Problem 12 - Algebra 2

Question

A downtown parking lot charges \(\$ 2\) for the first hour and \(\$ 1\) for each additional hour or part of an hour. Draw a graph of a function that represents this situation.

Step-by-Step Solution

Answer

Define a piecewise function for cost and plot as a step graph.

Step 1: Understand the Problem

First, we need to grasp the charging scheme of the parking lot. It charges a fixed rate of \( \\(2 \) for the first hour and \( \\)1 \) for each hour or part of an hour thereafter.

Step 2: Define the Function

We define the cost function \( C(h) \), where \( h \) is the number of hours parked. The cost function will be piecewise, with different expressions for different time durations.

Step 3: Determine Function Rules

For \( h \leq 1 \), the cost is \( \\(2 \). For every additional hour \( h > 1 \), the cost is \( \\)2 + (h - 1) \times \$1 \). Simplifying, this is \( C(h) = 2 + (h - 1) \) or \( C(h) = h + 1 \) for \( h \gt 1 \).

Step 4: Define the Piecewise Function

The complete piecewise function is defined as follows: \[ C(h) = \begin{cases} 2, & \text{if } 0 < h \leq 1 \ h+1, & \text{if } h > 1 \end{cases} \] This represents the cost in dollars based on the number of hours parked.

Step 5: Graph the Function

Draw the graph based on the piecewise function. For \( 0 < h \leq 1 \), plot a horizontal line at \( C(h) = 2 \). For \( h > 1 \), plot a line with a slope of 1 starting at \( h = 1 \), \( C(h) = 2 \). The graph will have a point at (1, 3) indicating the jump for additional hours.

Key Concepts

Function Graphing

When we talk about function graphing, what we are essentially doing is creating a visual representation of a mathematical function. This allows us to see how changes in input values affect the output. In this context, we're dealing with a parking cost problem, so the graph will show how the cost changes as you park for more hours.

To graph a function, start by understanding the relationship or rule that your graph will depict. For instance, with our parking lot problem:

• The first hour of parking costs a fixed amount of \(\\(2\), regardless of exact time spent within that hour.
• For every additional hour or fraction of an hour, the cost increases by \(\\)1\).

The graph of this function will help quickly illustrate these cost changes.

• When graphing, plot your axes first - typically, the horizontal axis (x-axis) represents the input (\

Piecewise Function Definition

A piecewise function is a function composed of multiple sub-functions, each of which applies to a specific interval of the function's domain. They are especially helpful for representing situations where a rule or condition changes at certain points.

For the parking lot problem, the cost function is piecewise because the cost calculation changes after the first hour. Here's how it breaks down:

• For the first hour (\(0 < h \leq 1\), the cost is constant at \(\\(2\).
• Beyond the first hour (\(h > 1\)), the cost increases linearly at a rate of \(\\)1\) per hour. So, if you park for, say, 3.5 hours, the cost will be \(\\(2 + 2.5 \times \\)1 = \$4.5\).

Clearly defining each segment of a piecewise function is critical, as it dictates how we graph the cost correctly. Remember, each segment usually appears on the graph as a different line or curve, depending on the function’s rule for that interval.

Using piecewise functions, we can succinctly describe complex real-life scenarios like this parking cost problem.

Parking Lot Cost Function

The parking lot cost function is unique due to how it charges. It combines a fixed initial cost with a variable cost as more time is spent in the lot. This function demonstrates how piecewise functions work in the real world.

We set the cost function, \(C(h)\), where \(h\) is the number of hours parked. Initially:

• For \(0 < h \leq 1\), the cost is straightforward and constant at \(\\(2\).
• As you park longer, \(h > 1\), the cost increases by \\)1 for each additional hour. Mathematically, this becomes \(C(h) = h + 1\) after the first hour.

The piecewise function expressed by \(C(h)\) closely models how parking fees accumulate on a real lot.

Understanding this function means you can always predict parking costs based on time, an essential skill for budgeting and planning. This function not only represents a real-world scenario but is also an excellent example of how mathematics translates into practical applications.