Question

a company starts with 500 units of a product in stock and sells 30 units per day. the function f(x)=500-30x describes the remaining stock after x days. which of the following is the meaning of the y-intercept?

the y-intercept is the initial stock of the product before any sales.

the y-intercept is the number of days when half of the stock has been sold.

the y-intercept is the number of days until the stock is depleted.

the y-intercept is the remaining stock after 10 days.

Explanation:

In the linear function \(y = f(x)=500 - 30x\), the \(y\) - intercept is found when \(x = 0\). Substituting \(x = 0\) into the function gives \(y=f(0)=500-30\times0 = 500\). This value represents the initial amount of stock before any days (\(x = 0\)) have passed, i.e., before any sales (\(30x\) term with \(x = 0\) means no units sold). The other options:

• For "the number of days when half of the stock has been sold", we would solve \(250=500 - 30x\) for \(x\) (not related to \(y\) - intercept).
• For "the number of days until the stock is depleted", we solve \(0 = 500-30x\) for \(x\) (not \(y\) - intercept).
• For "the remaining stock after 10 days", we substitute \(x = 10\) into \(y=500-30x\) (not \(y\) - intercept).

Answer:

The \(y\) - intercept is the initial stock of the product before any sales.