College Algebra K/DC Monday, 17 February 2014 • OBJECTIVE

TSW (1) graph quadratic functions and (2) determine (a) domain & range, (b) the vertex, and (c) the axis of symmetry.

• TEST: Sec. 2.7 – 2.8 is not graded.

• Sec. 3.1: pp. 312-313 (12-25 all) – Due tomorrow, Tuesday, 18 February 2014.

1

3.1

Quadratic Functions and Models

Quadratic Functions ▪ Square ▪ Graphing Techniques The Vertex Formula ▪ Completing the 3-2

Graphing Quadratic Functions

Graph

f

(

x

) =

x

2 + 4

x

domain and range.

– 4 by plotting points. Give the Domain: Range: 3-3

Graphing a Parabola by Completing the Square

Graph

f

(

x

) =

x

2 + 2

x

– 5 by completing the square, locating the vertex, and determining the axis of symmetry (AOS).

Complete the square.

Add and subtract 1.

Factor and simplify.

Vertex ( –1, –6) Axis of symmetry:

x

= –1 The axis of symmetry is ALWAYS a line!!! Do NOT give the AOS as only one number – it is ALWAYS

x

= something!

3-4

Graphing Quadratic Functions

Graph

F

(

x

) = –2(

x

+ 3) 2 + 5 and compare to the graph of

g

(

x

) = −2

x

2 . Give the domain and range.

The graph of

F

(

x

) is the graph of

g

(

x

) translated 3 units to the left and 5 units up.

Domain: Range: 3-5

Graphing a Parabola by Completing the Square

Graph

f

(

x

) = 2

x

2 +

x

– 6 by completing the square and locating the vertex.

Factor 2 from the first two terms.

Add and subtract Distributive property Vertex: AOS: Factor and simplify.

3-6

Finding the Axis and the Vertex of a Parabola Using the Vertex Formula For the quadratic function

f

(

x

) =

ax

2 +

bx

+

c

, the

x

-coordinate of the vertex is

x

 

b

This is also the axis of symmetry (AOS) 2

a

The

y

-coordinate is

y



f

   2

b a

  3-7

Finding the Axis and the Vertex of a Parabola Using the Vertex Formula Find the axis and vertex of the parabola

f

(

x

) = –3

x

2 + 12

x

– 8 using the vertex formula.

a

= –3,

b

= 12,

c

= –8 Axis:

x

= 2 Vertex: (2,

f

(2)) Vertex: (2, 4) 3-8

Assignment

• Sec. 3.1: pp. 312-313 (12-25 all) –

Write the problem, graph, give the vertex, axis, domain, and range.

–

Due tomorrow, Tuesday, 18 February 2014.

3-9

CLASS PROBLEMS: Sec. 3.1 (02/17/14)

leave today.



wire basket Due before you

For questions 1-8, find the vertex and AOS. Write the problem and show all work.

For questions 1-4, complete the square.

1) 3)

f

(

x

) =

x

2 + 4

x

– 9

f

(

x

) =

x

2 – 5

x

+ 3 2) 4)

f f

( (

x x

) = – ) = 2

x x

2 2 – 6

x

+ 4

x

+ 3 – 1 For questions 5-8, use the formula

x

5)

f

(

x

) = 3

x

2 +6

x

6) = –

b

/ 2

a

.

f

(

x

) = –2

x

2 + 8

x

+ 9 7)

f

(

x

) = –5

x

2 + 15

x

– 2 8)

f

(

x

) = 4

x

2 + 4

x

+ 4 3-10

Assignment: Sec. 3.1: pp. 312-313 (12-25 all)

tomorrow, Tuesday, 18 February 2014.

12:

Get from the book (matching) Due

13-25: Graph each equation. Give the vertex, axis, domain, and range.

13)

f x x

 2



2 1 4)

f x x

 4



2 1 5)

f x x

 3



2  4 1 6)

f x x

 5



2  4 17)

 

  1



x

2 1 9)

 



x

2  2

x

21 )

 

  1



2  3  3

x

2  1 0

x

 21 1 8 )

 

  3



x

 2



2  1 2 0)

 



x

2  6

x

 5 22 )

 

 2

x

2  4

x

 5 23)

 

  2

x

2  12

x

25)

 

 16  

x

2 24)  6

x

 5

 

  3

x

2  24

x

 46 3-11

JVHS F

ALCONS

College Algebra K/DC Tuesday, 18 February 2014 • OBJECTIVE

TSW apply quadratic models to real world applications.

• ASSIGNMENT DUE – Sec. 3.1: pp. 312-313 (12-25 all)  wire basket • TODAY’S ASSIGNMENT – Sec. 3.1: p. 315 (52-55 all)  the period.

Due tomorrow at the beginning of 13

3.1

Quadratic Functions and Models

Quadratic Models and Curve Fitting 14

Quadratic Modeling

Many phenomena in the real world are best modeled with a quadratic function.

These quadratic functions can be used to answer questions regarding the phenomena.

Sometimes the equations will be given, sometimes you will have to write them yourself.

3-15

Solving a Problem Involving Projectile Motion

A ball is thrown directly upward from an initial height of 75 ft with an initial velocity of 112 ft per sec. Give the function that describes the height of the ball in terms of time

t.

The projectile height function is where

s

(

t

) is the height (in terms of

t

)

v

0 is the initial velocity

s

0 is the initial height –16 is the gravitational constant (in feet) 3-16

Solving a Problem Involving Projectile Motion

After how many seconds does the ball reach its maximum height? What is the maximum height?

s

The graph of this is an upside down parabola, so the maximum height occurs at the vertex.

 

 

2

t

  

b

2

a

   2



112  16



75 The ball reaches its maximum height, 271 ft, after 3.5 seconds.

3-17

Solving a Problem Involving Projectile Motion

For what interval of time is the height of the ball greater than 200 ft?

Solve the quadratic inequality . Use the quadratic formula to find the values of

t

that satisfy .

a

= –16,

b

= 112,

c

= –125 or 3-18

Solving a Problem Involving Projectile Motion

The two numbers divide a number line into three regions, Choose test values to see which interval satisfies the inequality.

 16

t

2  112

t

 125  0

— + —

1 1.39

2 5.61

6 

 

2 

 

2   

 

2   125   29  125  35  125   29 The ball will be greater than 200 ft above ground level between 1.39 and 5.61 seconds after it is thrown. 3-19

Solving a Problem Involving Projectile Motion

After how many seconds will the ball hit the ground?

The ball hits the ground when the height (

s

(

t

) ) is zero.

Use the quadratic formula to find the positive solution of Reject The ball hits the ground after about 7.62 sec.

3-20

Assignment

• Sec. 3.1: p. 315 (52-55 all) – – –

You do not have to write the problem.

Answers should be in complete sentences.

Due on Wednesday/Thursday, 20/21 February 2013 at the beginning of the period.

3-21

Sec. 3.1: p. 315 (52-55)

Due at the beginning of the period on Wednesday/Thursday.

52) If an object is projected upward from ground level with an initial velocity of 32 ft per sec, then its height in feet after

t

seconds is given by

s

(

t

) = –16

t

2 + 32

t

.

Find the number of seconds it will take to reach its maximum height. What is this maximum height?

3-22

Sec. 3.1: p. 315 (52-55)

Due at the beginning of the period on Wednesday/Thursday.

53) A toy rocket is launched straight up from the top of a building 50 ft tall at an initial velocity of 200 ft per sec.

a) Give the function that describes the height of the rocket in terms of

t

.

b) Determine the time at which the rocket reaches its maximum height, and the maximum height in feet.

c) For what time interval will the rocket be more than 300 ft above ground level?

d) After how many seconds will the rocket return to the ground?

3-23

Sec. 3.1: p. 315 (52-55)

Due at the beginning of the period on Wednesday/Thursday.

54) A rocket is projected directly upward from ground level with an initial velocity of 90 ft per sec.

a) Give the function that describes the height of the rock in terms of time

t

.

b) Determine the time at which the rock reaches its maximum height, and the maximum height in feet.

c) For what time interval will the rock be more than 120 ft above ground level?

d) After how many seconds will it return to the ground?

3-24