Real-World Applications of Graphing Principles and Functions

This assignment focuses on illustrating the application of graphing principles, specifically quadratic, exponential, and logarithmic functions, in real-world scenarios. The solution provides two detailed examples: depreciation of an item's value over time, modeled using a linear function, and the exponential growth of a bacteria population. For each example, the assignment clearly defines the domain and range of the function, crucial components for understanding the function's behavior and limitations within the context of the problem. The depreciation example shows how a linear model can be used to calculate the decreasing value of an asset over a specific period, while the bacteria growth example demonstrates how exponential functions describe population increase. The assignment also includes visual representations (graphs) and references to support the concepts discussed, making it a comprehensive guide for students studying graphing principles and their practical relevance.

Q. This unit primarily focuses on graphing quadratic, exponential, and
logarithmic functions using a coordinate plane.Can you think of any real-
world examples of where graphing principles are used,like the domain and
range? (Make sure to include the domain and range in your response)
Answer
Definition(s):Function,loosely,is a relation that maps a set ofinputs to
a set ofoutputs. Eg: {(1, 3), (2, 4), (3, 5), . . . , (7, 9)} is a function,y = x2
is also function.One additionalrequirement for a relation to be a func-
tion is that each input should exactly map to one output,so a relation like
{(1, 2), (2, 3), (2, 4), . . . , (5, 6)} is not a function.A set of inputs over which
the function is well defined is called domain of a function.Domain of y = x2
is the set of all real numbers.Sometimes the domain can be set to a particu-
lar set of inputs over which the function is defined, eg.:y = x2 ∀ 0 ≤ x ≤ 2,
here the domain is the interval [0, 2].Range of a function is the range of val-
ues a function attains over a specified domain.Range of the function y = x2
over the domain of real numbers is:0 to +∞ while in the case when domain
being 0 ≤ x ≤ 2, the range is 0 to 4, which is also expressed as:0 ≤ y ≤ 4.
Real world applications of graphing principles:
Example 1) Depreciation:
A company usually calculates the depreciation cost ofits purchased items
using linear models.
Suppose a company buys an item for 1000 $.It anticipates that the value of
the item will depreciate to nil in 10 years.It assumes that the value depre-
ciates linearly.The domain for the value function is the period of time over
0 to 10 years while the range is the item’s value:0 to 1000.Graph of such a
linear model is shown in figure-1 below.
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Figure 1:Depreciation of cost over time
Example 2) Bacteria growth curve:
In a lab,under suitable conditions,a growing bacteria population increases
at regular intervals.Growth in this phase is exponential.Actually though,
exponential growth is just one part of bacteria’s life cycle and not a complete
representative of bacteria’s growth pattern observed in nature.
A biologist is studying a bacteria culture in a laboratory.At some initial
time t equal to 0 hours, he puts N1 bacteria into a dish which has a favorable
growth medium for the bacteria culture, and after n hours he finds that the
population has grown to N2.
Such bacteria growth problemsare modelled using exponentialfunction,
which generally take the following form:
N2 = N1ekn
where, k is termed as growth constant.If the population grows with time t
then k is positive.
This growth function has a domain:0 ≤ t ≤ n
It’s range is:N1 ≤ N ≤ N2.
A typical plot for the above bacteria growth model is shown in figure-2 below.
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Figure 2:Exp. function for bacteria growth
References:
Blitzer, R. (2019).Thinking mathematically (6th ed.).Boston:Pearson.
Stewart, J., Redlin, L., Watson, S.,Panman, P. (2017).Precalculus:Math-
ematics for Calculus.Australia:Cengage Learning.
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