Solve each problem. During the course of a year, the number of volunteers available to run a food bank each month is modeled by $V (x)$ , where $V (x) = 2 x^{2} - 32 x + 150$ between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, $V (x)$ is modeled by $V (x) = 31 x - 226$ . Find the number of volunteers in each of the following months. Sketch a graph of $y = V (x)$ for January through December. In what month are the fewest volunteers available?

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Identify the piecewise function for the number of volunteers, $V(x)$, where $x$ represents the month number: for January to August ($x=1$ to $x=8$), use $V(x) = 2x^2 - 32x + 150$, and for September to December ($x=9$ to $x=12$), use $V(x) = 31x - 226$.

Calculate the number of volunteers for each month by substituting the month number $x$ into the appropriate formula: for $x=1$ to $8$, compute $V(x) = 2x^2 - 32x + 150$, and for $x=9$ to $12$, compute $V(x) = 31x - 226$.

List the values of $V(x)$ for all months from January ($x=1$) through December ($x=12$) to have a complete set of volunteer numbers for the year.

Sketch the graph of $y = V(x)$ by plotting the points $(x, V(x))$ for each month $x=1$ to $12$. Note that the graph will have two parts: a quadratic curve from January to August and a linear function from September to December.

Analyze the calculated values or the graph to determine the month with the fewest volunteers by identifying the minimum value of $V(x)$ over the entire range $x=1$ to $12$.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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12m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. In this problem, V(x) is given by one quadratic formula from January to August and a linear formula from August to December. Understanding how to evaluate and interpret piecewise functions is essential for finding values and analyzing the behavior over specified intervals.

Evaluating Functions

Evaluating a function means substituting a given input value into the function's formula to find the output. Here, to find the number of volunteers in each month, you substitute the month number (x) into the appropriate formula for V(x). Accurate substitution and calculation are key to determining the correct volunteer counts.

Graphing and Analyzing Functions

Graphing a function involves plotting points (x, V(x)) to visualize its behavior over a domain. For this problem, sketching V(x) from January to December helps identify trends and the month with the fewest volunteers. Understanding how to interpret graphs, including identifying minima, is crucial for answering questions about the function's real-world implications.