Question

the food for lucia’s quinceañera costs $18.50 per person. lucia’s abuela helped with the cooking, so the caterers gave a $50 discount. the equation, $y = 18.5x - 50$ can be used to find the total cost, including the discount. the variable $x$ represents the number of guests. the variable $y$ represents the total cost, including the discount. the total cost is a function of the number of guests. what does the initial value represent? the amount of the discount in dollars what does the rate of change represent? the number of guests the cost per guest the total cost, including the discount the amount of the discount in dollars

Explanation:

For "What does the initial value represent?":

The equation is in slope - intercept form \(y = mx + b\), where \(b\) is the initial value. In the equation \(y=18.5x - 50\), the initial value (the constant term) is \(- 50\). But when we consider the context, the discount is a reduction in cost. The initial value here (the \(b\) term, but with a negative sign) represents the amount of the discount in dollars. Because if there were \(0\) guests (\(x = 0\)), \(y=- 50\), which shows the discount (a $50 reduction from a hypothetical cost of $0 when there are no guests).

In the linear equation \(y = mx + b\), \(m\) is the rate of change (slope). In the equation \(y = 18.5x-50\), the coefficient of \(x\) is \(18.5\). Since \(x\) is the number of guests and \(y\) is the total cost, the rate of change \(18.5\) represents the cost per guest. For each additional guest (\(\Delta x = 1\)), the total cost \(y\) increases by \(18.5\) dollars, which is the cost for one guest.

Answer:

the amount of the discount in dollars

For "What does the rate of change represent?":