Solving Simple Quadratics 1.0 Topic: Solving Quadratics

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1 Ns Solving Simple Quadratics 1.0 Topic: Solving Quadratics Date: Objectives: SWBAT (Solving Simple Quadratics and Application dealing with Quadratics) Main Ideas: Assignment: Square Root Property If x 2 = N, then x = + N or x = N If N > 0, then there are 2 real solutions. If N = 0, then there is 1 real solution. If N < 0, then there are two imaginary solutions. x 2 = 25 x 2 = 0 x 2 = 81 x 2 81 = 0 x = 0 Difference and Sum of Squares 4x 2 25 = 0 9x = 0 (x + 9) 2 = 0 (2x 7) 2 = 9 Perfect Square Double Root (3x + 11) 2 = 5 (x + 7) 2 = 4

2 We are going to fence in a rectangular field and we know that for some reason we want the field to have an enclosed area of 75 ft 2. We also know that we want the width of the field to be 3 feet longer than the length of the field. What are the dimensions of the field? Jennifer hit a golf ball from the ground and it followed the projectile h(t) = 16t t, where t is the time in seconds, and h is the height of the ball. Find when it hits the ground again. Application? s A square and a rectangle have the same area. The length of the rectangle is five inches more than twice the length of the side of the square. The width of the rectangle is 6 inches less than the side of the square. Find the length of the side of the square. The product of two positive integers are 66. The larger integer is 5 more than the smaller integer. Find the two integers.

3 Ns Standard Form of Quadratic 1.1 Topic: Standard Form of a Quadratic Date: Objectives: SWBAT (Find all key features when given a Quadratic in Standard Form) Main Ideas: Assignment: Standard Form of a Quadratic Function: y = ax 2 + bx + c, where a 0 or f(x) = ax 2 + bx + c Standard Form of a Quadratic Parts or Variables: a = LC (leading coefficient) - ax 2 quadratic part b = middle terms coefficient - bx linear part c = constant (integer without a variable) c constant part Shape of Quadratic Graph is called a parabola: Key Features of a Quadratic Function: f(x) = x 2 (Parent) Vertex (h. k) or (c, d): how to find h = b and k = f(h) 2a Axis of Symmetry or Line of Symmetry (LoS): y-int: f(0) = a(0) 2 + b(0) + c f(0) = c y int: (0, c) Goes through vertex so is the VUX x = h or x = b 2a Plug 0 in for x or find f(0) Maximum or Minimum: Will be the Vertex but depends on if parabola is opening up or down x-int/roots/zeros/solutions: Ways to solve (Quadratic Formula) Equation Plug 0 in for y x 2 + 2x + 5 = 0 Discriminant Discriminant b 2 4ac Graph of Related Function 2 2 4(1)(5) = (1)(25) = 0 ( 7) 2 4(2)(2) = 33 Real Solutions 0 x-intercepts 1 x-intercept 2 x-intercepts 0 1 2

4 Find all the key features of x 2 + x 30 = y and then graph the quadratic. What is the LC? Finding Key Features What is the pattern compared to parent? What is the y int? What is the vertex? What is the LoS? How many real solutions are there? What are the x ints? Find all the key features of x 2 + x 30 = y and then graph the quadratic. What is the LC? What is the pattern compared to parent? Your Turn What is the y int? What is the vertex? What is the LoS? How many real solutions are there? What are the x ints?

5 Standard Form of Quadratic 1.1 For the following questions, note if the question is asking for the positive zero (root) coordinate, the y-intercept, the vertex x-coordinate, or the vertex y-coordinate. Then find what they are asking for. An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the object's height s at time t seconds after launch is s(t) = 4.9t t , wheres is in meters. a. How long until the object is at its highest point? b. How long is the object in the air? What are they asking for? c. When will the object hit the ground? d. How high did the object get? e. When will the object be at its highest point? f. How high is the object at its highest point? g. At what height was the object launched at?

6 Ns Topic: Factoring Polynomials Factoring Again 1.2 Date: Objectives: SWBAT (Factor different types of Polynomials) Main Ideas: Assignment: Factor Factor = Product Polynomials and Their Factors Quadratic (x 2 ) = (linear factor)(linear factor) or (linear factor) 2 x 2 + 7x + 12 = (x + 3)(x + 4) or x 2 + 6x + 9 = (x + 3) 2 Cubic (x 3 ) = (linear)(linear)(linear) or (linear) 3 or (linear)(quadratic) or (linear) 2 (linear) x 3 7x 2 + 7x + 15 = (x + 1)(x 3)(x 5) Quartic (x 4 ) = Quantic (x 5 ) = Perfect Square Trinomials Square-Double-Square 25x x + 81 #1) To check if this trinomial is a Perfect Square..find the square root of the first term (quadratic term) and the last term (constant term) 25x 2 = 81 = #2) If they both check out..find the product of both and double and see if it matches the middle term (5x)(9)2 = #3) Then fill in the one binomial factor that is a double root (squared) with the two values you got from step #1 (5x + 9) 2 100x 2 140x + 49 x 2 7x + 16 Your Turn

7 Square-Minus-Square 49x 2 25 Sum and Difference of Squares #1) To check if this binomial is a Diff of Squares..find the square root of the first term (quadratic term) and the last term (constant term) 49x 2 = 25 = #2) If they both work out, fill in your two linear binomial factors with one being a sum and one being a difference (they are conjugates of each other) Square-Plus-Square (7x + 5)(7x 5) 49x #1) To check if this binomial is a Sum of Squares..find the square root of the first term (quadratic term) and the last term (constant term) 49x 2 = 25 = #2) If they both work out, fill in your two linear binomial factors with one being a sum and one being a difference (they are conjugates of each other) (7x + 5i)(7x 5i) 196x 2 9 x 4 81 Your Turn x x x 6 16 x

8 Sum and Difference of Cubes Factoring Again 1.2 Cube-Minus-Cube x 3 27 #1) To check if this binomial is a Diff of Cubes..find the Cube root of the first term (cubic term) and the last term (constant term) 3 x 3 3 = 27 = #2) If they both work out, fill in your linear binomial factor and quadratic trinomial factor following the pattern below a 3 b 3 = (a b)(a 2 + ab + b 2 ) x 3 27 = (x 3)(x 2 + 3x + 9) Cube-Plus-Cube x #1) To check if this binomial is a Sum of Cubes..find the Cube root of the first term (cubic term) and the last term (constant term) 3 x 3 3 = 64 = #2) If they both work out, fill in your linear binomial factor and quadratic trinomial factor following the pattern below a 3 + b 3 = (a + b)(a 2 ab + b 2 ) x = (x + 4)(x 2 4x + 16) 8x x Your Turn x x 6 y 6 x 3 + 5x 2 4x 20 x 5 9x 3 + 8x 2 72 Grouping

9 Ns Factoring Again to find Roots 1.3 Topic: Factor Form to find Roots Date: Objectives: SWBAT (Use Factored Form to Find Roots of a Polynomial) Main Ideas: Assignment: Multiply out to see ending polynomial: f(x) = (x + 1)(x 2)(x + 5) 2 Standard Form vs. Factored Form What do you notice about the overall degree of your polynomial? (Hint: compare it to the degrees of factors) Find the y int for this polynomial function? Using Standard Form Using Factored Form f(x) = (x 7)(x + 2) 2 (x 3) 3 Finding Roots/X-int Use ZZP to find all the zeros to this function. If we were to multiply this out, what do you think will be the overall degree? Find the y int:

10 Finding Zeros/Solutions x 3 = 27 x 3 5x 2 4x = 20 18x 3 6x 2 60x = 0 x = 0 Your Turn x 4 + 8x = 0 x 6 5x = 0

11 Ns Completing the Again 1.4 Topic: Solving By Completing the Square Date: Objectives: SWBAT (Solve Quadratics by completing the square) Main Ideas: Assignment: y = 2x 2 8x + 11 #1) Move the constant to the back and put the x 2 and the x terms in parenthesis with a space at the end. y = (2x 2 8x + ) + 11 Completing the Square a Different Way #2) Factor out the LC from the two terms inside the parenthesis. y = 2(x 2 4x + ) + 11 #3) Complete the Square to make the polynomial inside parenthesis a perfect square. ( b 2 ) 2 = ( 4 2 ) 2 = 4 #4) Add this to the inside but you will subtract this to the constant and you will multiply by the LC you factored in #2. y = 2(x 2 4x + 4) (4) #5) Simplify. y = 2(x 2) y = 2(x 2) #6) If you want to solve/find zeros/find roots/find x-intercepts put zero in for y and solve Be Careful y = 2(x 2) What Form does this look like of a Quadratic?

12 y = 3x 2 12x + 8 y = x 2 + 2x 2 Your Turn f(x) = 3x x 11

13 Ns Topic: All Forms of Quadratics Switching Forms 1.5 Date: Objectives: SWBAT (Transform any form into any other form of Quadratics) Main Ideas: Assignment: y = (2x 3)(x 7) Standard to Factored Factored to Standard y = 2x 2 + 7x + 3 y = 3x 2 6x + 7 Standard to Vertex

14 y = 7(x 1) Vertex to Factored Form Vertex to Standard y = (x 2) 2 25 y = (3x 5)(x + 1) Factored to Vertex

15 Ns Analyzing Quadratics (Part 1) 1.6 Topic: Part 1 of Analyzing Quadratics Date: Objectives: SWBAT (Sketch Graphs of Quadratics from Vertex Form) Main Ideas: Assignment: Exploring What we Know y = x 2 or f(x) = x 2 What function is this? What is the name of the graphs shape? What is the Vertex of the parent? What form is it in? Vertex Factored Standard What is its LoS? What is its Domain? What is its Range? What is its y-int and x-ints? What is the pattern of the parent? What are some good reference points? What do you know about the other side with respect to the vertex of your reference points? f(x) = a(bx c) 2 + d f(x) = x and f(x) = x 2 2 f(x) = 2x 2 and f(x) = 1 2 x2 Changing the Parent Vertex: Does is change anything? Vertex: Does it change anything?

16 f(x) = (x 3) 2 and f(x) = (x + 3) 2 f(x) = x 2 Vertex: Does is change anything? Vertex: Does is change anything? Shifts: y = 2(x + 2) 2 5 Increasing/Decreasing Intervals: Reflections: +/ Function: Together Scaling: Vertex: LoS: y-int: x-int(s): End Behaviors: Left Algebraic Proof on y-int: Domain: Right Algebraic Proof on x-int(s): Range:

17 Analyzing Quadratics (Part 1) 1.6 f(x) = 3(x 4) 2 Shifts: Range: Reflections: Increasing/Decreasing Intervals: Scaling: +/ Function: Vertex: y-int: Algebraic Proof on y-int: Your Turn LoS: Domain: Shifts: x-int(s): End Behaviors: y = 1 3 (x 1)2 + 1 Range: Algebraic Proof on x-int(s): Reflections: Increasing/Decreasing Intervals: Scaling: +/ Function: Vertex: y-int: Algebraic Proof on y-int: LoS: x-int(s): Algebraic Proof on x-int(s): Domain: End Behaviors: One More What is the equation of the parabola that passes through the point (2, 1) and has a vertex of ( 1, 4)?

18 Ns Analyzing Quadratics (Part 2) 1.7 Topic: Part 2 of Analyzing Quadratics Date: Objectives: SWBAT (Sketch Graphs of Quadratics from Other Forms) Main Ideas: Assignment: y = 2x 2 16x 17 Work Space to Change to Vertex Form Different Form Vertex: y-int: x-int(s): Changes fromm Parent Function? f(x) = 3x x + 7 Work Space to Change to Vertex Form Your Turn Vertex: y-int: x-int(s): Changes from Parent Function?

19 Write a quadratic function for the given graph. Upper Level Given Roots Write a quadratic function, in vertex and standard form, which contains ( 4, 1) and vertex ( 2, 5). Write a quadratic function in standard form that contains (2, 1) and vertex (5, 8).