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8th Grade Math: Bacteria Population Function and Linear Relationship Question
Mathematics
Grade 8 (Junior High School)
Question Content
Lin counts 5 bacteria under a microscope. She counts them again each day for four days, and finds that the number of bacteria doubled each day—from 5 to 10, then from 10 to 20, and so on. Is the population of bacteria a function of the number of days? If so, is it linear? Explain your reasoning.
Correct Answer
1. Yes, the bacteria population is a function of the number of days. 2. No, it is not linear.
Detailed Solution Steps
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Step 1: Determine if it is a function. A function is a relationship where each input (number of days) has exactly one output (bacteria population). For day 0: 5 bacteria, day 1: 10 bacteria, day 2: 20 bacteria, day 3: 40 bacteria, day 4: 80 bacteria. Each day has only one corresponding population, so it meets the definition of a function.
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Step 2: Determine if it is linear. A linear function has a constant rate of change (the difference between consecutive outputs is the same). Calculate the rate of change: From day 0 to 1: 10-5=5; day 1 to 2:20-10=10; day 2 to 3:40-20=20; day 3 to 4:80-40=40. The rate of change increases each day, so it is not constant. Alternatively, the equation for the population is $P(d)=5\times2^d$, which is an exponential function, not a linear function with the form $y=mx+b$.
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Step 3: Summarize the reasoning: It is a function because each day maps to exactly one population, but it is not linear because the rate of change is not constant.
Knowledge Points Involved
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Definition of a Function
A function is a relation between a set of inputs (domain) and a set of possible outputs (range) where each input is associated with exactly one output. It can be tested using the vertical line test on a graph, or by verifying that no input has multiple distinct outputs in a table/equation.
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Linear Function Characteristics
A linear function is a function whose graph is a straight line, represented by the equation $y=mx+b$ where $m$ is the constant slope (rate of change) and $b$ is the y-intercept. The key feature is a constant rate of change between all pairs of consecutive points.
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Exponential Growth
Exponential growth occurs when a quantity increases by a constant factor over equal intervals of time, modeled by $y=a\times r^x$ where $a$ is the initial amount, $r$ is the growth factor, and $x$ is time. Unlike linear growth, the rate of change increases (or decreases for decay) over time.
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