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The quadratic equation h = −16 t 2 + 128 t + 32 is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height? Round answers to the nearest tenth.

It will take 4 seconds to reach the maximum height of 288 feet.

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A toy rocket shot upward from the ground at a rate of 208 ft/sec has the quadratic equation of h = −16 t 2 + 208 t . When will the rocket reach its maximum height? What will be the maximum height? Round answers to the nearest tenth.

It will take 6.5 seconds to reach the maximum height of 676 feet.

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Access these online resources for additional instruction and practice graphing quadratic equations:

Key concepts

  • The graph of every quadratic equation is a parabola.
  • Parabola Orientation For the quadratic equation y = a x 2 + b x + c , if
    • a > 0 , the parabola opens upward.
    • a < 0 , the parabola opens downward.
  • Axis of Symmetry and Vertex of a Parabola For a parabola with equation y = a x 2 + b x + c :
    • The axis of symmetry of a parabola is the line x = b 2 a .
    • The vertex is on the axis of symmetry, so its x -coordinate is b 2 a .
    • To find the y -coordinate of the vertex we substitute x = b 2 a into the quadratic equation.
  • Find the Intercepts of a Parabola To find the intercepts of a parabola with equation y = a x 2 + b x + c :
    y -intercept x -intercepts Let x = 0 and solve for y . Let y = 0 and solve for x .
  • To Graph a Quadratic Equation in Two Variables
    1. Write the quadratic equation with y on one side.
    2. Determine whether the parabola opens upward or downward.
    3. Find the axis of symmetry.
    4. Find the vertex.
    5. Find the y -intercept. Find the point symmetric to the y -intercept across the axis of symmetry.
    6. Find the x -intercepts.
    7. Graph the parabola.
  • Minimum or Maximum Values of a Quadratic Equation
    • The y - coordinate of the vertex of the graph of a quadratic equation is the
    • minimum value of the quadratic equation if the parabola opens upward.
    • maximum value of the quadratic equation if the parabola opens downward.

Practice makes perfect

Recognize the Graph of a Quadratic Equation in Two Variables

In the following exercises, graph:

In the following exercises, determine if the parabola opens up or down.

y = −2 x 2 6 x 7

down

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y = −9 x 2 24 x 16

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Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find the axis of symmetry and the vertex.

y = x 2 + 8 x 1

x = −4 ( −4 , −17 )

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y = x 2 + 2 x + 5

x = 1 ( 1 , 6 )

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y = −2 x 2 8 x 3

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Find the Intercepts of a Parabola

In the following exercises, find the x - and y -intercepts.

y = x 2 + 7 x + 6

y : ( 0 , 6 ) ; x : ( −1 , 0 ) , ( −6 , 0 )

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y = x 2 + 8 x 19

y : ( 0 , 19 ) ; x : none

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y = 4 x 2 20 x + 25

y : ( 0 , 25 ) ; x : ( 5 2 , 0 )

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y = x 2 14 x 49

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Graph Quadratic Equations in Two Variables

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.

y = x 2 + 6 x + 5

y : ( 0 , 5 ) ; x : ( −1 , 0 ) , ( −5 , 0 ) ;
axis: x = −3 ; vertex : ( −3 , −4 )
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, -4) and the intercepts (-5, 0), (-1, 0) and (0, 5). 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 = x 2 + 4 x + 3

y : ( 0 , 3 ) ; x : ( −1 , 0 ) , ( −3 , 0 ) ;
axis: x = −2 ; vertex: ( −2 , −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 (-2, -1) and the intercepts (-1, 0), (-3, 0) and (0, 3). 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 = 9 x 2 + 12 x + 4

y : ( 0 , 4 ) x : ( 2 3 , 0 ) ;
axis: x = 2 3 ; vertex: ( 2 3 , 0 )
This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 5. The parabola has points plotted at the vertex (-2 thirds, 0) and the intercept (0, 4). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2 thirds.

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y = x 2 + 8 x 16

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y = x 2 + 2 x 7

y : ( 0 , −7 ) ; x : none ;
axis: x = 1 ; vertex : ( 1 , −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 -15 to 5. The parabola has points plotted at the vertex (1, -6) and the intercept (0, -7). 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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Questions & Answers

Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon?
Mohamed Reply
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
muhammad Reply
4x-7y=8 2x-7y=1 what is the answer?
Ramil Reply
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
true
bismark
factoriz e
usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
Peggie
Ok cool answer peggie
Frank
thanks
Ramil
copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x
Sandra Reply
What is c+4=8
Penny Reply
2
Letha
4
Lolita
4
Rich
4
thinking
C+4=8 -4 -4 C =4
thinking
I need to study
Letha
4+4=8
William
During two years in college, a student earned $9,500. The second year, she earned $500 more than twice the amount she earned the first year.
Nicole Reply
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
Josh Reply
Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.
Ida Reply
hey
Juan
Sup
patrick
The sum of two numbers is 155. The difference is 23. Find the numbers
Michelle Reply
The sum of two numbers is 155. Their difference is 23. Find the numbers
Michelle
The difference between 89 and 66 is 23
Ciid
Joy is preparing 20 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution?
Amber Reply
hello
bismark
I need a math tutor BAD
Stacie
Me too
Letha
me too
Xavier
ok
Bishal
teet
Bishal
Sue and Deb work together writing a book that takes them 90 days. If Sue worked alone, it would take her 120 days. How long would it take Deb to write the book alone?
Hailey Reply
Tell Deb to write the book alone and report back on how long it took her. Deb could get bored and stop, she could get sick next week and prolong the time, she could die next month. This question is impossible to answer.
vV
right*
tyler
is there a thumbs up mathematical symbol in algebra? 👍²= vV 90days x 120days =10800 120days - 90days = 30 10800days/30days= 360days? (ad) derall=1440 mins (90x ad )+(120 x ad )/(120 x ad ))-((90d x ad)= you go glen coco! I'll see my self out
Nick
Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
genevieve Reply
6.25
Ciid
6.25
Big
?
Twila
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Katherine Reply
complexed. could you please help me figure out?
Ciid
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
help
Twila
draw a diagram first
John
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
Marisol Reply
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid
I'm mathematics teacher from highly recognized university.
Mzo Reply
here a question professor How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths as many sailors as soldiers? can you write out the college you went to with the name of the school you teach at and let me know the answer I've got it to be honest with you
tyler
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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