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y = 2 x 2 4 x + 1

y : ( 0 , 1 ) ; x : ( 1.7 , 0 ) , ( 0.3 , 0 ) ;
axis: x = 1 ; vertex : ( 1 , −1 )
This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, -1) and the intercepts (1.7, 0), (0.3, 0) and (0, 1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.

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y = 2 x 2 4 x + 2

y : ( 0 , 2 ) x : ( 1 , 0 ) ;
axis: x = 1 ; vertex: ( 1 , 0 )
 This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, 0) and the intercept (0, 2). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.

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y = −4 x 2 6 x 2

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y = x 2 4 x + 2

y : ( 0 , 2 ) x : ( −4.4 , 0 ) , ( 0.4 , 0 ) ;
axis: x = −2 ; vertex: ( −2 , 6 )
This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-2, 6) and the intercepts (-4.4, 0), (0.4, 0) and (0, 2). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2.

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y = 5 x 2 10 x + 8

y : ( 0 , 8 ) ; x : none ;
axis: x = 1 ; vertex : ( 1 , 3 )
This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, 3) and the intercept(0, 8). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.

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y = −16 x 2 + 24 x 9

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y = 3 x 2 + 18 x + 20

y : ( 0 , 20 ) x : ( −4.5 , 0 ) , ( −1.5 , 0 ) ;
axis: x = −3 ; vertex: ( −3 , −7 )
This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-3, -7) and the intercepts (-4.5, 0) and (-1.5, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.

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y = −2 x 2 + 8 x 10

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Solve Maximum and Minimum Applications

In the following exercises, find the maximum or minimum value.

y = 2 x 2 + x 1

The minimum value is 9 8 when x = 1 4 .

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y = −4 x 2 + 12 x 5

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y = x 2 6 x + 15

The minimum value is 6 when x = 3 .

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y = −9 x 2 + 16

The maximum value is 16 when x = 0 .

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In the following exercises, solve. Round answers to the nearest tenth.

An arrow is shot vertically upward from a platform 45 feet high at a rate of 168 ft/sec. Use the quadratic equation h = −16 t 2 + 168 t + 45 to find how long it will take the arrow to reach its maximum height, and then find the maximum height.

In 5.3 sec the arrow will reach maximum height of 486 ft.

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A stone is thrown vertically upward from a platform that is 20 feet high at a rate of 160 ft/sec. Use the quadratic equation h = −16 t 2 + 160 t + 20 to find how long it will take the stone to reach its maximum height, and then find the maximum height.

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A computer store owner estimates that by charging x dollars each for a certain computer, he can sell 40 x computers each week. The quadratic equation R = x 2 + 40 x is used to find the revenue, R , received when the selling price of a computer is x . Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

20 computers will give the maximum of $400 in receipts.

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A retailer who sells backpacks estimates that, by selling them for x dollars each, he will be able to sell 100 x backpacks a month. The quadratic equation R = x 2 + 100 x is used to find the R received when the selling price of a backpack is x . Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

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A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using 240 feet of fencing. The quadratic equation A = x ( 240 2 x ) gives the area of the corral, A , for the length, x , of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the corral.

The length of the side along the river of the corral is 120 feet and the maximum area is 7,200 sq ft.

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A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A = x ( 100 2 x ) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.

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Everyday math

In the previous set of exercises, you worked with the quadratic equation R = x 2 + 40 x that modeled the revenue received from selling computers at a price of x dollars. You found the selling price that would give the maximum revenue and calculated the maximum revenue. Now you will look at more characteristics of this model.
Graph the equation R = x 2 + 40 x . Find the values of the x -intercepts.


  1. This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 60. The y-axis of the plane runs from -50 to 500. The parabola has a vertex at (20, 400) and also goes through the points (0, 0) and (40, 0).
  2. ( 0 , 0 ) , ( 40 , 0 )
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Questions & Answers

Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Jose
interesting
saw
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
Eric here. I'm a parent. 53 years old. I have never taken algebra. I want to learn.
Eric
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
they are 92 candies in the bag
POP
rectangular field solutions
Navin Reply
What is this?
Donna
t
muqtaar
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from $1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
how
muqtaar
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Practice Key Terms 6

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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