1. Home
  2. College algebra
  3. Polynomial and rational functions
  4. Quadratic functions

5.1 Quadratic functions  (Page 5/15)

<< Chapter < Page
  College algebra     Page 5 / 15
Chapter >> Page >

Finding maximum revenue

The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?

Revenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity. We can introduce variables,   p   for price per subscription and   Q   for quantity, giving us the equation   Revenue = p Q .

Because the number of subscribers changes with the price, we need to find a relationship between the variables. We know that currently   p = 30   and   Q = 84,000.   We also know that if the price rises to $32, the newspaper would lose 5,000 subscribers, giving a second pair of values,   p = 32   and   Q = 79,000.   From this we can find a linear equation relating the two quantities. The slope will be

m = 79,000 − 84,000 32 − 30 = −5,000 2 = −2,500

This tells us the paper will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the y -intercept.

Q = −2500 p + b Substitute in the point Q = 84,000  and  p = 30 84,000 = −2500 ( 30 ) + b Solve for b b = 159,000

This gives us the linear equation   Q = −2,500 p + 159,000   relating cost and subscribers. We now return to our revenue equation.

Revenue = p Q Revenue = p ( −2,500 p + 159,000 ) Revenue = −2,500 p 2 + 159,000 p

We now have a quadratic function for revenue as a function of the subscription charge. To find the price that will maximize revenue for the newspaper, we can find the vertex.

h = − 159,000 2 ( −2,500 ) = 31.8

The model tells us that the maximum revenue will occur if the newspaper charges $31.80 for a subscription. To find what the maximum revenue is, we evaluate the revenue function.

maximum revenue = −2,500 ( 31.8 ) 2 + 159,000 ( 31.8 ) = 2,528,100
Got questions? Get instant answers now!

Finding the x - and y -intercepts of a quadratic function

Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the   y - intercept of a quadratic by evaluating the function at an input of zero, and we find the   x - intercepts at locations where the output is zero. Notice in [link] that the number of   x - intercepts can vary depending upon the location of the graph.

Three graphs where the first graph shows a parabola with no x-intercept, the second is a parabola with one –intercept, and the third parabola is of two x-intercepts.
Number of x -intercepts of a parabola

Given a quadratic function   f ( x ) ,   find the   y - and x -intercepts.

  1. Evaluate   f ( 0 )   to find the y -intercept.
  2. Solve the quadratic equation   f ( x ) = 0   to find the x -intercepts.

Finding the y - and x -intercepts of a parabola

Find the y - and x -intercepts of the quadratic   f ( x ) = 3 x 2 + 5 x − 2.

We find the y -intercept by evaluating   f ( 0 ) .

f ( 0 ) = 3 ( 0 ) 2 + 5 ( 0 ) − 2 = −2

So the y -intercept is at   ( 0 , −2 ) .

For the x -intercepts, we find all solutions of   f ( x ) = 0.

0 = 3 x 2 + 5 x − 2

In this case, the quadratic can be factored easily, providing the simplest method for solution.

0 = ( 3 x − 1 ) ( x + 2 )
h = − b 2 a k = f ( −1 ) = − 4 2 ( 2 ) = 2 ( −1 ) 2 + 4 ( −1 ) − 4 = −1 = −6

So the x -intercepts are at   ( 1 3 , 0 )   and   ( − 2 , 0 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
Theophilus Reply
what's lochia albra
Kizito
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 7

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask