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Many real-life situations, such as buying a cellphone, making an investment, or studying the population growth of a particular species, can be modeled using functions. This lesson will review some situations and identify whether they can be fitted into a linear, quadratic, or exponential function to make optimal decisions.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Distinctive Characteristics of a Quadratic Function

The graphs of a linear function, a quadratic function, and an exponential function can be seen in the applet below.
Linear, Quadratic, and Exponential graphs button applet
Is it possible to name at least three characteristics of a quadratic function that neither a linear nor an exponential function has?
Example

Classifying Functions

Magdalena has just found an old printer toy she used to play with when she was a child. She discovered that the toy prints random functions as labels.

Magdalena and Linear, Quadratic, and Exponential function labels
Magdalena ran the printer and got three labels with the following functions.
She wants to determine whether each of the functions is linear, quadratic, or exponential. Help Magdalena to accomplish this task!

Hint

Review the definitions of linear, quadratic, and exponential functions.

Solution
Consider the first given function.
Note that the independent variable is written as an exponent in the function and the base is a non-negative number. Therefore, this is an exponential function.
Now, consider the second function.
This function only contains one linear term and one constant term. This means that it is a linear function.
Finally, consider the third function.
Notice that this is a polynomial function and the highest exponent of the independent variable is Therefore, it can be concluded that this is a quadratic equation.
This information is summarized in the following table.
Linear Function
Quadratic Function
Exponential Function
Pop Quiz

Determining if a Function Is Linear, Quadratic, or Exponential

In the following applet, determine whether the given function is linear, quadratic, or exponential.

Identify whether the given function is linear, quadratic, or exponential.
Discussion

Finding the Pattern of a Table of Values to Determine Its Model

When considering a table of values, the trend of the data can be determined by observing how the dependent variable changes over equal intervals. To do so, the following situations need to be analyzed.

  1. If the values have a common difference, the data follows a linear model.
  2. If the values have a common ratio, the data follows an exponential model.
  3. If the second differences — the differences of the first differences — are constant, the data follows a quadratic model.

The applet below illustrates each situation.

An applet showing a linear, quadratic, and exponential model in a table of values
Example

Determining the Pattern Describing Real-Life Data Sets

Magdalena's father works as a manager for a fish farming company. This morning, Magdalena and her brother Vincenzo poked around in her father's briefcase. They found some papers containing information about the fish population in three nearby lakes.

Fish Population (Thousands)
Time (Months) Lake Verdastro Lake Rumoroso Lake Mezzo
Magdalena told her brother that determining if the fish growth follows some pattern will help their father make better decisions for the company. However, they are not sure how to do this task. Help them determine whether the collected data for each lake can be modeled by a linear, quadratic, or exponential function.

Answer

Lake Verdastro: Linear Function
Lake Rumoroso: Exponential Function
Lake Mezzo: Quadratic Function

Hint

The values have a common difference in a linear function, while in an exponential function, the values have a common ratio. Furthermore, in a quadratic function, the second differences are constant.

Solution
The recorded data for each lake will be analyzed to determine if they have a common difference, a common ratio, or a constant second difference. So, the data for Lake Verdastro will be analyzed first.
Verdastro data in a roc table showing that it has common difference
Because the fish population in Lake Verdastro has a common difference of it can be said that it follows a linear model. The data for Lake Rumoroso will be analyzed in a similar fashion.
Rumoroso data in a roc table showing that it has common ratio
Note that the data of Lake Rumoroso has a common ratio of Therefore, it can be fitted into an exponential function. Finally, the model for the data of Lake Mezzo will be determined.
Mezzo data in a roc table showing that it has second constant difference
Since the second differences of the data of Lake Mezzo are constant, the fish population for the lake over time can be modeled with a quadratic function. With this, the growth model of the fish population for each lake has been determined.
Lake Model
Verdastro Linear Function
Rumoroso Exponential Function
Mezzo Quadratic Function

Note that the analysis suggests that Lake Rumoroso is the best for fish farming. The children gave this information to their father, who was delighted by this excellent analysis and took them for ice cream as a reward.

Example

Saving for a New Computer

Magdalena will start college soon, so she decided to start saving money to buy a new computer. The computer she wants costs around Thanks to her help with the analysis for the fish farming company, her father gave Magdalena an initial amount of She plans to add to her savings each week.

Magdalena looking at the computer she desires

Magdalena wants to buy the computer as soon as possible. She thinks she will be able to afford it after weeks of saving the money but she isn't sure. Answer the following questions to help her discover if she will save enough money for the new computer by then.

a Can this situation be modeled by a linear, quadratic, or exponential function? Write the function that models the situation.
b How much money will Magdalena have saved after weeks? Interpret the results.

Answer
a Model: Linear Function
Function:
b Answer:
Interpretation: Because the amount of money Magdalena will save after weeks is greater than the computer's price, she will be able to buy the computer.

Hint
a After weeks of saving money, the total amount of savings can be determined by adding the product of and the amount of weekly savings to the initial amount.
b Substitute for into the function found in Part A. Compare the result with the price of the computer.

Solution
a Magdalena starts with the initial that her father gave her. Each week she plans to add to this amount. Therefore, her balance after weeks can be calculated by adding the product of and to the initial amount of
Looking at the function, it can be concluded that this situation is modeled by a linear function.
c The amount of money that Magdalena will have after weeks can be found by substituting for in the function found in Part A.
Substitute

Multiply

Multiply

AddTerms

Add terms

If everything goes according to plan, Magdalena will have after weeks. Recall that the price of the computer is around The amount that Magdalena is expected to have saved is greater. Therefore, she will be able to buy the new computer and use it for her school projects by that time.
Magdalena enjoying her brand new computer
Example

Finding the Optimal Increase to Get the Maximum Revenue

Magdalena is happy with her new computer. She is also using it for her part-time job as a designer to help pay for her school expenses. She sells designs per month at each. However, because creating each design is complex, she plans to increase the price of each design. She estimates that for each increase, two fewer designs will be sold per month.

Magdalena wants to know what price will maximize her revenue, or the amount of money she makes. Identify the following information to help Magdalena make the best decision about her small business.

a Identify and write the function that models this situation.
b What is the maximum amount of money that Magdalena could make each month?
c What price would allow Magdalena to sell her designs so that she could make the most money each month?

Answer
a Model: Quadratic Function
Function:
b
c

Hint
a Begin by writing a function for the new price of a design and another function for the number of designs sold.
b Remember that the axis of symmetry intersects the parabola at its vertex.
c Substitute the value of the vertex into the expression of the new price of a design.

Solution
a Magdalena has estimated how each increase will affect the number of designs sold. Let be the number of price increases. This means the new price of a design will be the current price of plus times
Increasing the price of the designs means that Magdalena will not sell as many designs. For each increase, fewer designs will be sold. This means that the number of designs sold will be given by the current number of designs sold per month minus times .
Finally, the amount of money Magdalena can make each month can be determined by multiplying the number of designs sold by the new price of a design
Substitute the expression of each function and simplify to find the model describing the amount of money Magdalena can make.
SubstituteII

,

▼
Simplify right-hand side
Distr

Distribute

CommutativePropMult

Commutative Property of Multiplication

Distr

Distribute

AddTerms

Add terms

CommutativePropAdd

Commutative Property of Addition

Therefore, Magdalena's situation can be modeled with a quadratic function.
b Consider the quadratic function found in Part A.
The function will be modeled with a parabola. Since the leading coefficient is negative, the parabola opens downward and reaches its maximum at its vertex. The maximum revenue is given by the value of the vertex. First, the value of the vertex will be determined. Consider the formula for the axis of symmetry of a parabola.
Since the axis of symmetry intersects the parabola at the vertex, the value of the vertex is also given by this expression. In this quadratic function, and
SubstituteII

,

▼
Evaluate right-hand side
The axis of symmetry and the coordinate of the vertex is To find the coordinate of the vertex, substitute into the quadratic function and simplify.
Substitute

▼
Simplify right-hand side
CalcPow

Calculate power

Multiply

Multiply

AddTerms

Add terms

The vertex of the parabola is This means that the maximum amount of money that Magdalena can expect to make each month is It is worth noting that does make sense in this context, since represents the number of increases of needed to be made to the design price.
c It was determined that the maximum revenue per month is reached when This means that Magdalena will need to make increases of to the price of a design to make the most amount of money each month. Substitute this value into the function to find the new price for a design.
Substitute

▼
Simplify right-hand side
Multiply

Multiply

AddTerms

Add terms

Therefore, Magdalena should sell her designs for each to make the most amount of money each month.
Example

Determining the Storage Needed for a School Assignment

Vincenzo, Magdalena's little brother, loves dinosaurs. He wants to make a minute film for a school assignment, explaining the history of this extinguished group of reptiles. He asked Magdalena to help him make and edit the video on her computer. However, the siblings are worried because Vincenzo has a USB drive with only to store the film on.

Video-editing-demo.jpg

After some research, Magdalena found that any high-quality video file requires an initial storage space of This required storage space doubles for each additional minute. The children now have to investigate whether the USB drive has enough space for Vincenzo's completed film. Find the following information to help them.

a Identify the model and write a function representing the size of an minute long video.
b Does the USB have enough storage space for Vincenzo's completed video? Explain.

Answer
a Model: Exponential

Function:

b No, because the video size will be which is larger than the USB storage.

Hint
a Consider that exponential functions grow by equal factors over equal intervals.
b Substitute for in the function found in Part A. Then divide the result by to convert it from megabytes to gigabytes.

Solution
a It is given that the size of the video grows by a factor of for every minute of the video. This means this situation is represented by an exponential growth and it can be modeled with an exponential function. Consider the general form of an exponential function.
In this form, represents the growth factor and is the intercept, which is sometimes referred to as the initial value. Since the video file needs an initial storage space of and it doubles for every additional minute, substituting for and for will give the expression representing the video size after minutes.
b To determine if the USB storage is large enough for Vincenzo's minute film, evaluate the expression written in Part A for
Substitute

CalcPow

Calculate power

Multiply

Multiply

The size of a minute video is To compare this value with the total capacity of the USB drive, it needs to be converted to from megabytes to gigabytes. Because equals dividing the size of the video by will give the size in gigabytes.
Therefore, the minute film will have a size of This is larger than the total storage of the USB drive. Unfortunately, the children cannot save the file onto the drive and have to buy a new one with storage of at least
Closure

Identifying Some Remarkable Characteristics of Quadratic Functions

Many situations involving linear, quadratic, and exponential functions have been covered throughout this lesson. It is now time to point out some remarkable characteristics of quadratic functions. Consider the applet presented at the beginning of the lesson.
Linear, Quadratic, and Exponential graphs button applet
Is it possible to name at least three characteristics of a quadratic function that neither a linear nor an exponential function has?

Answer

See solution.

Hint

Begin by analyzing the behavior of the functions. Do all functions have a turning point? What is the maximum number of intercepts each function could have?

Solution

Three distinctive characteristics of quadratic functions will be identified one at a time.

First Characteristic

To identify the first characteristic, look at the behavior of the graphs of linear, quadratic, and exponential functions.
An applet showing the behavior of linear, quadratic, and exponential functions. The parabola opens upward.
In the applet above, it can be seen that both linear and exponential functions increase throughout the graph. Conversely, the quadratic function is a parabola that shifts from decreasing to increasing at the vertex. Now consider another three functions.
An applet showing the behavior of linear, quadratic, and exponential functions. The parabola opens downward.
This time both the linear and exponential functions only decrease. The parabola of the quadratic function opens downward now but it still changes directions, this time from increasing to decreasing. With this information, the following conclusion can be made.

Linear and exponential functions either only increase or only decrease. Conversely, quadratic functions always decrease and increase. The order depends on the direction in which the parabola opens.

Second Characteristic

A second unique characteristic can be determined directly from the first. The vertex of a parabola is its turning point. At this point, the function reaches its absolute maximum or minimum. If the parabola opens upward, the vertex represents the absolute minimum. If it opens downward, the vertex is the absolute maximum.
An applet showing the absolute maximum of a parabola opening downward and the absolute minimum when the parabola opens upward
Conversely, linear and exponential functions approach positive infinity or negative infinity since they only decrease or increase, meaning that they do not have an absolute maximum or minimum. The second characteristic for quadratic functions can now be noted.

Because a parabola either opens upward or downward, there is always one point that is the absolute minimum or absolute maximum of the function. This point is called the vertex.

Third Characteristic

The intercepts of a function are known as the zeros of the function. They are found by setting the function rule equal to zero and simplifying.
Because a quadratic equation has at most solutions, the quadratic function can have one, two, or no intercepts.
An applet showing the zeros of a parabola
A linear functions whose graph is not horizontal has exactly one intercept. Exponential functions can have one or no intercepts.
Linear, Quadratic, and Exponential graphs button applet

Using this information, a third distinctive characteristic of quadratic functions can be determined.

While non-horizontal linear functions and exponential functions have at most one intercept, quadratic functions can have up to two intercepts.

Conclusion

Only three distinctive characteristics of quadratic functions are presented here. Please note that there are a few others.

Three Distinctive Characteristics of Quadratic Functions
Linear and exponential functions either only increase or only decrease. Conversely, quadratic functions always decrease and increase. The order depends on the direction in which the parabola opens.
Because a parabola either opens upward or downward, there is always one point that is the absolute minimum or absolute maximum of the function. This point is called the vertex.
While non-horizontal linear functions and exponential functions have at most one intercept, quadratic functions can have up to two intercepts.


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