chapter 4: quadratic functions and factoring 4.1 graphing...

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Chp 4.notebook 1 October 23, 2012 4.1 Graphing Quadratic Functions in Stand a quadratic function in standard form is written y = ax 2 + bx + c, where a 0 A quadratic Function creates a U-shaped graph called a parabola The vertex is the lowest/highest point of the parabola Axis of symmetry is the line that divides the graph into two symmetric parts The roots of the function (answers) are the two points that cross the x-axis (x-intercepts), when y = 0 VOCAB: &KDSWHU 4XDGUDWLF )XQFWLRQV DQG )DFWRULQJ

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Page 1: Chapter 4: Quadratic Functions and Factoring 4.1 Graphing ...mrsmasonmath.weebly.com/uploads/5/9/6/3/5963714/... · 4.1 Graphing Quadratic Functions in Standard form ... 4.2 Graphing

Chp 4.notebook

1

October 23, 2012

4.1 Graphing Quadratic Functions in Standard forma quadratic function in standard form is written y = ax2 + bx + c, where a ≠ 0A quadratic Function creates a U-shaped graph called a parabola The vertex is the lowest/highest point of the parabola

Axis of symmetry is the line that divides the graph into two symmetric parts

The roots of the function (answers) are the two points that cross the x-axis (x-intercepts), when y = 0

VOCAB:

Chapter 4: Quadratic Functions and Factoring

Page 2: Chapter 4: Quadratic Functions and Factoring 4.1 Graphing ...mrsmasonmath.weebly.com/uploads/5/9/6/3/5963714/... · 4.1 Graphing Quadratic Functions in Standard form ... 4.2 Graphing

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Practice Graphing Graph of a quadratic functionif a is positive, the parabola opens up

if a is negative, the parabola opens down

The vertex has an x­coordinate of

The axis of symmetry is written x = 

b2a­

b2a­

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Practice Graphing Graph of a quadratic function

STEP 1:  find the x­coordinate of the vertex by using

STEP 2: Make a table of values

STEP 3: Plot the points and create the parabola

b2a­vertex =

Page 4: Chapter 4: Quadratic Functions and Factoring 4.1 Graphing ...mrsmasonmath.weebly.com/uploads/5/9/6/3/5963714/... · 4.1 Graphing Quadratic Functions in Standard form ... 4.2 Graphing

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October 23, 2012

Graph: y = x2 ­2x ­3

Page 5: Chapter 4: Quadratic Functions and Factoring 4.1 Graphing ...mrsmasonmath.weebly.com/uploads/5/9/6/3/5963714/... · 4.1 Graphing Quadratic Functions in Standard form ... 4.2 Graphing

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October 23, 2012

Graph: y = ­2x2 ­x +2

Page 6: Chapter 4: Quadratic Functions and Factoring 4.1 Graphing ...mrsmasonmath.weebly.com/uploads/5/9/6/3/5963714/... · 4.1 Graphing Quadratic Functions in Standard form ... 4.2 Graphing

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October 23, 2012

Graph: y = 2x2 ­7x­8

Page 7: Chapter 4: Quadratic Functions and Factoring 4.1 Graphing ...mrsmasonmath.weebly.com/uploads/5/9/6/3/5963714/... · 4.1 Graphing Quadratic Functions in Standard form ... 4.2 Graphing

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4.2 Graphing Quadratic Functions in Vertex or Intercept form

vertex form: standard form: 

The graph of y = a(x­h)2 + k is the parabola y = ax2 translated horizontally h units and vertically k units

Characteristics of the graph:• the vertex is (h,k)• The axis of symmetry is x = h• The graph opens up if a>0 and down if a<0

y

x

y = ax2

(0,0)

y = a(x­h)2+k

(h,k)

hk

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Step 1: Identify h, a and k. Use a to determine if the parabola opens up or down

Graph a quadratic function in vertex form: y = a(x-h) 2+k

Step 2: Plot the vertex (h,k) and draw the axis of symmetry

Step 3: Evaluate the function for two values of x.

Step 4: Draw parabola through plotted points

STEP 1

STEP 2

STEP 3

STEP 4

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Example 2

Step 1: Identify h, a and k. Use a to determine if the parabola opens up or down

Graph a quadratic function in vertex form: y = a(x-h) 2+k

Step 2: Plot the vertex (h,k) and draw the axis of symmetry

Step 3: Evaluate the function for two values of x.

Step 4: Draw parabola through plotted points

STEP 1

STEP 2

STEP 3

STEP 4

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Characteristics of the graph y=a(x­p)(x­q):• The x­intercepts are p and q• The axis of symmetry is halfway between (p,0) and (q,0) it has equation• The graph opens up if a>0 and down if a<0

y

x

y = ax2

(p,0) (q,0)

Intercept form: y=a(x­p)(x­q)

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Example 1

Step 1: Identify the x-intercepts

Graph a quadratic function in intercept form: y = a(x-p)(x-q)

Step 2: Find the coordinates of the vertex

Step 3: Draw parabolua through the vertex and x-intercepts

STEP 1

STEP 2

STEP 3

STEP 4

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Example 1

Step 1: Identify the x-intercepts

Graph a quadratic function in intercept form: y = a(x-p)(x-q)

Step 2: Find the coordinates of the vertex

Step 3: Draw parabolua through the vertex and x-intercepts

STEP 1

STEP 2

STEP 3

STEP 4

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Example: Change intercept form to standard form

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Example: Change intercept form to standard form

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Example: Change vertex form to standard form

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Example: Change vertex form to standard form

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4.3 Solve x2 + bx + c = 0 by factoring

ac

b

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8

6

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x2 ­ 5x + 6Factor

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x2 + 7x + 12

Factor

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October 23, 2012

x2 ­ 1x ­ 6

Factor

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Solve

x2 + 3x ­ 4 = 0

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October 23, 2012

Solvex2 + 5x ­14 = 0

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October 23, 2012

Solvex2 + 12x = ­27

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Example 1:

Special Factoring Patterns

Difference of Squares (DOS): 

Perfect Square Trinomial:

a2 ­ b2 = (a­b)(a+b)

a2 + 2ab + b2 = (a+b)2

a2 ­ 2ab + b2 = (a­b)2

4x2 ­ 9

Example 2:

Example 3:

x2 + 6x + 9

x2 ­ 8x + 16

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You try:

1.  x2 + 16x + 64

2. 9x2 ­ 49

3. x2 ­ 18x + 81

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4.4 Solve ax2 + bx + c = 0 by factoring

2x2 ­ 3x ­ 20Factor

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3x2 + 11x ­ 4Factor

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October 23, 2012

7x2 ­ 31x + 12Factor

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6x2 ­ 11x + 3Factor

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October 23, 2012

5x2 ­ 8x + 3 = 0Solve

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October 23, 2012

3x2 + 14x + 15 = 0Solve

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October 23, 2012

7x2 + 11x ­ 30 = 0Solve

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October 23, 2012

2x2 + 3x ­ 5 = 0Solve

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Factor out monomials first

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4.5 Solve Quadratic Equations by Finding Square RootsSQUARE ROOT OF A NUMBERif b2 = a, then b is a square root of a

example: if 32 = 9, then 3 is a square root of 9.

VOCAB: positive square root: √ negative square root: -√ positive/negative square root: ±√ radicand: the number or expression inside a radical symbol.

MATH IS √ DUDE

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Simplifying radicalsProperties of Radicals

Product property              √ab = √a√b

Quotient property √ = a

b√a

√b

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An expression with radicals is insimplest form if the following are true 1.) No perfect square factors other 

than 1 are in the radicand

2.) No fractions are in the radicand

3.) No radicals appear in the denominator of a fraction

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Examples

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Examples

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Examples:rationalize denominators of fractions

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Examples: solve a quadratic

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Examples: solve a quadratic

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4.6 Perform Operations with Complex Numbers

Not all quadratic equations have real-number solutions. For example: x2 = -1 has no real-number solutions because the square of any real number x is never a negative number

To overcome this problem mathematicians created an expanded system of numbers using the imaginary unit i. Defined as not that i2 = -1. The imaginary unit can be used to write the square root of any negative number.

The square root of a negative numberProperty Example

1. If r is a positive real number, then

2. By Property (1), it follows that

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example:

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example:

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Complex Numbers:A complex number written in standard form is a number a+bi where a and b are real numbers.  The number a is the real part of the complex number and the bi is imaginary part.

If b≠0 then a + bi is an imaginary number.  If a = 0 and b ≠ 0, then a+bi is a pure imaginary number.

RealNumbers

ImaginaryNumbers

Pure ImaginaryNumbers

Complex Numbers: a + bi

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The sum and difference of complex numbersTo add (subtract) two complex numbers, add (subtract) their real parts and their imaginary parts separately.

1. Sum of complex numbers:

2. Difference of complex numbers:

examples:a:

b:c:

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examples:1:

2:

3:

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Multiply Complex Numbers:

a: 

b:

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Multiply Complex Numbers:

1: 

2:

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Complex Conjugates:Two complex numbers of the form a+bi and a­bi are called complex conjugates.  The product of conjugates is always a real number.

Divide Complex Numbers: use complex conjugates

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Divide Complex Numbers: use complex conjugates

1:

2:

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Plot Complex NumbersPlot the complex numbers in the same complex plane.

a.

b.

c.

d.

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Find absolute values of complex numbersAbsolute Vale of a Complex Number:

a.

b.

The absolute value of a complex number z = a + bi, denoted

is a nonnegative real number defined as

This is the distance between z and the origin in the complex plane

examples:

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4.7 Complete the Squarex2 + bx +( )2 = (x + )2

b2

b2

To complete the square for the expression x2 + bx, add

Diagram:

x

x

b

xx2 bxx

x

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example 1: Solve a quadratic by finding square roots

step 1: Write left side as binomial squared

step 2: Take square root of each side.

step 3: Solve for x

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example 2: Make a perfect square trinomial

Step 1: Find half of the coefficient of x

Step 2: square the result of step 1

Step 3: Replace c with the result from step 2

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example 3: Solve x 2 + bx + c (a = 1)

step 1: write left side in the form x 2+bx

step 2: Complete the square and add it to both sides

step 3: Write left side as a binomial squared

step 4: Take square root of each side

step 5: Solve for x (simplify if necessary)

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example 4: Solve ax 2 + bx + c (a ≠ 1)

step 1: divide everything by a

step 2: write left side in the form x2 + bx

step 3: complete the square and add it to both sides

step 4: write the left side as a binomial squared

step 5: take square root of both sides

step 6: solve for x, simplify if necessary

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Extra ExamplesSolve the equation by completing the square

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Write a quadratic function in vertex form

step 1: complete the square and add it to both sides

step 2: write beginning part as binomial squared

step 3: solve for y

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Write a quadratic function in vertex form

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4.8 Use the quadratic formula and the discriminant

Quadratic Formula: The solutions of thequadratic equation ax2+bx+c=0 are

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Proving the Quadratic Formula!

ax2+bx+c = 0 where a≠0

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Example 1:x2 ­ 8x + 15 = 0

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Example 2:

2x2 + 6x + 2 = ­1

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Example 3: One solution

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Example 4:One solution

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Example 5: Imaginary Solution

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Example 6: Imaginary Solution

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Word ProblemsObjects Dropped:

Objects Launched/Thrown:initial velocity

initial height

A juggler tosses a ball into the air.  The ball leaves the juggler's hand 4 feet above the ground and has an initial vertical velocity of 40 feet per second.  The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air?

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Word ProblemsObjects Dropped:

Objects Launched/Thrown:initial velocity

initial height

A basketball player passes the ball to a teammate.  The ball leaves the player's hand 5 feet above the ground and has an initial vertical velocity of 55 feet per second.  The teammate catches the ball when it returns to a height of 5 feet.  How long is the ball in the air?

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Word ProblemsThe equation h = ­16t2 + 20t + 6 gives the height, h, in feet of a basketball as a function of t, in seconds

a) What is the maximum height the ball reaches?

b) At what time does the ball hit the ground?

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Applications of the discriminantIn the quadratic formula the expression inside the radical is the discriminant

Discriminantb2-4ac

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Number of solutions of a quadraticConsider the quadratic equation ax2 + bx + c = 0:

• If b2­4ac is positive, then the equation has two solutions• if b2­4ac is zero, then the equation has one solution• if b2­4ac is negative, then the equation has no real solutions

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Example:a. x2 ­ 3x ­ 4

b. ­x2 + 2x ­ 1

c. 2x2­2x + 3

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Example:a. ­3x2 + 5x ­ 1

b. ­x2 + 10x ­ 25

c. x2 ­2x + 4

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Match the graph with the discriminant:

a. b2­4ac = 2

b. b2­4ac = 0

c. b2­4ac = ­3

c

A

b

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4.10 Write Quadratic Functions and Models

Example 1: Write a quadratic function in vertex form y = a(x-h)2+K

Vertex: (1, ­2)  Point: (3, 2)

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Example 2: Write a quadratic function in vertex form y = a(x-h)2+K

Vertex: (4, ­5)  Point: (2, ­1)

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Example 3: Write a quadratic function in intercept form y = a(x-p)(x-q)

x­intercepts: ­2 and 5  Point: (6, 2)

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Example 4: Write a quadratic function in intercept form y = a(x-p)(x-q)

x­intercepts: ­1 and 4  Point: (3, 2)

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Example 5: Write a quadratic function in Standard form y = ax 2 + bx + c

points:  (­1, ­3), (0, ­4), (2, 6)

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Example 6: Write a quadratic function in Standard form y = ax 2 + bx + c

points:  (­1, 5), (0, ­1), (2, 11)