Problem 25 - Algebra 2

Question

A high school spends \(\$ 750\) each year on student accuemic achievement awards. The amount spent per award depends on how many awards are given. Write and graph a function that models the relationship between the number \(a\) of awards given and the cost \(c\) of each award. What are a reasonable domain and range for the function?

Step-by-Step Solution

Answer

The function to model the relationship between the number of awards and the cost of each award is \(c = 750/a\). The graph will start at the point (1,750) and will approach the x-axis as a increases, without crossing it. The domain of this function is all real numbers \(a > 0\), and the range is all real numbers \(c > 0\).

Step 1: Define the Relationship

We are given that the school spends \$750 per year on awards and that the cost of each award depends on how many awards are given. This means that the more awards are given, the cheaper each award will be. Therefore, it can be said that the cost of each award is inversely proportional to the number of awards.

Step 2: Write a Function

This relationship can be written as a function. If \(a\) represents the number of awards and \(c\) represents the cost of each award, then \(c = 750 / a\).

Step 3: Graph the Function

Next, create a graph with two axes, where the x-axis represents the number of awards (a) and the y-axis represents the cost of each award (c). Plot the function \(c = 750 / a\). The graph will start at the point (1,750) and will approach the x-axis as a increases, but will never cross it.

Step 4: Define Domain and Range

The domain (possible values for a) is all real numbers greater than zero as we cannot give negative number of awards and giving zero awards would result in undefined cost. So, domain = \(a > 0\). The range (possible values for c) is also all real numbers greater than zero because the cost of each award cannot be zero or less. So, range = \(c > 0\)

Key Concepts

Function Graphing

When we talk about function graphing, we are visualizing mathematical relationships. Here, the first axis (x-axis) is used for the number of awards, represented as \(a\). The second axis (y-axis) is used for the cost per award, represented as \(c\). The relationship between these variables is expressed with the equation \(c = \frac{750}{a}\). This equation shows inverse variation; as more awards are given, the cost per award decreases.

To graph this function, set up a coordinate plane:

• Have the x-axis for the awards (\(a\))
• Have the y-axis for the award cost (\(c\))

Choose some values of \(a\) to calculate \(c\), and plot these points. Notice how as \(a\) increases, the points form a curve that gets closer to the x-axis but never touches it. This curve is known as a hyperbola, a characteristic shape of inverse variation functions.

Understanding this graph helps visualize how the cost per award is affected by the number of awards given out.

Domain and Range

The domain and range of a function help define its possible values. For our function \(c = \frac{750}{a}\), understanding these values is crucial.

The **domain** refers to all potential values that \(a\) (the number of awards) can take. In this scenario, awards can't be negative or zero, so the domain is all real numbers greater than zero or \((a > 0)\). This represents that at least one award must be given.

The **range** involves all possible values for \(c\) (the cost). Since the school must spend some money on the awards and the cost can't be zero, the range is all positive real numbers or \((c > 0)\). This highlights that as the number of awards increases, the cost approaches zero but never actually becomes zero.

Being aware of the domain and range ensures clarity about what the function can represent, avoiding mathematical missteps such as division by zero or negative costs.

Algebraic Functions

Algebraic functions are equations made up of variables and constants that are combined with operations like addition, subtraction, multiplication, division, etc. In the case of our exercise, the function \(c = \frac{750}{a}\) is an example of an algebraic function, specifically an inverse variation.

Inverse variation occurs when one variable increases while the other decreases. In this case, as the number of awards \(a\) increases, the cost \(c\) per award decreases. The term "750" in the function represents the total budget allocation for awards, showcasing how the function is tailored to a specific situation.

Understanding algebraic functions and their behavior is fundamental in figuring out real-world problems. These functions are mathematical models that help in predicting and explaining occurrences, just like how this model relates the number of awards to their costs.