PreCalculus

Bell Ringer: pg 129 #46

HW Requests: pg 152 #1-10 a. The average of three numbers is 70. When the smallest of the three numbers is replaced by 75, the average is increased by 5. What is the number that was replaced by 75?

In class: inherited domains for inverses pg 129 #48. (pg 140 #35-38, 47-50)Homework: pg 175 # 1-6, 8, 10, 12

Announcements: Binders due 12/2

Date: 11/14/11 Obj: SWBAT algebraically and graphically represent translations, reflections, stretches, and shrinks of functions, understand Inherited domains from inverses and recognize polynomial functions.

Slide 2- 3

Polynomial Function 0 1 2 1

1 2

1 2 1 0

Let be a nonnegative integer and let , , ,..., , be real numbers with

0. The function given by ( ) ...

is a .

The

n n

n n

n n n

n a a a a a

a f x a x a x a x a x a

polynomial function of degree

leading coeffi

n

is .n

acient

Objective: SWBAT find the vertex and axis of symmetry and write the equation of quadratic function in standard and vertex form.

Bell Ringer: pg 139 #13-16;

HW Requests: pg 175 # 1-6, 8, 10, 12

Exit Ticket: (p. 176 # 23-35 odds). Students will sketch problems 23 and 25.

Homework: p. 176 #24-32 evens

Parking Lot: a. The average of three numbers is 70. When the smallest of the three numbers is replaced by 75, the average is increased by 5. What is the number that was replaced by 75?

Announcements: Binders due 12/2

Date: 11/15/11

Slide 1- 5

Stretches and Shrinks

Let be a positive real number. Then the following transformations result in

stretches or shrinks of the graph of ( ):

a stretch by a factor of

c

y f x

xy f

c

Horizontal Stretches or Shrinks

if 1

a shrink by a factor of if 1

a stretch by a factor of if 1 ( )

a shrink by a factor of if 1

c c

c c

c cy c f x

c c

Vertical Stretches or Shrinks

Watch your fractions

Horizontal Stretches or Shrinksa stretch by a factor of if 0 < c < 1

a shrink by a factor of if c > 1

𝑦= 𝑓 (𝑐𝑥)

But c > 0

Slide 2- 7

Vertex Form of a Quadratic Equation

To find hh = To find k; Substitute h into standard quadratic function equationk = f(h) = a(h)2 +b(h) + c

Axis of symmetry x= y intercept = f(0)

Objective: SWBAT find the vertex and axis of symmetry and write the equation of quadratic function in standard and vertex form.

Bell Ringer: Correct problems from “Properties of Parabolas” worksheet;

HW Requests: pg 176 #24-32 evens

Exit Ticket: Complete Properties of Parabolas handout from yesterday

Homework: p. 176 #34, 36, 38 6-6 Practice – Analyzing Graphs of Quadratic Functions Worksheet 1-15 ; Read pg 172 Free Fall

Announcements: Make-up Wednesday Sect. 1.4-1.5 Quiz6th Per. Asia Clark, D. Lee, J. Reed, D. Weathersby, Chamone WilliamsBinders due 12/2

Date: 11/16/11

Slide 2- 9

Example Finding the Vertex and Axis of a Quadratic Function

2

Use the vertex form of a quadratic function to find the vertex and axis

of the graph of ( ) 2 8 11. Rewrite the equation in vertex form. f x x x

2

2

The standard polynomial form of is ( ) 2 8 11.

So 2, 8, and 11, and the coordinates of the vertex are

82 and ( ) (2) 2(2) 8(2) 11 3.

2 4The equation of the axis is 2, the vertex

f f x x x

a b c

bh k f h f

ax

2

is (2,3), and the

vertex form of is ( ) 2( 2) 3.f f x x

1. Rewrite the equation but use the method of completing the square to find vertex form.

2. Write an equation for the parabola with vertex (1, 3), point (0, 5).

Characterizing the Nature of a Quadratic Function

Graph Vertex Form of a Quadratic Equation

h = k = f(h) = a(h)2 +b(h) + c

Vertex (h, k); Axis of symmetry x= ; y intercept = f(0)

Graph the parabola. Use a straight edge.

1. Graph the vertex and the y intercept.

2. Find 4 additional points that are on the parabola. Graph those points. Remember because of symmetry (even) 4 points becomes 8 points. (Check your table.)

3. Using the quadratic formula find the zeros. Plot those points.

4. Graph the axis of symmetry.

5. Find the max. and min. points.

Objective: SWBAT solve problems such as projectile motion (free fall) problems using quadratic functions.

Bell Ringer: Go over Exit TicketHW Requests: 6-6 Practice – Analyzing Graphs of Quadratic Functions Worksheet 1-15

Exit Ticket: Complete yesterday’s Properties of Parabolas handout add #3, 4 (4 min.)

In class worksheetHomework: p. 176 #34, 36, 38, 61-64

Announcements: Make-up Wednesday Sect. 1.4-1.5 Quiz6th Per. Lee, WeathersbyQuiz Tuesday 2.1Binders due 12/2

Date: 11/17/11

Vertex Form of a Quadratic Equation

Completing the Square Move from quadratic function in standard form to vertex form.• Group the x terms in one parentheses.• Factor out terms from the parentheses so that what

is left is a(x2 + bx)• Add inside parentheses and subtract from the rest

of the expression.

Pdemsy 117, 118

Vertical Free-Fall Motion

2

0 0 0

2 2

The and vertical of an object in free fall are given by

1( ) and ( ) ,

2where is time (in seconds), 32 ft/sec 9.8 m/sec is the

,

s v

s t gt v t s v t gt v

t g

height velocity

acceleration

due to gravity0 0

is the of the object, and is its

.

v initial vertical velocity s

initial height

An object is tossed upward with an initial velocity of 15 ft/sec. from a height of 4 ft. What is the object’s maximum height? How long does it take the object to reach its maximum height?

A ball is thrown across a field. Its path can be described by the equationy = -0.002 x2+.2x + 5. where x is the horizontal distance (in feet) and y is the height (in feet).What is the ball’s maximum height? How far had it traveled horizontally to reach itsmaximum height?

Max values at the vertex (h,k)