7.1: Getting to Know You Each student gets a card: either a table, graph, rule, or situation Consider what you know about the card Find other students.

7.1: Getting to Know You Each student gets a card: either a table, graph, rule, or situation Consider what you know about the card Find other students in your class that have a representation of the same line Find a group of desks to sit in.

Day 34: October 25th Objective: Strengthen your knowledge of y=mx+b and the relations of linear graphs and their equations. Also, learn that for data to be linear, the data must have constant growth and that for a point to lie on the graph, it must make the equation true. THEN Review your understanding of y=mx+b as you guess and check an equation for a trend line to represent data that is roughly linear. Also, use the equation and a graph to make and justify predictions. 7-1 (pg 275, RscrcPg) Homework Check 7-2 (pgs ) Wells Time 7-10 to 7-12 (pgs ) Conclusion Homework: 7-4 to 7-9 (pgs ) AND 7-13 to 7-18 (pg 280)

Day 35: October 26th Objective: Gain an abstract understanding of slope as they discover that slope is the change in y (Δy) divided by the change in x (Δx). THEN Assess Chapter 6 in a team setting. Homework Check 7-19 to 7-24 (pgs ) Wells Time Chapter 6 Team Test Conclusion Homework: 7-25 to 7-30 (pg 284)

Growth Triangle Vertical Segment Horizontal Segment

Finding the Change with Growth Triangles
What is the growth factor for this line? Change in y direction 9 9 1 9 1 Change in x direction 1

Slope of a Line The slope is a measure of steepness. It is the ratio of the vertical change to the horizontal change OR the ratio of the change in y to the corresponding change in x. “Delta y” = Change in y “Delta x” = Change in x Example: Slope Triangle What is the slope of the line? = 2 = 4 What is the equation of the line?

Day 36: October 27th Objective: Use slope triangles both to compare the relative steepness of lines and to build intuition about positive, negative and zero slopes. THEN Explore what information is needed to determine a line. Learn that parallel lines have the same slope, develop an algorithm for finding the slope of a line through two points without graphing, and discover the slope of vertical lines. Homework Check 7-31 to 7-36 (pgs , RscrcPg) Wells Time 7-43 to 7-46 (pgs ) Conclusion Homework: 7-37 to 7-42 (pgs ) AND 7-47 to 7-52 (pgs )

Steepness of a Line What makes a line steeper?
What makes a line less steep? The slope is further away from 0. The slope is closer to 0.

Different Values of Slope
Negative Zero Positive Decreasing Horizontal Increasing Negative Zero Positive Always “run” to the right Positive Positive Positive

NOT ENOUGH INFORMATION
What’s my Line? If possible, graph the line below and find its equation. If there is not enough information to draw one specific line, draw at least two lines that fit the given criteria. Line A goes through the point (2,5). NOT ENOUGH INFORMATION

What’s my Line? If possible, graph the line below and find its equation. If there is not enough information to draw one specific line, draw at least two lines that fit the given criteria. Line B has a slope of -3 and goes through the origin ( the point (0,0) ). ENOUGH INFORMATION

Line C goes through points (2,8) and (3,10).
What’s my Line? If possible, graph the line below and find its equation. If there is not enough information to draw one specific line, draw at least two lines that fit the given criteria. Line C goes through points (2,8) and (3,10). ENOUGH INFORMATION

NOT ENOUGH INFORMATION
What’s my Line? If possible, graph the line below and find its equation. If there is not enough information to draw one specific line, draw at least two lines that fit the given criteria. Line D has a slope of 4. NOT ENOUGH INFORMATION

Line E goes through the point (8,-1) and has a slope of -¾.
What’s my Line? If possible, graph the line below and find its equation. If there is not enough information to draw one specific line, draw at least two lines that fit the given criteria. Line E goes through the point (8,-1) and has a slope of -¾. ENOUGH INFORMATION

Necessary Information to Determine a Line
To graph a line and find its equation, what information do you need? First Option: The slope and a point Second Option: Two Points (3,10) (2,8) (8,-1) -4 1

Parallel Lines equal. ... The slopes of parallel lines are Example:
The rate of change of parallel lines is the same.

The graph is not changing in the x-direction
Vertical Lines The slope of a vertical line is undefined. ... Example: Find the slope of the line below. The graph is not changing in the x-direction You can not divide by 0.

Slope Formula The slope of the line through the points (x1, y1) and (x2, y2) is given by: Ex: Find the slope between (2, -14) and (10,30)

Day 37: October 28th Objective: Develop an algorithm for finding the slope of a line through two points without graphing, and discover the slope of vertical line. THEN Understand slope as a rate. Connect m and b in y=mx+b with contextual meaning. Homework Check Finish: 7-43 to 7-46 (pgs ) Wells Time 7-62, 7-64 (pgs ) Slope Show Worksheet Conclusion Homework: 7-56 to 7-61 (pg 295) AND 7-65 to 7-42 (pgs )

Day 38: October 31st Objective: Practice finding slopes and linear equations while solving a challenging team puzzle. Homework Check 7-71 to 7-72 (pg 300) Conclusion Homework: 7-73 to 7-78 (pgs )

First plot the y-intercept on the y-axis
Slope-Intercept Form Graph: Next, use rise over run to plot new points You can go backwards if you need! First plot the y-intercept on the y-axis Now connect the points with a line!

Cover-Up Method Plot : -2x + 5y = -10 Find the intercepts X Y -2 5

Day 39: November 1st Objective: Employ multiple methods to find the y-intercept of a line given its slope and one point on it. Learn how to solve for the y-intercept to find the equation of a line algebraically. THEN Discover that the slopes of perpendicular lines are opposite reciprocals. Use this to find the equation of a line given a point and a line perpendicular to it Homework Check 7-79 to 7-84 (pgs ) Wells Time 7-91 to 7-94 (pgs ) Conclusion Homework: 7-85 to 7-90 (pg 306) AND 7-96 to (pg 309)

7-79: Algebraically Finding the y-intercept
Colleen wants to know how much a chick weighs when it is hatched. Colleen tracked one of her chickens and found it grew steadily by about 5.2 grams each day since it was born. Nine days after it hatched, the chick weighed 98.4 grams. Algebraically determine how much the chick weighed the day it was hatched. When the chick was hatched, it was day 0. Thus, we need to find the y-intercept. Since the growth is constant, the situation in linear: The growth (slope) is 5.2 grams per day The chick weighed 98.4 grams (y) after 9 days (x) The equation now has one distinct variable. Solve it. 51.6 grams

7-79: Algebraically Finding a x-Value
Now Colleen wants to know when the chicken will weigh 140 grams. Algebraically find the answer. Use the slope and y-intercept to write an equation. Use the Slope-Intercept form: The 140 grams represents a y value. Substitute 140 for y The equation now has one distinct variable. Solve it. 17 Days

Example: 7-83 (a) Algebraically find the equation for a line with a slope of -3, passing through the point (15,-50). Find the y-intercept. Use Slope-Intercept Form: The slope is -3 A point on the graph is x=15 and y=-50 The equation now has one distinct variable. Solve it. Substitute back into Slope-Intercept Form:

Perpendicular Lines A line is perpendicular to another if it meets or crosses at right angles (90°). For instance, a horizontal and a vertical line are perpendicular lines.

Slopes of Perpendicular Lines
Complete the following assuming Line A and Line B are perpendicular. Make Line A have a slope of What is the slope of Line B (the line perpendicular to Line A)? A B A B A B A B -3 2 2 3

Slopes of Perpendicular Lines
Two lines are perpendicular if their slopes are opposite reciprocals of each other. In other words, if the slope of a line is then the perpendicular line has a slope of Example: What is the slope of a line perpendicular to each equation below.

Example Find the y-intercept.
Algebraically find the equation of the line that goes through the point (2,3) and is perpendicular to y = -4x – 2. This y-intercept does not matter. Find the y-intercept. Use Slope-Intercept Form: The slope is ¼ A point on the graph is x=2 and y=-3 The equation now has one distinct variable. Solve it. Substitute back into Slope-Intercept Form:

Day 40: November 2nd Objective: Learn and apply an algebraic method to find the equation of a line passing through two given points. THEN Use your knowledge of y=mx+b to find equations of lines from a graph. Homework Check 7-102 to (pgs ) Wells Time 7-111 (pgs 313, RscrcPg) Conclusion Homework: to (pg 312) AND to (pgs )

Example: Parallel Lines
Algebraically find the equation of the line that goes through the point (16,4) and is parallel to 3x + 4y = 8. Find the Slope Find the y-intercept. Use Slope-Intercept Form: The slope is -3/4 A point on the graph is x=16 and y=4 The equation now has one distinct variable. Solve it. Since Parallel Lines have the same slope, our new equation also has slope -3/4. (This y-intercept does not matter) Substitute back into Slope-Intercept Form:

Example: Algebraically Finding the equation of a Line with Two Points
Algebraically find the slope-intercept equation of a line that contains the points (14,-1) and (-21,14). (14,-1) (x1,y1) (-21,14) (x2,y2) Find Slope Find the y-intercept. Use Slope-Intercept Form: The slope is -3/7 A point on the graph is x=14 and y=-1 (it does not matter which one you pick) The equation now has one distinct variable. Solve it. Substitute back into Slope-Intercept Form:

Day 41: November 3rd Objective: Learn how to find the equation of a line when given any point and the slope with the Point-Slope form of a line. Also, continue to find the equation of a line given different representations or information. Homework Check Notes: Point-Slope Formula What is My Equation? Worksheet Wells Time 7-118 (pgs 316, RscrcPg) Conclusion Homework: to (pgs 320 to 322)

Point-Slope Form In most textbooks the graphing form for a linear equation is called Point-Slope Form and is the following: The equation of a line that contains the point (x1,y1) and whose slope is m is: (x1,y1) Point: Slope:

Substitute into point-slope
Example 1 Algebraically find the equation of a line that passes through the point (1,-2) and has a slope of ½. (1,-2) (x1,y1) ½ m Substitute into point-slope

Substitute into point-slope
Example 2 Algebraically find the slope-intercept equation of a line that contains the points (-1,4) and (-4,-2). (-1,4) (x1,y1) (-4,-2) (x2,y2) Substitute into point-slope Find Slope 2 m

Substitute into slope-intercept form
Example 3 Algebraically find the equation of a line that passes through the point (0,8) and has a slope of 5. (0,8) b y-intercept 5 m Substitute into slope-intercept form

What is my Equation? y=-1/3x+3 y=3/2x-1 y=2x+5 y+2=-1/5(x-2)
DABC

Day 42: November 4th Objective: Continue to find the equation of a line given different representations or information. THEN Assess Chapters 6-7 in an individual setting. Homework Check 7-118 (pg 316, RscrcPg) Wells Time Chapters 6-7 Individual Test Conclusion Homework: Finish (pg 316)

Day 43: November 7th Objective: Review how to build rectangles with tiles and learn shortcuts for finding the dimensions of a completed generic rectangle. Discover that the products of the terms in each diagonal of a generic rectangle are equal. THEN Develop an algorithm to factor quadratic expressions without algebra tiles. Homework Check 8-1 to 8-4 (pgs ) Wells Time 8-12 to 8-15 (pgs ) Conclusion Homework: 8-6 to 8-11 (pgs ) AND 8-16 to 8-21 (pg 333)

Specific Expressions Trinomial – Binomial – Monomial –
Consisting of three terms (Ex: 5x3 – 9x2 + 3) Consisting of 2 terms (Ex: 2x6 + 2x) Consisting of one term (Ex: x2)

ax2 + bx +c Quadratic Expression
An expression in x that can be written in the standard form: ax2 + bx +c Where a, b, and c are any number except a ≠ 0.

Factoring The process of rewriting a mathematical expression involving a sum to a product. It is the opposite of distributing. Example: SUM PRODUCT

Solutions to 8-3

Finding the Dimensions of a Generic Rectangle
Mr. Wells’ Way to find the product for a generic rectangle: Make sure to Check Second, find missing WHOLE NUMBER dimensions on the individual boxes. First, find the POSITIVE Greatest Common Factor of two terms in the bottom row. 5 10x -15 2x 4x2 -6x 2x -3 Lastly, write the answer as a Product:

A Pattern with Generic Rectangles
The product of one diagonal always equals the product of the other diagonal. Example: 10x . -6x = -60x2 10x -15 4x = -60x2 4x2 -6x

Factoring with the Box and Diamond
c is always in the top right corner Because of our pattern, the missing boxes need to multiply to: Fill in the results from the diamond and find the dimensions of the box: 3 3x 4x +6 (2x2)(6) 12x2 GCF 2x 2x2 4x x ___ Diamond Problem 7x x 2 ax2 is always in the bottom left corner Write the expression as a product: The missing boxes also have to add up to bx in the sum ( 2x + 3 )( x + 2 )

Factoring Example Factor: (3x2)(-10) -30x2 5 15x -2x -10 -2x 15x GCF x
Product c (3x2)(-10) -30x2 5 15x -2x -10 ax2c -2x x GCF x 3x2 ___ bx ax2 13x 3x -2 Sum ( x + 5 )( 3x – 2 )

Day 44: November 9th Objective: Continue to practice factoring skills while learning special cases: quadratics with missing terms, quadratics that are not in standard form, and quadratics with more than one possible factored form. Homework Check 8-22 to 8-25 (pgs ) Conclusion Homework: 8-27 to 8-32 (pg 336)

Factoring: Different Order
Rewrite in Standard Form: ax2 + bx + c Product c (15x2)(-77) -1155x2 11 33x -35x -77 ax2c -35x x GCF 5x 15x2 ___ bx ax2 -2x 3x -7 Sum ( 5x + 11 )( 3x – 7 )

Factoring: Perfect Square
Product c (x2)(9) 9x2 3 3x 9 ax2c 3x x GCF x x2 ___ bx ax2 6x x 3 Sum ( x + 3 )( x + 3 ) ( x + 3 )2

Factoring: Missing Terms
Product c (9x2)(-4) -36x2 2 6x -6x -4 ax2c -6x x GCF 3x 9x2 ___ bx ax2 3x -2 Sum ( 3x + 2 )( 3x – 2 )

Factoring: Which Expression is correct?
Notice that every term is divisible by 2 Factor: If you use the box and diamond, the following products are possible: x2 ÷2 Which is the best possible answer?

Day 45: November 10th Objective: Complete the focus on factoring by considering expressions that can be factored first with a common factor and then again using the quadratic factoring method. Homework Check 8-33 to 8-36 (pgs ) Conclusion Homework: 8-37 to 8-42 (pg 339)

Solutions to 8-33 9x2 – 12x + 4 = (3x – 2)2
81m2 – 1 = (9m + 1)(9m – 1) 28 + x2 – 11x = (x – 4)(x – 7) 3n2 + 9n + 6 = (3n + 3)(n + 2) OR = (n + 1)(3n + 6)

Factoring: Factoring Completely
Ignore the GCF and factor the quadratic Reverse Box to factor out the GCF 5( x + 3 )( 2x – 1 ) Product Don’t forget the GCF c (2x2)(-3) -6x2 3 6x -x -3 ax2c -x x GCF x 2x2 ___ bx ax2 5x 2x -1 Sum

Factoring: Factoring Completely
Ignore the GCF and factor the quadratic Reverse Box to factor out the GCF 3x(x + 3)(x – 5) Product Don’t forget the GCF c (x2)(-15) -15x2 3 3x -5x -15 ax2c -5x x GCF x x2 ___ bx ax2 -2x x -5 Sum

Factoring: Just Reverse Box
Reverse Box to factor out the GCF There is no longer a quadratic, it is not possible to factor anymore.

Day 46: November 11th Objective: Describe a parabola using its intercepts, vertex, symmetry, and whether it opens up or down. Homework Check 20 Minutes: Factoring a Quadratic Worksheet Wells Time 8-43 (pgs ) Conclusion Homework: 8-45 to 8-50 (pgs ) AND Finish Worksheet

Factoring: Ensuring “a” is Positive
When the x2 term is negative, it is difficult to factor. Reverse Box to factor out the negative Ignore the GCF and factor the quadratic -( x + 6 )( x + 7 ) Product Don’t forget the GCF c (x2)(42) 42x2 6 6x 7x 42 ax2c 6x x GCF x x2 ___ bx ax2 13x x 7 Sum

Factoring a Quadratic (x+9)(x–4) (x–6)(x+5) (6x–1)(x–9) (2x+9)(x–8)

Quadratic Vocabulary Parabola: The graph of a quadratic equation.
x-intercept: The value of x when y=0. y-intercept: The value of y when x=0. Line of Symmetry: The imaginary line where you could fold the image and both havles match exactly. Parabola Opens Up: Resembles “valley” OR holds water. Parabola Opens Down: Resembles “hill” OR spills water. Vertex: The lowest point of a parabola that opens up and the highest point of a parabola that opens down.

Investigating a Parabola
x = 1 x = 0 Line of Symmetry: Vertex: y–intercept(s): x–intercept(s): Shape: x = 1 Line of Symmetry: Vertex: y–intercept(s): x–intercept(s): Shape: x = 0 (1,-9) (0,4) (0,-8) (0,4) (-2,0) and (4,0) (-2,0) and (2,0) Opens Up Opens Down

x = 2 x = 1 Line of Symmetry: Vertex: y–intercept(s): x–intercept(s): Shape: x = 2 Line of Symmetry: Vertex: y–intercept(s): x–intercept(s): Shape: x = 1 (2,1) (1,0) (0,5) (0,1) None (1,0) Opens Up Opens Up

x = 3 x = 1.5 Line of Symmetry: Vertex: y–intercept(s): x–intercept(s): Shape: x = 3 Line of Symmetry: Vertex: y–intercept(s): x–intercept(s): Shape: x = 1.5 (3,-4) (1.5,~6.25) (0,5) (0,4) (1,0) and (5,0) (-1,0) and (4,0) Opens Up Opens Down

x = -2.5 x = 1 Line of Symmetry: Vertex: y–intercept(s): x–intercept(s): Shape: x = 1 Line of Symmetry: Vertex: y–intercept(s): x–intercept(s): Shape: x = -2.5 (1,0) (-2.5,~ -5.25) (0,-1) (0,1) (1,0) (~ -4.75,0) & (~ -.25,0) Opens Down Opens Up

Quadratic Characteristics
The standard form of a quadratic equation (ax2+bx+c) has the following connections to the graph: Does it open up or down? If a is positive, the parabola opens up and the vertex is a minimum. If a is negative, the parabola opens down and the vertex is a maximum. What is the y-intercept? c represents the y-intercept ( 0, c ) Since “c” is +7, the y-intercept is (0,7) Example: Since “a” is negative (-2), the parabola opeds down

Day 47: November 14th Objective: Identify connections between different representations of quadratics: an equation, a table, a context, and a graph. Also connect the intercepts and vertex of a parabola to a context: the launch, maximum height, and landing of a water balloon. THEN Sketch the graph of a quadratic rule quickly, using its intercepts. Also how to find the x-intercepts of a parabola by factoring the corresponding quadratic equations and applying the Zero Product Property. Homework Check 8-51 to 8-53 (pgs ) Wells Time 8-59 to 8-64 (pgs ) Conclusion Homework: 8-54 to 8-58 (pgs ) AND 8-65 to 8-70 (pgs )

Quadratic Function Web
Table Non-Algebraic Rule or Equation Graph Algebraic Situation

What do you Need to Sketch a Parabola?
Can you sketch a parabola if you only know where its y-intercept is? For example, if the y-intercept of a parabola is (0,-15) can you sketch the graph? Why or why not? NO!

Can you sketch a parabola if you only know where its x-intercept are? For example, if the x-intercepts of a parabola are (-3,0) and (5,0) can you sketch the graph? Why or why not? NO!

Can you sketch a parabola if you only know where its x-intercept and y-intercept are? For example, if the x-intercepts of a parabola are (-3,0) and (5,0) and the y-intercept is (0,-15), can you sketch the graph? Why or why not? The vertex must lie on the line of symmetry. YES!

Algebraically Finding x- and y-intecepts
Find the x- and y-intercept(s) of y = 2x2 + 5x – 12. y-intercept(s) x-intercept(s) We do not have a method to solve this equation. (0,-12)

x–Intercepts, Solutions, Roots, and Zeros in Quadratics
x-intercept(s): Where the graph of y=ax2+bx+c crosses the x-axis. The value(s) for x that makes a quadratic equal 0. Solution(s) OR Roots: The value(s) of x that satisfies 0=ax2+bx+c. Zeros: The value(s) of x that make ax2+bx+c equal 0.

Zero Product Property If a . b = 0, then a and or b is equal to 0
Ex: Solve the following equation below. 0 = ( x + 14 )( 6x + 1 ) OR Would you rather solve the equation above or this: 0 = x2 + 25x + 14 ?

Solving a Quadratic Solve: (2x2)(-12) -24x2 4 8x -3x -12 8x -3x GCF
Product c (2x2)(-12) -24x2 Factor to rewrite as a product 4 8x -3x -12 ax2c 8x x GCF 2x2 x ___ bx ax2 5x Use the Zero-Product Property 2x -3 Sum

Sketching a Quadratics
Find the y-intercept Sketch: y = 2x2 + 5x – 12 Pay attention to Symmetry 8 Factor to find the x-intercepts -6 6 Draw the Parabola -16

Day 48: November 15th Objective: Review using the Zero Product Property to find the x-intercepts of a parabola and practice solving quadratic equations by factoring. Also learn how to write a quadratic rule from a table. THEN Practice moving from a quadratic rule to its graph (and vice versa) for parabolas with one or two x-intercepts. Homework Check 8-71 to 8-75 (pgs ) Wells Time 8-82 to 8-84 (pgs ) Conclusion Homework: 8-76 to 8-81 (pg 352) AND 8-85 to 8-90 (pgs )

Algebraically Finding a Vertex
Use an algebraic method to find the vertex of: 1. Find the x-intercepts 2. Average the x-intercepts 4. Substitute x into the rule for y WARNING: This method does not work if there are no x-intercepts

Finding a Quadratic Equation with the x-intercepts and Another Point
Find a rule for the quadratic table below. Since we know the x-intercepts x -2 -1 1 2 y -15 9 12 Sketch: Plot all the points Substitute into Factored Form: Plug in the x-intercepts Use symmetry to find any missing x-intercepts (-1,0) (3,0) Plug in another point Solve for a Equation:

Day 49: November 16th Objective: Practice moving from a quadratic rule to its graph (and vice versa) for parabolas with one or two x-intercepts. THEN How to simplify square roots by rewriting numbers in factored form so that one of the factors is a perfect square. Homework Check Finish: 8-82 to 8-84 (pgs ) Wells Time Notes: Simplifying Square Roots Conclusion Homework: Simplifying Radicals Worksheet

Finding a Quadratic Equation with the x-intercepts and Another Point
Find a rule for the quadratic table below. x -7 -6 -5 -4 -3 Y 20 -10 Since we know the x-intercepts Substitute into Factored Form: There may be a GCF (adjustment value) Plug in the x-intercepts We need to solve for “a” but there are 3 variables Plug in another point Solve for a Equation:

Irrational Numbers Radical numbers are typically irrational numbers (unless they simplify to an integer). Our calculator gives: But the decimal will go on forever because it is an irrational number. For the exact answer just use: Some radicals can be simplified similar to simplifying a fraction.

Radical Product Property
ONLY when a≥0 and b≥0

The square of whole numbers.
Perfect Squares The square of whole numbers. 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , 121, 144 , 169 , 196 , 225, etc

Simplifying Square Roots
Check if the square root is a whole number Find the biggest perfect square (4, 9, 16, 25, 36, 49, 64) that divides the number in the root Rewrite the number in the root as a product Simplify by taking the square root of the perfect square and putting it outside the root CHECK! Note: A square root can not be simplified if there is no perfect square that divides it. Just leave it alone. ex: √15 , √21, and √17

Write the following as a radical (square root) in simplest form: Simplify. 36 is the biggest perfect square that divides 72. Rewrite the square root as a product of roots. Ignore the 5 multiplication until the end.

Simplify these radicals:

Adding and Subtracting Radicals
Simplify the expressions: Always simplify a radical first. Treat the square roots as variables, then combine like terms ONLY.

Fractions and Radicals
Simplify the expressions: There is nothing to simplify because the square root is simplified and every term in the fraction can not be divided by 10. Make sure to simplify the fraction.

Radical Quotient Property
ONLY when a≥0 and b>0

The Square Root of a Fraction
Write the following as a radical (square root) in simplest form: Take the square root of the numerator and the denominator Simplify.

Rationalizing a Denominator
The denominator of a fraction can not contain a radical. To rationalize the denominator (rewriting a fraction so the bottom is a rational number) multiply by the same radical. Simplify the following expressions:

WARNING In general: For Example: Not Equal

Day 50: November 17th Objective: Learn how to use the Quadratic Formula to solve quadratic equations that are not factorable. THEN Learn how to solve quadratic equations that have no solution, have only one solution, or are not in standard form. Homework Check 8-91, 8-92, 8-94 (pgs ) Wells Time 8-103 to (pg 360) Conclusion Homework: 8-96 to (pg 359) AND to (pg 362)

But this parabola has two zeros.
Example Use the Zero Product Property to find the roots of: Product But this parabola has two zeros. (x2)(-7) -7x2 c -7 ax2c IMPOSSIBLE x2 bx ___ ax2 -3x Sum Just because a quadratic is not factorable, does not mean it does not have roots. Thus, there is a need for a new algebraic method to find these roots.

Quadratic Formula For ANY 0 = ax2 + bx +c (standard form) the value(s) of x is given by: MUST equal 0 Plus or Minus Opposite of b “All Over”

Solving a Quadratic with the Quadratic Formula
Algebraically solve: Must equal 0 Find the values of “a,” “b,” “c” a = b = c = 1 -3 -7 Simplify the expression in the square root first Since the square root can not simplified, this is an acceptable EXACT answer Substitute into the Quadratic Formula Or you can approximate the expressions (don’t forget parentheses). This is NOT exact. Or you can write two expressions. One with addition in the numerator and other with subtraction. This is also an EXACT answer.

Solving a Quadratic with the Quadratic Formula
Algebraically solve: Must equal 0 Find the values of “a,” “b,” “c” a = b = c = 2 6 -5 Simplify the expression in the square root first The square root can be simplified. Substitute into the Quadratic Formula The GCF of every term is 2

Solving a Quadratic with the Quadratic Formula: No Solutions
Algebraically solve: Must equal 0 Find the values of “a,” “b,” “c” a = b = c = 1 -5 9 Simplify the expression in the square root first Substitute into the Quadratic Formula This can not be calculated because you can not square root a negative. The graph of the quadratic has no x-intercepts. NO SOLUTIONS

Day 51: November 18th Objective: Learn how to solve quadratic equations that have no solution, have only one solution, or are not in standard form. THEN Practice using the Zero Property and the Quadratic Formula by deciding which method is best to try first for different types of equations. Also solve quadratics in an application problem and equations that are not in standard form Homework Check FINISH to (pg 360) Wells Time 8-112 to (pgs ) Conclusion Homework: to (pg 362) AND to (pg 366)

Solving a Quadratic with the Quadratic Formula: Two Solutions
Algebraically solve: Must equal 0 Find the values of “a,” “b,” “c” a = b = c = 4 -121 Simplify the expression in the square root first The square root can be simplified. Substitute into the Quadratic Formula Or Since the answers will be rational, it is best to list both.

Solving a Quadratic with the Quadratic Formula: One Solution
Algebraically solve: Must equal 0 Find the values of “a,” “b,” “c” a = b = c = 36 -60 25 Simplify the expression in the square root first The square root of 0 is 0. Substitute into the Quadratic Formula The is no difference to adding zero or subtracting 0. This expression will result in only one answer.

The Discriminant of a Quadratic
For ANY 0 = ax2 + bx +c (standard form) the value given by: If the Discriminant (the value underneath the square root in the quadratic formula) is…. Greater than zero there are two roots Equal to zero there is one root Less than zero there are no roots

Solving a Quadratic: Make Sure to Isolate 0
Solve: Factor to rewrite as a product Product c (x2)(-4) -4x2 -4 Solve for 0 first! -4x x -4 Distribute ax2c x2 x -4x x GCF ___ ax2 bx x 1 -3x Use the Zero-Product Property Sum

Solving a Quadratic: Make Sure to Isolate 0
Solve: Find the values of “a,” “b,” “c” Solve for 0 first! a = b = c = 1 -3 -4 Distribute Simplify the expression in the square root first The square root can be simplified. Substitute into the Quadratic Formula Or Since the answers will be rational, it is best to list both.

Choosing a Strategy To Solve Quadratics
Zero Product Property Quadratic Formula When the quadratic is in Factored form When the quadratic contains decimals When the quadratic is not factorable

Day 52: November 21st Objective Learn how to solve linear inequalities with one variable and how to represent the solutions on a number line. THEN Continue to develop how to solve linear, one-variable inequalities by finding a boundary point and testing a value in the inequality. Also, use an inequality to solve a word problem. Homework Check 9-1 to 9-6 (pgs ) Wells Time 9-14 to 9-16 (pgs ) Conclusion Homework: 9-7 to 9-13 (pgs ) AND 9-17 to 9-22 (pg 381)

Inequality Notation Not Included
The interval does NOT include the endpoint(s) Inequality Notation Graph < Less than > Greater than Open Dot Included The interval does include the endpoint(s) Inequality Notation Graph ≤ Less than or equal to ≥ Greater than or equal to Closed Dot

9-1: a

Answer on a Graph x -5 -4 -3 -2 -1 1 2 3 4 5

9-1: b

9-1: c

9-1: d

9-1: e

9-1: f

9-1: g

9-1: h

Solving an Inequality In order to find the points that satisfy an inequality statement: Find the boundary Test every region to find which one(s) satisfies the original statement

Finding an Inequality Boundary
Boundary Point: A solution(s) that makes the inequality true (equal). It could be the smallest number(s) that make it true. Or it is the largest number(s) that makes it NOT true. EX: Find the boundary point of To find a boundary replace the inequality symbol with an equality symbol.

Solving a 1 Variable Inequality
Represent the solutions to the following inequality algebraically and on a number line. Closed or Open Dot(s)? Graphical Solution Find the Boundary Test Every Region x Change inequality to equality Pick a point in each region x = 0 x = 2 Solve Substitute into Original 3 < 1 -1 < 1 False True Shade True Region(s) Plot Boundary Point(s) Algebraic Solution

Solving a 1 Variable Inequality: The Answer is All Numbers
Represent the solutions to the following inequality algebraically and on a number line. Closed or Open Dot(s)? Graphical Solution Find the Boundary Test Every Region x Change inequality to equality Solve All Numbers “Algebraic” Solution Since every value of k satisfies the equation, every Point is a Boundary Point

Solving a 1 Variable Inequality: No Solutions
Represent the solutions to the following inequality algebraically and on a number line. Closed or Open Dot(s)? Graphical Solution Find the Boundary Test Every Region x Change inequality to equality Solve No Solution Since every value of k satisfies the equation, every Point is a Boundary Point “Algebraic” Solution

Reminder: Compound Inequalities
The following are examples to algebraically write the following graphs: 0≤x<4 x<-1 or x>2

Day 53: November 22nd Objective: Continue to develop how to solve linear, one-variable inequalities by finding a boundary point and testing a value in the inequality. Also, use an inequality to solve a word problem. THEN Assess Chapter 8 in a team setting. Homework Check Notes: Simplifying Square Roots Finish: 9-14 to 9-16 (pgs ) Wells Time Chapter 8 Team Test Conclusion Homework: Finish Simplifying Radicals Worksheet

Day 54: November 23rd Objective: Go over how to calculate key points of a graph on the calculator. THEN Learn how to graph linear inequalities with two variables. Homework Check Calculator Commands Wells Time 9-23 to 9-33 (pgs , RscPg) Conclusion Homework: Finish Classwork

Examining the Solutions of a Linear Equation
Are the following points solutions to the equation y = -2x + 3? Justify each conclusion with both the graph and the equation? a. (-1,5) Yes. b1. (2,-1) Yes. b2. (0,0) No. The points on the line make the equation true.

Your team will be given a list of points to test in the inequality y ≥ -2x +3. For each point that makes the inequality true, click on the point on the class graph.

Graphing a 2 Variable Inequality
Graphically represent the solutions to the following inequality: Solid or Dashed? Find the Boundary Plot points for the equality Test Every Region Pick a point in each region (0,0) (3,0) Substitute into Original 0 > -3 0 > 1.5 Shade True Region(s) True False

Graphically represent the solutions to the following inequality: Solid or Dashed? Find the Boundary Plot points for the equality Test Every Region (0,0) Pick a point in each region (-4,0) Substitute into Original 0 ≤ 6 8 ≤ 6 Shade True Region(s) True False

Day 55: November 28th Objective: Learn how to graph linear and non-linear inequalities with two variables. Also use the graph of a two-variable inequality to solve a word problem. THEN Learn how to use the absolute-value operation and how to graph an absolute-value equation and inequality. Homework Check 9-34 to 9-38 (pgs , RscrPg) Wells Time 9-45 to 9-50 (pgs ) Conclusion Homework: 9-39 to 9-44 (pg 387) AND 9-51 to 3-56 (pgs )

Graphically represent the solutions to the following inequality: Solid or Dashed? Find the Boundary Plot points for the equality Test Every Region Pick a point in each region (0,0) (0,5) Substitute into Original 0 < 3 5 < 3 Shade True Region(s) True False

Absolute Value Official Definition: Practical Definition:
Example: Evaluate Absolute value makes things positive. Read “The absolute value of -7.” 7 -10 -5 5 10

Arithmetic with Absolute Value
In order to evaluate an expression containing an absolute value, the absolute value part needs to be simplified first. Mr. Wells likes to extend the familiar PEMDAS acronym to APEMDAS (evaluate the absolute value expression before parentheses). Example: Evaluate Evaluate the expression inside the absolute value first. Do NOT use the Distributive Property. Absolute value makes things positive.

Absolute Value in the Calculator
The calculator will calculate Absolute Value. Instead of the absolute value bars the calculator uses abs( ). Hit MATH Hit the right arrow button Press enter on 1:abs(

All “linear” absolute value graphs will have a “V” shape.
Connect the points. x y -4 -3 -2 -1 1 2 3 4 4 3 2 1 Always make a table. 1 All “linear” absolute value graphs will have a “V” shape. 2 3 4

Graphically represent the solutions to the following inequality: Solid or Dashed? Find the Boundary Plot points for the equality Test Every Region Pick a point in each region (0,0) (0,2) Substitute into Original 0 < 1 2 < 1 Shade True Region(s) True False

Day 56: November 29th Objective: Continue to develop the ability to graph two-variable inequalities by graphing systems of inequalities. THEN Continue to learn how to graph systems of inequalities and apply this understanding to solve problems. Homework Check 9-57 to 9-61 (pgs , RscrPg) Wells Time 9-68 to 9-70 (pgs , RscrPg) Conclusion Homework: 9-62 to 9-67 (pg 394) AND 9-71 to 9-77 (pgs )

System of Inequalities
The Solution region is where the shadings overlap. For instance the following point is in the solution region because it satisfies both inequalities: 0 ≤ 5 True (0,0) 0 > -1 True Points are solutions to this system if they make both inequalities true.

Graph the solution region of the system of inequalities. Plot each inequality individually. Find out which side to shade for each inequality. (0,0) (0,1) 0 ≥ 1 1 < 0 False False Find where the shadings overlap.

Graph the solution region of the system of inequalities. Plot each inequality individually. Find out which side to shade for each inequality. (0,0) (0,0) 0 < 3 0 ≤ 4 True True Find where the shadings overlap.

Graph the solution region of the system of inequalities. Plot each inequality individually. Find out which side to shade for each inequality. (0,0) (0,0) 0 ≤ 4 0 ≥ 4 True False Find where the shadings overlap.

Graph the solution region of the system of inequalities. Plot each inequality individually. Find out which side to shade for each inequality. (0,0) (0,1) 0 ≥ 3 1 ≤ 0 NO SOLUTION False False Find where the shadings overlap.

9-78: United Nations to the Rescue
Read 9-78 on page 398. New Directions: Each group will receive Lesson 9.3.3A Resource Page that describes the budget for 5 countries and 1 special assignment. For all 5 countries AND the special assignment, write an inequality expressing how many food and medicine packages each country (and special assignment) is able to give. Let x equal the number of food packages and y equal the number of medicine packages. Each group will also receive Lesson 9.3.3B Resource Page that contains a graph. Graph all 6 inequalities on the same set of axes. Highlight the solution region representing the options of medicine and food packages that can be donated by each country and the special assignment simultaneously.

Day 58: December 1st Objective: Learn a quick way to factor perfect square trinomials and quadratics that are a difference of squares. Homework Check 12-1 to 12-4 (pgs ) Wells Time Quadratics Worksheet Conclusion Homework: Finish Classwork AND Inequalities Worksheet

Special Quadratics Discuss any patterns noticed and arrange examples into groups based on those patterns noticed.

Special Quadratics Perfect Square Trinomials Difference of Squares a=1
Arrange examples into groups based on any patterns noticed. Perfect Square Trinomials Difference of Squares a=1

Factoring a Quadratic when a=1
If the coefficient to x2 is 1, the quadratic easily factors: x2 + bx + c = ( x + r )( x + s ) if Completely factor the following: Ex 1: x2 – 2x – 8 Ex 2: x2 + 13x +42 Ex 3: x2 – 10x +9 The product of r and s is c and The sum of r and s is b = ( x + 2 )( x – 4 ) The product of 2 and -4 is -8 The sum of 2 and -4 is -2 = (x + 7)(x + 6) The product of 7 and 6 is 42 The sum of 6 and 7 is 13 =( x – 1 )( x – 9 ) The product of -1 and -0 is 9 The sum of -1 and -9 is -10

Difference of Squares If you have the difference of two quantities squared, it easily factors: a2 – b2 = ( a + b )( a – b ) Completely factor the following: Ex 1: x2 – 144 Ex 2: 25x2 – 121 Ex 3: 36x2 – y2z2 = x2 – 122 = ( x + 12 )( x – 12 ) =(5x)2 – 112 = ( 5x + 11 )( 5x – 11 ) =(6x)2 – (yz)2 =( 6x + yz )( 6x – yz )

Perfect Square Trinomial
If the coefficient to x2 and the constant are perfect squares AND the coefficient to x is twice the product of those perfect squares, the quadratic easily factors: a2x2 + 2abx + b2 = ( ax + b )2 or a2x2 – 2abx + b2 = ( ax – b )2 Completely factor the following: Ex 1: x2 + 24x + 144 Ex 2: 4x2 – 20x + 25 Ex 3: 16x2 + 56x + 49 = 12x x+122 = ( x + 12 )2 = 22x2 – 2.2.5x +52 = ( 2x – 5 )2 = 42x x+72 = ( 4x + 7 )2

Day 59: December 2nd Objective: Learn how to simplify algebraic fractions. Homework Check Notes: Simplifying Rational Expressions Wells Time Rational Expressions Worksheet Conclusion Homework: Finish Classwork AND 10-7 to (pg 411)

Rational Expressions A Rational Expression is an algebraic fraction: a fraction that contains a variable(s). Our goal is to simplifying rational expressions by “canceling” off common factors between the numerator and denominator. Similar to simplifying a numeric fraction. Example:

Simplifying Rational Expressions
Simplify the following expressions by finding a common factor:

The Major Requirement for Simplifying Rational Expressions
A fellow student simplifies the following expressions: Which simplification is correct? Substitute two values of x into each to justify your answer. Equal. Not Equal. MUST BE MUITLIPLICATION! It can be simplified if the numerator and denominator are single terms and are product of factors.

Which is Simplified Correctly?
Which of the following expressions is simplified correctly? Explain how you know. X Left Right -5 -11.5 25 -1 1.5 1 4 3.29 16 7 5.9 49 X Left Right -5 -3 -1 1 2 4 6 7 9 The left side of the equation has to equal the right. MUST BE MUITLIPLICATION! It can be simplified if the numerator and denominator are single terms and are product of factors.

Example 1 State the values that make the denominator zero and then simplify: Half the work is done. It is already factored. Rewrite any factors if they are raised to a power Make the Denominator 0: CAN cancel since the top and bottom have common factors. 2 and -7. Don’t forget about cancelling common numeric Factors. These Make the ORIGINAL denominator equal 0. We assume that x can never be these values.

Example 2 State the values that make the denominator zero and then simplify: Can NOT cancel since its not in factored form. Also it is not obvious what values of x make the denominator 0. The denominator is factored, so it is obvious what values of x make it 0 Always Factor Completely Make the Denominator 0: 4, -4, and 0. CAN cancel since the top and bottom have a common factor These Make the ORIGINAL denominator equal 0. We assume that x can never be these values.

Example 3 State the values that make the denominator zero and then simplify: Can NOT cancel any factors since its not in factored form If they are not quadratics, find a common factor. Make the Denominator 0: a=0 or b=0 CAN cancel common factors since the top and bottom have a common factor These Make the ORIGINAL denominator equal 0. We assume that a & b can never be these values.

Day 60: December 5th Objective: Understand how to multiply and divide rational expressions and continue to learn how to simplify rational expressions . Homework Check Notes: Multiplying and Dividing Rational Expressions Wells Time Rational Expressions: Multiply and Divide Worksheet Conclusion Homework: Finish Classwork AND to (pg 411)

Multiplying and Dividing Fractions
Multiply Numerators Multiply Denominators Divide: Multiply by the reciprocal (flip) Remember to Simplify!

Example 1 Simplify the following expressions:

Example 2 Simplify: Half the work is done. It is already factored.
Combine the fractions by multiplying Cancel common factors Rewrite any factors if they are raised to a power

Example 3 Simplify: Almost the same as x – 5
Can NOT cancel since its not in factored form. Simplify: Multiply by the reciprocal (Flip the fraction) Almost the same as x – 5 Factor Make sure to Factor Completely TRICK: Factor out -1 to make it the same. Cancel common factors

Day 61: December 6th Objective: Assess Chapters 8 and 9 in an individual setting. Homework Check Chapters 8-9 Individual Test Conclusion Homework: to (pg 417)

Day 62: December 7th Objective: Learn how to solve complicated equations (ones with large numbers, fractions, or decimals) by writing and solving simpler equivalent equation. THEN Continue to learn how to solve complicated linear and quadratic equations that involve fractions by rewriting and solving an equivalent equation. Homework Check 10-24 to (pgs ) Wells Time 10-34 to (pgs ) Conclusion Homework: (pg 417) AND to (pg 420)

Solve an Equation by Rewriting
Solving this quadratic is bothersome because factoring and the quadratic formula is tedious due to the decimals. Multiply every term by 2 to cancel the decimals. Using the quadratic formula or trying to factor to solve the equation is easier for this equation.

Solving this quadratic is bothersome because factoring and the quadratic formula is tedious due to the large numbers. Divide every term by the GCF (3000) to make the numbers smaller Using the quadratic formula or trying to factor to solve the equation is easier for this equation.

Solving this equation is bothersome because of the fractions. Multiply every term by a common denominator. Solving this equation with our old techniques is easier.

Solving this equation is bothersome because of the fractions. Factor to find a common denominator. Multiply every term by a common denominator. Solving this equation with our old techniques is easier.

7.1: Getting to Know You Each student gets a card: either a table, graph, rule, or situation Consider what you know about the card Find other students.

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7.1: Getting to Know You Each student gets a card: either a table, graph, rule, or situation Consider what you know about the card Find other students.

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Presentation on theme: "7.1: Getting to Know You Each student gets a card: either a table, graph, rule, or situation Consider what you know about the card Find other students."— Presentation transcript:

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