Question

The student council at a large high school is having a fundraiser for a local charity. The council president suggests that they set a goal of raising \(\$ 4000\) a) Let \(x\) represent the number of students who contribute. Let \(y\) represent the average amount required per student to meet the goal. What function \(y\) in terms of \(x\) represents this situation? b) Graph the function. c) Explain what the behaviour of the function for various values of \(x\) means in this context. d) How would the equation and graph of the function change if the student council also received a \(\$ 1000\) donation from a local business?

Short answer

The function is \(y = \frac{4000}{x}\). With a \$1000 donation, it becomes \(y = \frac{3000}{x}\).

Step-by-step solution

- Define the function

Let us start by defining the relationship between the number of students, represented by \(x\), and the average amount per student, represented by \(y\). To raise \$4000, each student should contribute enough so that the total amount is \$4000. Hence, the function can be represented as: \[ y = \frac{4000}{x} \]

- Graph the function

To graph the function \(y = \frac{4000}{x}\), create a set of values for \(x\) (number of students) and compute the corresponding values of \(y\) (average amount per student). Plot these points on a graph. The graph will show a hyperbolic curve, indicating that as the number of students increases, the average amount per student decreases.

- Analyze the behavior of the function

The function shows that for small values of \(x\), the average amount each student must contribute \((y)\) is high. As \(x\) increases, the value of \(y\) decreases, meaning each student needs to contribute less. This inverse relationship is important in understanding how participation (number of students) affects the amount each student needs to give.

- Incorporate the additional donation

If the student council receives a \$ 1000 donation from a local business, the goal for the students changes from \$4000 to \$3000. Thus, the new function becomes: \[ y = \frac{3000}{x} \] The graph of this new function will be similar to the original but will be lower, representing the reduced amount needed from each student.

Key concepts

Linear Functions

Linear functions are mathematical expressions that establish a direct relationship between two variables. In its simplest form, a linear function can be represented as: \( y = mx + b \) Here, \( y \) represents the dependent variable, \( x \) the independent variable, \( m \) the slope or rate of change, and \( b \) the y-intercept. In the context of our fundraising problem, the relationship between the total amount required and the number of students isn't linear but rather follows a different pattern. However, understanding linear functions helps to grasp how direct relationships work.

Elementary characteristics of linear functions include:

• A constant rate of change
• Straight-line graphs
• Easily interpretable slopes and intercepts

While linear functions provide a useful foundation, our fundraiser problem will use a different type of function to represent the relationship between student contributions and total amount raised.

Hyperbolic Functions

In our fundraising scenario, the relationship between the number of students contributing and the amount each student needs to contribute follows a hyperbolic pattern. A hyperbolic function is one of the forms: \( y = \frac{k}{x} \), where \( k \) is a constant.

Here, we have \( y = \frac{4000}{x} \), meaning if 100 students contribute, each would need to give \( \frac{4000}{100} = 40 \). If 500 students contribute, each needs to give \( \frac{4000}{500} = 8 \).

Key properties of hyperbolic functions:

• Inverse relationship: As one variable increases, the other decreases.
• Asymptotes: The graph approaches the axes but never touches them.
• Hyperbolic curve: Unlike linear graphs, these curves reflect changing rates.

Incorporating an additional donation from a business changes the equation to \( y = \frac{3000}{x} \), reducing the burden on each student, and resulting in a similar, but lower, hyperbolic graph.

Fundraising Problems

Fundraising problems often involve figuring out how to meet a financial goal using contributions from various sources. In our example, the student council wants to raise $4000. They first need to understand how much each student needs to contribute, depending on the number of students participating. This involves setting up an equation or function that represents this relationship.

Important steps in solving fundraising problems:

• Define the total amount needed (goal).
• Identify the contributors and their possible contributions.
• Create an equation that describes the relationship between contributors and the amount contributed.
• Adjust for additional sources of funds if available.

Through these steps, we can determine whether the goal is realistic and how best to reach it.

Graphing Functions

Graphing functions helps visualize the relationships between different variables. In our case, plotting the function \( y = \frac{4000}{x} \) shows the inverse relationship between the number of contributors (x) and the average amount each needs to contribute (y).

Steps to graph this function:

• Select a range of values for \( x \), like 10, 20, 50, 100.
• Calculate the corresponding \( y \) values using the function.
• Plot the (x, y) pairs on a coordinate plane.
• Draw a smooth curve through the points.

The resulting hyperbolic curve will help illustrate how increasing the number of students decreases the average contribution. Comparatively, the new function after an additional donation \( y = \frac{3000}{x} \) will be lower, reflecting the reduced target amount for students.