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Intermediate Algebra
Unit 6: Quadratic Equations
1
0 Intermediate Algebra ^ > c f ^ 5 , c p y V ^ ' ^ ^ v i n ^ o . unii6: Quadratic Equations
Q u a d r a t i c g r a F U N
Find tl ie vertex of the graph of the function and write its coordinates in the outlined cells of V * the table. Then find points on each side of the vertex. Plot the points and draw the graph.
0 y = - 4 x +1
y
-\ 1
- 2 -
1 (/
0 y
X y
7
- 3
-If
' 1 - 1 o f 7
@ y = - x ^ + 2x + 5 X y
- J
O rr z *r 3
X y
- ( 0
O jSBSSSS 1^
- I
3
2 + 6x — 1
\ f \
1 /
1 1 O Suppose you have 20 meters of fence to go aroimd a rectangular garden. The width and len^h of the garden are represented in the figure below, where w = width. 10 -w w The area of the garden is giyen by the formula: U = lOw - w^. :;^omplete the table and graph to show how area depends on width.
w (m) A (m2) 1 2 Kf, 3 4 5 ^> 6 7 2-1 8 9 L
0 y = x^ + 6x X y X y
-u o -u o J 1 1
1-3
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O o
y = i x 2 - 3 x + 2
X
O
y
1 - . 5 - 1 1
- 2 . 1
—u
-2-
0 2 3 4 5 6 7 6 0 10 Width (m)
2
-)C Ay, ^ c?f S j.-vin^J^ •. v •= |^ L'n/f 6: Quadratic Equations
Graphing Quadratic Equations: (^^^' '^^^ e . } ^ ^ x ^ . Hx . 3 ^
Graph each quadratic equation, determine the direction of the opening, state the coordinates of ^ the vertex point, and state the equation of the axis of symmetry:
1. y = x^ + 12X + 32
i
1 A X
K f
f
\ 1 \
I 1
1
\ / f *
s
f.
/
(
J 1
2. y = 2x^ - 4x
A
y •
V z X /
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Graph each quadratic equation, determine the direction of the opening, state the coordinates of the vertex point, and state the equation of the axis of symmetry:
3. y = x2-1
Quadratic Equations: i^^a~y.^-t hy: ^ c
Examples: "f^v g^eocA^ ^ i ^ u ^ v ^ - ^ c -e-^u^^bia>v. ^ ^vvA : na^-^/fvuV
u - 1^ ' ^
Examples:
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6
Properties of Quadratic Equations:
Unit 6: Quadratic Equations
SHOW ALL ALGEBRA AND RELEVANT WORK TO COIVIPLETE THE FOLLOWING:
For each quadratic equation, determine whether each function has a m a x i m u r p ^ r a minimum value. Then aloebraicallv find the maximum or minimum value, the y-inteivept, the coordinates of the vertex, and the equation of the axis of symmetry.—^ ^ ~,
1. y = 6x^ 2. y = -8x^
- - o
X - o
2.
V o
3. y = x^ + 2x 4. y = -x^ + 4x - 1
7
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SHOW ALL ALGEBRA AND RELEVANT WORK TO COMPLETE THE FOLLOWING:
For each quadratic equation, determine whether each function has a maximum or a minimum value. Then alqebraicallv find the maximum or minimum value, the y-intercept, the coordinates of the vertex, and the equation of the axis of symmetry.
5. y = x^ + 2x - 3 6. y = -2x^ + 4x - 3
X - I
7. y = 3x^ + 12x + 3
r-
3
8. y = 2x^ + 4x + 1
y
8
Quadratic Equations:
Unit 6: Quadratic Equations
y.c . -2. , V ^ 3
M c^^r^ ^ ^ ^ ^ ^ ^ - t y V ^
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Writing Quadratic Equations:
T h e fol lowing graph grids have x-intercepts plotted. For each question: • state the roots • state the factors • write the quadratic equation • • complete the corresponding graph T
Unit 6: Quadratic Equations
f /
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a /
- ^ • • o c h ^ : x - Z - - o ^ x - ( o — o
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f
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rT>o-V3 I x - X ^
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- f . ^ + ^ - . V ' S ^ c ,
r o o H ' . y ^ - ' ? ^ y.--\
c o
• / ? t c - U ^ ' . V ' ^ - O
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Introduction to Projectile Motion:
Unit 6: Quadratic Equations
d
r
-6
= o
1 2
Projectiie Motion:
(1) The height in feet of a rocl<et launched straight up in the air can be modeled by the function f(x) = -16x^ + 96x, as seen in the accompanying diagram, where x represents the time in seconds.
(a) State the rocket's maximum height: l^"^ ^ejiJz.
Unit 6: Quadratic Equations
(b) State the t ime it takes to reach the max imum height: 3 s<-^
(c) State how long the rocket was in the air: (p , s € ^
(2) The height in feet of a dolphin as it jumps out of the water at an aquarium show can be modeled by the function f(x) = -16x^ + 32x, where x is the t ime in seconds after it exits the water. Use this function to complete the graph on the given set of axes.
Dolphin Jump
6, f t (a) State the dolphin's maximum height:
(b) State the t ime it takes to reach the max imum height: \
(c) State how long the dolphin was in the air: Time (Seconds)
(3) The height in feet of a soccer ball that is kicked can be modeled by the function f(x) = -8x^ + 24x, where x is the time in seconds after it is kicked. Use this function to complete the graph on the given set of axes.
(a) State the soccer ball's max imum height: } ^
(b) State the time it takes to reach the m a x i m u m height: i •^'jC-c
(c) State how long the soccer ball was in the air: ^
SocewKIck
(4) The height in feet of an acrobat who j u m p s from a trampoline 10 feet in the air to a large mat on the ground can be modeled by the function f(x) = -8x^ + 16x + 10, where x is the t ime in seconds after the acrobat jumps . Use this function to complete the graph on the given set of axes.
(a) State the acrobat's max imum height:
(b) State the t ime it takes to reach the m a x i m u m height: [
(c) State how long the acrobat was in the air:
Acrobatic Jump
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(ft) 0 g o 1 2 3 4 S o 5 o 2 3 4 5
Time (e)
0 - b -
t - zkl.
© WU")^ ^(^'iC^') ^
• t - 5 s e c
ifiM. vV* .Vi» i - o^\A)a^A ^\xyci^ - » \^
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( Write the letter of the correct answer in each box containing the exercise number.}
in Exercises 1-4, use the graphs at the right to find the following: S ^ The equation of the axis
of symmetry for Graph A.
> © The coordinates of the vertex for Graph A.
^ The equation of the axis of symmetry for Graph B.
^ ^ X - 3 (j-j 9 The coordinates of the
vertex for Graph B . r i i i i x ^ ^ i i . i i. • j , i i i i i i i ^ ,
i^^ni M M I I I \ j I I • I I M M ] ^ ® ( 3 . 4 ) In Exercises 5-12, find the equotlon of the axis of symmetry and the coordinates of the vertex point of the function. (Only the vertex point is given in the answer column.)
C © y = x 2 - 4 A : + 1 ^ ® S(x) = + 6x + 5
y A *f
\ 1 \
X
A
o >
/ r I 4 I
o /
/
+ 8 x - 3 (2. © y = 2 x ^ - 9 VV © y = 2x^
^ © /l-^) = ~ 3 x ^ + 6x + 4 i n ® y = - 2 x ^ + l O x - 7
f © y = ^ x 2 + 4 x + 1 <s| ® / (x ) = - ^ x 2 + 3 x - 2
In Exercises 13-16, use the vertical motion formula given in the box below.
Ans-wers 5-12 U ® ( - 4 , - 7 ) © - ^ r = 4 ) r
•% © ( -2 . -11) © (2. - 3 ) 5
© ( 3 . 6 . 8 ) © ( 1 . 7 ) ^
( ^ © ( - 3 . - 4 ) © ( 2 . 9)
© - ( - 3 r - 7 ) - © ( 3 . 2 . 5 ) >7-
< 0 ® (2.5.5.5) © ( 0 . - 9 ) - 7
© ( - 1 . - 3 ) © ( 3 , 2 )
If an object is thrown upward, its approximate height h (In feet) is given by the formula: h = -16t^ + ut + c. where t is the time in motion (in seconds), u is the Initial upward velocity (in feet per second), and c is the initial height (in feet).
Zen throws a ball upward with an initial upward velocity of 64 ^/^^Z^hn^esi 15-16 The ball is 5 ft above the ground when it leaves Zen's ha^d. ^ ^ ^ - ^ j ^ ^ ^ ^ ^ j , ^ 2.5 sec
1 3 © 2 sec
- © 2.0 ace
V<^© In how many seconds will the ball reach its maximum h e i ^ t ? ^
L © What is the ball's maximum height?
A fireworks rocket is shot upward with an initial velocity of 80 ft/s. Q g g The rocket is 3 ft above the groimd when it is fired, y^^.,^^^^ ^ ^ ' u A '
f\n how many seconds will the rocket reach its maximimi height? ' ^® ^
^ © What is the rocket's maximum height? \oZJ{X ^'^^^ ) G? © 103 ft
11 6 14 14 2 12 4 d 2 16 12 15 10 6 1 15 5 13 9 15 7 3
? U I 6, 1 S r4 / \ 6 6 4 C K CO fV S
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Projectile Motion:
((1)}An object is fired straiglnt up from the top of a 200-foot tower at a velocity of 80 feet per second. T h e height, h{t), of the object t seconds after firing is given by the equation h(t) = -16t^ + 80t + 200. (Answer tlie following questions, rounding each to the nearest hundredth.) (a) Find the t ime when the object reaches its maximum height. Then find the maximum height. (b) After how many seconds will it take the object to reach the ground?
- b
a , 3 - s e c
fcn>w^6. ocL (p. sec
L . : - —
W U2mhe height, h{t) in feet of an object t seconds after it is propelled straight up from the ground ^ ^ w i t h an initial velocity of 60 feet per second is modeled by the equation h(t) = -16t^ + 60t.
' O '^1 (Answer the following questions, rounding each to the nearest hundredth.) ' ' (a) At what t ime(s) will the object be at a height of 56 feet? H (b) How would the equation change if the object was propelled from a platform that was 10 Q feet in the air? Write the new equation.
' (c) W h e n would the object reach its max imum height, and what would the m a x i m u m height be, assuming it was shot from this platform?
(d) W h e n would the object shot from the platform reach the ground?
Y - 2 ^ t c ^ '
^pnxAA<^ ^ - b 'Z.^\o
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1 6
(3) A juggler throws a ball into the air from a height of 5 ft with an initial velocity of 16ft/s. The motion of the ball can be modeled by the equation h(t) = -16t^ + 16t + 5.
' 2 -1 (Answer the following questions, rounding each to the nearest hundredth.) y ' (a) How long does the juggler have to catch the ball before it hits the ground? "
^ (b) Will the ball ever reach a height of 10 feet? Explain your answer in a complete sentence X o.toA and provide relevant algebraic work to support your reasoning.
(4) A baseball player hits a ball toward the outfield. The height h of the ball in feet is modeled by h(t) =-16t^ + 22t + 3, where t is the time in seconds. If no one catches the ball, how long will
I o^Z A jt stay in the air? (Round your answer to the nearest hundredth.)
' 'J rotA ^ ode- l . ^ O S€o
(5) The quadratic function that approximates the height of a javelin thrown is h(t) = -0.08t^ + 4.48, I where t is the t ime in seconds after it is thrown and h is the javelin's height in feet.
n (Answer the following questions, rounding each to the nearest hundredth.) 1^1^^^ (a) What is the m a x i m u m height the javelin reaches and how long does that take?
V i (b) How long will it take for the javelin to hit the ground?
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Review Questions:
(1) T h e height h (in feet) of a certain rocl<et t seconds after it is shot into the air is modeled by the function h(t) = -16t^ + 88t + 200 .
a) Find the t ime when the rocket reaches the maximum height a/gefera/ca/Zy. b) Find the max imum height of the rocket algebraically.
(2) T h e height h (in feet) of a certain rocket t seconds after it is shot into the air is modeled by
the function h(t) = - 1 6 t ' + 1 2 0 t + 3 0 . Use your calculator to answer the following questions and round your answers to the nearest tenth, if applicable.
a) W h e n will the rocket reach a height of 180 feet? ^ b) How many seconds after the rocket is fired will it hit the ground?
'^^ * c) Will the rocket ever reach a height of 275 feet? Explain your reasoning.
(3) Determine whether each function has a maximum or minimum value. Then find the max or min value of each function algebraically.
a) y = x ^ + 6 x + 9 b) y = 3 x ^ - 1 2 x - 2 4 c) y = - x ^ + 4 x
18
(4) Graph the following quadratic functions and identify the • direction of opening • equation of the axis of symmetry \ • coordinates of the vertex • y-intercept
Be able to find algebraically
a) y = x^ - 2x + 5 b) y = 3x^ + 9x + 6
r I
/ s —
•1 V
-1 r*—
2-i
c) y = - 2 x ^ + 1 2 x - 9
y
X •
fwOS^* X =
X = "-1; ( p ; - / . S "
\ > 1
i \ 1 \ 1 \
1 1 X w <•- i
7 f 1
\ \ 1 I !
i
(5) Solve each equation by graphing in the calculator. State the roots, rounding to tiie nearest fiundredtii if applicable.
a) x ' - 36 = 0
X - 6.
^d) - x ' - 4 0 x - 80 = 0
— » X - - 2 . HI^SZp
^ ^ = - 3 7 . - S ^ s ^ f
X ^ - 2 . - 1 1
X ^ - 3 - 7 , ^ < ?
b) - x ' - 3 x + 10 = 0
x ^ 2 ,
e) - 3 x ' - 6 x - 2 = 0
— ^ x = ' 1 . i r y i l S '
c) 2 x ' + X - 3 = 0
X ^ »
f) 1 0 x ^ + 3 x - 1 = 0
X - ' 0 . 4 - 2 . 1 6 ^ ' ! 7
>< ' - 0 . ^ 2 _ 1 9
x = - o ^
x ^ o . ^
2 0