Questions and Answers
Which graph shows exponential growth?
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?
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?
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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?
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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?
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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?
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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?
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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?
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Which is the graph of f(x) = 5(2)^x?
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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?
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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?
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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?
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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?
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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?
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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?
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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.
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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.
