncing toy?
Which explicit function represents the geometric sequence of heights of a bouncing toy?
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What will the graph show for the sequence 640, 160, 40, 10?
What will the graph show for the sequence 640, 160, 40, 10?
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What is the initial value of the geometric sequence represented on the graph?
What is the initial value of the geometric sequence represented on the graph?
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What value should Isaak use as the common ratio in the formula for the sequence 64, 112, 196, 343?
What value should Isaak use as the common ratio in the formula for the sequence 64, 112, 196, 343?
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Study Notes
Exponential Growth and Decay
- Exponential growth is represented by functions where the output increases at a constant percentage rate, such as f(x) = 500(2)^x or f(x) = 4(1.5)^x.
- The equation f(x) = 500(2)^x indicates an initial value of 500, and a multiplicative rate of change of 2.
- Exponential decay occurs when quantities decrease at a rate proportional to their current value, exemplified by f(x) = 5,000(0.4)^x, where 40% remains after each time period.
Graphical Representations
- Graphical representations of exponential functions reveal their behavior:
- Functions like f(x) = 4(1.5)^x show increasing y-values for increasing x-values, indicating exponential growth.
- Functions such as f(x) = (1/4)(4)^x exhibit behavior characteristic of exponential decay.
- The specific graph of f(x) = 3(2/3)^x represents decay, while f(x) = 3(4)^x shows growth.
Key Points on Functional Characteristics
- For functions f(x) = 4(3)^x and f(x) = 5(4)^x, understanding how to derive equations that pass through specific points (like (2, 36) or (2, 80)) assists in real-world applications of exponential models.
- Initial values and factors of increase or decrease considerably influence the shape of exponential functions.
Multiplicative Rate of Change
- The multiplicative rate of change for exponential functions reflects how quickly the function's output is changing, with examples of rates such as:
- A rate of change of 5 indicates rapid growth.
- In cases of decay, values like 0.4 or 0.97 reflect slower diminishment in quantity.
Sequences and Common Ratios
- Geometric sequences are characterized by a consistent multiplication factor known as the common ratio, such as -3 in the sequence -2, 6, -18, 54.
- An explicit formula for a geometric sequence can be identified, such as the height sequence of a bouncing toy represented by f(x) = 64(3/4)^(x-1).
Contextual Applications
- Real-world problems, such as population growth, investment values, or collector's items, can be modeled using exponential functions to predict future values based on initial quantities and growth rates.
- Equations like y = 18(1.15)^x help estimate the future worth of limited-edition items or changing town populations.
Summary of Graph Characteristics
- Determining whether a graph exhibits exponential growth or decay requires analysis of its slope and y-intercept; the domain and range also provide insight into the function's behavior over time.
- Visualizing the trends in population dynamics, bacterial decay, or financial growth helps clarify the importance of rate of change in exponential functions.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of exponential functions and their graphs with these flashcards. Each card presents a question to help reinforce key concepts related to exponential growth and corresponding equations. Perfect for students studying algebra or calculus concepts.
