Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each month is modeled by V(x), where V(x)=2x^2-32x+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 mod-eled by V(x)=31x-226. Find the number of volunteers in each of the following months. January

Verified step by step guidance

1. Identify the function that models the number of volunteers for January. Since January is between January and August, use the function V(x) = 2x^2 - 32x + 150.
2. Substitute x = 1 into the function V(x) = 2x^2 - 32x + 150 to find the number of volunteers in January.
3. Calculate the expression 2(1)^2 - 32(1) + 150.
4. Simplify the expression by performing the operations: first, calculate 2(1)^2, then subtract 32(1), and finally add 150.
5. The result of the simplified expression will give the number of volunteers in January.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form V(x) = ax^2 + bx + c. In this case, V(x) = 2x^2 - 32x + 150 represents the number of volunteers from January to August. Understanding how to evaluate quadratic functions is essential for determining the number of volunteers in specific months.

Function Evaluation

Function evaluation involves substituting a specific input value into a function to find its output. For example, to find the number of volunteers in January, we substitute x = 1 into the quadratic function V(x). This process is crucial for solving the problem and obtaining the correct number of volunteers for each month.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this scenario, V(x) is modeled by a quadratic function from January to August and a linear function from August to December. Recognizing how to work with piecewise functions is important for understanding the changes in the number of volunteers over the specified months.