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^

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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

- ) ?

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 /

1

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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

7

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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9

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 /

\ V J f

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

1 3

(ft) 0 g o 1 2 3 4 S o 5 o 2 3 4 5

Time (e)

0 - b -

t - zkl.

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• t - 5 s e c

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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.)

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(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?

1 7

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