algebra 1 – chapter 9 9.1 – solving quadratics by square roots objective: 1.know terminology of...

Algebra 1 – Chapter Algebra 1 – Chapter 99

9.1 – Solving Quadratics by Square Roots

Objective:1. Know terminology of square roots2. Find square roots3. Solve quadratic equations with square

roots4. Evaluate using square roots

Vocabulary: radical, radicand

DefinitionDefinitionSquare RootSquare Root

o The square root of a number N is the number(s) that multiplied by itself equals N• Positive numbers have two square

roots• Negative numbers have no square

roots• 0 has exactly one square root

Anatomy of a Square Anatomy of a Square RootRoot

54radical

radicand

9

4-

25

144-

196

16

Find these square roots:Find these square roots:

Solving Equations Solving Equations Using Square RootsUsing Square Roots

x2 = 4

x = 2 or -2

We are allowed to take the square root of both sides of an equation…as long as both sides are POSITIVEPOSITIVE!

Steps for Solving Steps for Solving Equations Using Equations Using

Square RootsSquare RootsNOTENOTE: This only works if there is only x2 terms and

constant terms!1. Move all x2 terms to the left side of the equation

(either add or subtract).2. Move all constant terms to the right side (either add

or subtract).3. Isolate the x2 (by dividing – make sure that x2 is

positive).4. Take square root of both sides.

1. x2 = 25

2. x2 = 81

3. x2 = 7

4. x2 – 144 = 0

5. x2 + 12 = 0

6. x2 – 11 = 0

Solve these equations:Solve these equations:

1. Move all x2 terms to the left side of the equation (either add or subtract).

2. Move all constant terms to the right side (either add or subtract).

3. Isolate the x2 (by dividing – make sure that x2 is positive).

4. Take square root of both sides.

7. 3x2 – 48 = 0

8. 12x2 – 60 = 0

9. 120 – 6x2 = -30

Solve these equations:Solve these equations:

1. Move all x2 terms to the left side of the equation (either add or subtract).

2. Move all constant terms to the right side (either add or subtract).

3. Isolate the x2 (by dividing – make sure that x2 is positive).

4. Take square root of both sides.

The DiscriminantThe Discriminant

b2 4ac• Part of the quadratic formulaquadratic formula

• Tells us how many solutions a quadratic equation has

• This will be really important, but first we need to get good at evaluating it ORDER OF ORDER OF OPERATIONSOPERATIONS

Evaluate for Evaluate for When a = 1, b = -2, c = -3.

b2 4ac

1. Plug in a, b, and c.2. Simplify the radicand.3. Take square root.

Evaluate for Evaluate for 10. a = 16, b = -8, c = 1

11. a = 3, b = -4, c = 3

b2 4ac1. Plug in a, b, and c.2. Simplify the radicand.3. Take square root.

Falling Object ModelFalling Object Model• When an object is dropped, the speed with which

it falls continues to increase. Ignoring air resistance, its height h can be approximated by the falling object model.

h = -16t² + sh= height in feett = time in seconds s = initial height

Falling Object ModelFalling Object Model

h = -16th = -16t² + s² + s• How long will it take a free-fall ride at an

amusement park to drop 121 feet. Assume there is no air resistance.

t = _____ seconds (Why not ±?)

If a bird egg fell out of a nest on the Sears Tower (1450ft), how long until it hits the ground? h = -16t2 + s

Brain BreakBrain Break• Take a few minutes to get up, stretch out, talk to a

neighbor, or try the following rebus puzzles…

A big “if”A big “if” Stay Stay overnightovernight

L 9.1 HomeworkL 9.1 Homework

Tonight’s HomeworkP. 507-508#’s 31-33, 39-41, 58-63, 83

Algebra T3 – Chapter 9Algebra T3 – Chapter 9Daily Warm-UpSolve Each Equation. 2x² - 8 = 0 x² + 25

= 0

Goals:1. Use properties of radicals to simplify radicals.2. Use quadratic equations to model real-life problems.

9.2 Simplifying Radicals9.2 Simplifying Radicals

Product Property:

The square root of a product equals the product of the square root of the factors.

n u m b e rs p o s i ti v e a re b a n d a w h e nbaa b

1 0 044 0 0 6 4 1 64

Quotient Property:

The square root of a quotient equals the quotient of the square root of the numerator and the denominator.

num b ers p ositive a re b a nd a w henb

aba

25

9259

25

100

25

100

Find these square roots:

3 22

2 73

81 05

Simplify these square roots:

48 50 9 6

Steps:1. Find the largest perfect square factor2. Split the radical3. Square root the perfect square

PLEASE NOTE: You can also use a Factor Tree!


54

85

5

9 8 1 5 0-

1 5 0


481

327

3664

Steps:1. Reduce the fraction2. Split the radical3. Square root perfect squares4. Look for perfect square factors and pull them out5. Reduce if possible


25

18

2572

450

420

5032

854


4

188

845 03 1021

20

8

332


Tonight’s HomeworkP. 514#’s 10-13, 18-20, 22-24, 43, 48

Algebra T3 – Chapter 9Algebra T3 – Chapter 9Daily Warm-UpSimplify the Expression.

3 6 3 4196

2

9.1/9.2 Check Up9.1/9.2 Check Up

9.3 Quadratic 9.3 Quadratic FunctionsFunctions

Goal:1. Sketch the graph of a quadratic function.2. Use quadratic models in real life settings.

9.3 Graphing Quadratic 9.3 Graphing Quadratic

FunctionsFunctions

Objective:1.Identify a quadratic equation2.Identify the vertex and axis of symmetry

of a parabola3.Graph a quadratic function

Vocabulary: quadratic equation, standard form, parabola, vertex, axis of symmetry

Quadratic Equation and Quadratic Equation and

FunctionFunction

Quadratic Equation : polynomial equation with a degree of 2. It can be written in standard form as follows:

ax² + bx + c = 0

Quadratic Function: a function that can be written in the standard form

y = ax² + bx + c

1.) y = 4x2 + x – 4

Is each statement a quadratic function? If yes, find a, b, c.

4.) y = 4x2 + 4

3.) y = x – 4

2.) y = -13x2 – x

5.) y = -2x3 + 3x2 – x + 12

Every quadratic function makes a u-shaped graph called a parabola.If the leading coefficient a is positive, the parabola opens up.

If the leading coefficient a is negative, the parabola opens down.

Example quadratic equation:

Graph of Quadratic Function

y = -x² + 4

The lowest or highest point of a parabola is called the vertex.

The vertical line through the vertex that splits the parabola in half is called the axis of symmetry.

Axis ofSymmetry


Vertex

Step 1: Find the x-coordinate of the vertex.x = -b 2a

Step 2: Find the y-coordinate of the vertex, by substituting the x-coordinate into the equation.

Step 3: Make a table of values, using x-values to the left and right of the vertex.

Step 4: Plot the points and connect them with a smooth curve to form a parabola.

Graphing of Quadratic Function

Graph y = 2xGraph y = 2x2 2 – 8x + 6– 8x + 6Step 1: Find x of vertex

a = and b=

Step 2: Substitute in xof vertex to find y.

Vertex is: ( , )

a

bx

2

Step 3: Make a table x y x y 00 11 22 -2-2 33 44

Axis of symmetry is vertical Axis of symmetry is vertical line through the vertex. line through the vertex. Same as x of vertex. x=2Same as x of vertex. x=2

Now you try one!Now you try one!Graph y = -2x² - 4x + 5

Graph AnalysisGraph AnalysisGraph y = x² – 2x – 3

Also list:Open up or down?Vertex?Axis of symmetry?

x -1 0 1 2 3

y 0 -3 -4 -3 0

Opens Up

Vertex is (1, -4).

The axis of symmetry is x

= 1

Check your info.Check your info.

9.3 Exit Questions9.3 Exit QuestionsSketch a graph of the function.y = x² - 2x + 4

1.Does graph open up or down?2.Give coordinates of the vertex.3.Write the of the axis of symmetry.


Tonight’s HomeworkP. 521-522#’s 33-34, 46-49

Objective:1.Solve quadratic equations by graphing2.Solve Real World problems involving

quadratic equations

Vocabulary: zeros, roots

9.4 Solving Quadratic Equations

by Graphing

1. Graph: y = x2 + 2x

2. Solve: 2x2 + 3 = 35

3. Solve: 2x2 + 3x = 35

9.3 Check Up

The lowest or highest point of a

parabola is called the vertex.

The vertical line through the vertex that splits the parabola in half is called the axis of symmetry.

The x intercepts of a graph are calledthe solutions or zeros. We also call them roots!

Axis ofSymmetry


Vertex

Solve by graphing 4xSolve by graphing 4x² ² = 16= 16

• Put in Standard Form: 4x² - 16 = 0a = b = c =

• Find vertex value = • Chart x values and find intercepts

x y

2 0

1 -12

0 -16

-1 -12

-2 0

Solution:

x = ± 2

2 0

1 -12

-1 -12

-2 0

0

-12

- -12

- 0

Solve by graphing: xSolve by graphing: x²² - 4x - 4x = 5= 5

Standard Form?

Step 1: Find x of vertex

Step 2: Substitute in xof vertex to find y.

Step 3: Make a table ofvalues until you find solutions.

Step 4: Plot points to form parabola and state solutions.

Solve by graphing: xSolve by graphing: x²² - - 4x = 54x = 5

y = x² - 4x – 5

Solution from graphing is x = 5, x = -1

What would factored equation be?

y = (x – 5)(x + 1)

How can we solve a quadratic How can we solve a quadratic

equation? equation?

• Graphing: Put in standard formo Fill out a table of values.o Find x intercepts.

• Factoring: Put in standard formo Factor by one of the four rules.o Use zero product property to solve.

Now you try one!Now you try one!

Solve y = 2x² - 4x – 6 by graphing

Open up or down?Vertex?Axis of symmetry?

Solutions? Factor to check?

(-1,0) (3,0)

(1,-8)

x=1

9.4 Exit Questions9.4 Exit Questions1. Solve by graphing: y = x² + 4x - 1

2. Solve by any method: 2x² = 32

9.4 Homework9.4 Homework

Tonight’s HomeworkP. 529-530#’s 18, 28-29, 33-34

Algebra T3 – Algebra T3 – Chapter 9Chapter 9

Daily Warm-UpSolve the function by graphing.

y = x² + 5x + 6

9.5 Solving Quadratic 9.5 Solving Quadratic Equations by the Quadratic Equations by the Quadratic

FormulaFormula

Goal: 1. Use the quadratic formula to solve a quadratic equation and use quadratic models for real life situations.

The Quadratic The Quadratic FormulaFormula

a

acbbx

2

42

The solutions of the quadratic equation ax² + bx + c = 0 are

When a ≠ 0 and b² - 4ac ≥ 0.

You can read this formula as “x equals the opposite of b, plus or minus the square root of b squared minus 4ac, all divided by 2a.”

x² + 5x – 6 x² + 5x – 6

= 0= 0

Step 1: Write is standard formx² + 5x – 6 = 0

Step 2: Identify a, b, ca = 1, b = 5, c = -6

Step 3: Plug into quadratic formula

a

acbbx

2

42

(1)

(1)(-6)55x

2

42

Solve 4x² - x – 7 = 0Solve 4x² - x – 7 = 0Step 1: Write is standard form

4x² - x - 7 = 0

Step 2: Identify a, b, c

a = 4, b = -1, c = -7

Step 3: Plug into quadratic formula

a

acbbx

2

42

YOU TRY - Solve 2x² - 3x + YOU TRY - Solve 2x² - 3x + 3= 03= 0

a

acbbx

2

42

YOU TRY - Solve 2x² - 3x – YOU TRY - Solve 2x² - 3x – 8 = 08 = 0

a

acbbx

2

42

Find the x intercepts of the Find the x intercepts of the

graph of y = x² + 3x - 8graph of y = x² + 3x - 8 Finding the x intercepts is the SAME as finding the

solutions to the equation. Use the Quadratic Formula to find the x intercepts

Vertical Motion Vertical Motion ModelsModels

Object is dropped: h = -16t² + s

Object is thrown: h = -16t² + vt + s h = height (feet) t = time in motion (seconds)s = initial height (feet) v = initial velocity (feet per second)

In these models the coefficient of t² is ½ the acceleration due to gravity. On the surface of Earth, this acceleration is about 32 feet per second per second.

Why is there no middle term for the dropped object model?

From a 40 foot cliff, you throw a stone From a 40 foot cliff, you throw a stone

downward at 20ft/sec into the water downward at 20ft/sec into the water

below. How long will it take to hit the below. How long will it take to hit the

water?water?


Tonight’s HomeworkP. 536-537#’s 32-35, 55-56, 79

Algebra T3 – Chapter Algebra T3 – Chapter 99

Daily Warm-UpFind the x-intercepts of the graph of the

Equation.y = x² - 11x + 24

Algebra T3Algebra T3

9.6 Applications of the Discriminant

Objective:1.Use the discriminant to find the number of

solutions to a quadratic equation2.Use the discriminant in real life

DiscriminantDiscriminant

x = -b ± √ b² - 4ac

2a

Discriminant

The discriminant can be used to find the number of solutions of the quadratic equation in the form of: ax² + bx +c

0 = x2 + 3x – 10

0 = x2 + 8x + 16

0 = x2 + 2x + 10

dis. is +, 2 real solutions

dis. = 0, 1 solution

dis. is -, no real solutions

Discriminant is b² - 4ac

How many solutions for: How many solutions for:

x x²² – 2x + 4 = 0 – 2x + 4 = 0

(-2)² - 4(1)(4) -12=

Since answer is negative there is no real solution

acb 42

How many solutions for:

-3x2 + 5x = 1

Put in standard form: -3x² + 5x -1

a= -3, b = 5, c = -1

Use discriminant: b² - 4ac =

How many solutions for:

-x2 = 10x + 25

•Remember to put in standard form before identifying a, b, and c.

How many zeros for:

y = x2 + 6x + 3

Remember the “zeros” are the x intercepts or solutions of the equation.

This means there 2 zero’s or x intercepts.

y = x² + 6x + 3y = x² + 6x + 3

Vertex x coordinate = -b/2a = -6/2(1) = -3

Y coord. = (-3)² + 6(-3) + 3 = 9 – 18 + 3 = -6

Vertex = (-3,-6)

Note because a is + graph opens upward.

This means there are 2 x intercepts

How many solutions for: y = x2 + 6x + 10

This means there are no zero’s or x intercepts.

y = x² + 6x + 10y = x² + 6x + 10


y coord. = (-3)² + 6(-3) + 10 = 9 – 18 + 10 = 1

Vertex = (-3, 1)


This means there are no x intercepts

How many zeros for: y = x2 + 6x + 9

This means there is one zero or x intercept.

y = x² + 6x + 9y = x² + 6x + 9


Ycoord. = (-3)² + 6(-3) + 3 = 9 – 18 + 9 = 0

Vertex = (-3,0)


This means there is 1 x intercept

YOU TRY - How many solutions for:

-3x2 + 6x - 3 = 0

•Remember to put in standard form before identifying a, b, and c.

L 9.6 HomeworkL 9.6 HomeworkI am a protector. I sit on a bridge. One person can see right through me, while others wonder what I hide.

What am I?

Tonight’s HomeworkP. 544#’s 9-17, *18-19

Algebra T3 – Chapter Algebra T3 – Chapter 99

Daily Warm-UpSketch a graph of the function.y = x² + 5x + 6

9.7 Graphing 9.7 Graphing Quadratic Quadratic InequalitiesInequalities

Objective: Sketch the graph of a quadratic inequality.

9.3 - Graphing a Quadratic 9.3 - Graphing a Quadratic

FunctionFunction• Step 1: Find the x-coordinate of the vertex.

• Step 2: Make a table of values, using x-values to the left and right of the vertex.

• Step 3: Plot the points and connect them with a smooth curve to form a parabola.

Sketching the Graph of a Quadratic Sketching the Graph of a Quadratic

InequalityInequalityStep 1) Replace inequality symbol(<,etc.) with = and sketch y

= ax² + bx +cDotted line for < or >Solid line for ≤ or ≥

Step 2) Pick a point clearly on inside or outside and test it in original inequality.

Step 3) If test point is a solution shade its region. If not, shade the other region.

***Can also shade based on symbol and direction of parabola (Mental Math)

Sketch y ≥ ½x² Sketch y ≥ ½x² + x+ x

Find Vertex

Find Solution

Dotted or Solid

Test Point to Shade/Mental Shading

Sketch y ≥ x² + 2xSketch y ≥ x² + 2x

YOU TRY YOU TRY

Sketch y ≤ -2x² Sketch y ≤ -2x²

YOU TRY YOU TRY

Sketch y > -x² - 2 Sketch y > -x² - 2

L 9.7 HomeworkL 9.7 HomeworkThere is a frog lying dead in the center of a

lilypad. The lily pad is sinking and there are twolily pads either side of him. Which lily pad should he jump to, the left or the right?

Tonight’s HomeworkP. 551-553#’s 13-16, 25-28, 40

algebra 1 – chapter 9 9.1 – solving quadratics by square roots objective: 1.know terminology of...

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