Algebra 2B - Chapter 6 Quadratic Functions

6-1: Graphing Quadratic Functions Learning Targets: I can graph quadratics using five points. I can find max. and min. values.

TermPicture/FormulaIn your own words: Quadratic Function Standard Form: Parabola Vertex Max/Min x-coordinate of vertex Axis of symmetry y-intercept Graph of a quadratic function: This is where the graph makes its turn. Plug x into the function to find y.

Using the TI to Graph Know these function keys on your calculator: y = set window table max and min

Example 1: Graph f(x) = 3x 2 - 6x + 7 Axis of symmetry: y-intercept: Direction of opening: Up / down? Maximum / Minimum? Value: _____________ Vertex: x f(x) “a” value? 1 4(1, 4)

Example 2: Graph f(x) = - x 2 + 6x - 4 Axis of symmetry: y-intercept: Direction of opening: Up / down? Maximum / Minimum? Value: _____________ Vertex: x f(x)

Lesson 6.1: Closing H.W. Practice 6.1 Can you graph quadratics using five points and/or your calculator? Can you find max. and min. values?

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ALGEBRA 2B - CHAPTER 6 LESSON 6.6 Analyzing graphs of Quadratic Functions

LESSON 6.6 LEARNING TARGETS: I can graph quadratic functions in vertex form. I can write quadratic functions in standard and/or vertex form.

TermPicture/FormulaIn your own words: Quadratic Function Vertex Form: Vertex Max/Min (h, k) Max = a > 0, min = a < 0 x-coordinate of vertex Axis of symmetry The “h” value. Standard Form Vertex Form

Exploration Activity

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Solving Quadratic Equations by factoring

Lesson 6.3 Objectives:  Solve quadratic equations by factoring  Write a quadratic equation with given its roots

Any trinomial: Example: Perfect square trinomial: Example: Difference of two squares: Example: Hint: always check for GCF first! 5x 2 -13x + 6X – Box Method x x + 36 y 4 – z 2

Vocabulary Zero Product Property: For any real numbers a and b, if ab = 0, then either a = 0 or b = 0, or both a and b equal zero. To write a quadratic equation with roots p and q: Write the pattern: (x – p)(x – q) = 0

Example 1: Solve by factoring 3x 2 = 15x 1.Set the equation equal to zero. 2.Factor out the GCF if possible. 3.Use another factoring method if possible. 4.Set the factor/s equal to zero. 5.Solve.

Example 2: Solve by factoring 4x 2 - 5x = 21 1.Set the equation equal to zero. 2.Factor out the GCF if possible. 3.Use another factoring method if possible. 4.Set the factor/s equal to zero. 5.Solve.

Write the quadratic equation given its roots Example 3: Write a quadratic equation with roots 3 and -5. Remember…(x – r)(x – p) = 0

Write the quadratic equation given its roots Example 4: Write a quadratic equation with roots -7/8 and 1/3.

Your Turn 1: Solve by factoring 5x x – 12 = 0 1.Set the equation equal to zero. 2.Factor out the GCF if possible. 3.Use another factoring method if possible. 4.Set the factor/s equal to zero. 5.Solve.

Your Turn 2: Solve by factoring 12x 2 - 8x + 1 = 0 1.Set the equation equal to zero. 2.Factor out the GCF if possible. 3.Use another factoring method if possible. 4.Set the factor/s equal to zero. 5.Solve.

Your Turn 3: Write a quadratic equation with roots -5.

Your Turn 4: Write a quadratic equation with roots -4/9 and -1

Lesson 6.3: Assessment Check for Understanding: Closure 6.3 Homework: Practice 6.3

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Algebra 2B - Chapter 6 Lesson 6.5 Solving Quadratic Equations by using the QuadraticFormula

Lesson 6.5 Learning Targets: I can:  Solve quadratic equations by using the Quadratic Formula  Use the discriminant to determine the number and types of roots

Vocabulary Quadratic Equation: Quadratic Formula: Discriminant: Two rational roots: Two irrational roots: __________________ ____________________ One rational root: Two complex roots _________________ __________________

Example 1: 2x 2 + 5x + 3 = 0 a = ___ b = ___ c = ___ Discriminant: _________________________ Number & type of roots: ________________

Example 2: 4x x + 29 = 0 a = ___ b = ___ c = ___ Discriminant: _________________________ Number & type of roots: ________________

Example 3: 25 x x = – 16 a = ___ b = ___ c = ___ Discriminant: _________________________ Number & type of roots: ________________

Example 4: x 2 - 8x = -14 a = ___ b = ___ c = ___ Discriminant: _________________________ Number & type of roots: ________________

Your Turn 1: x 2 + 2x - 35 = 0 a = ___ b = ___ c = ___ Discriminant: _________________________ Number & type of roots: ________________

Your Turn 2: x 2 - 6x + 21 = 0 a = ___ b = ___ c = ___ Discriminant: _________________________ Number & type of roots: ________________

Your Turn 3: 3x 2 +5x = 2 a = ___ b = ___ c = ___ Discriminant: _________________________ Number & type of roots: ________________

Your Turn 4: x x + 24 = 0 a = ___ b = ___ c = ___ Discriminant: _________________________ Number & type of roots: ________________

Lesson 6.5: Assessment Check for Understanding: __________________________________ Homework: ____6.5 Skills Practice________

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I can solve quadratic equations by graphing. (exact roots) I can estimate solutions of quadratic equations by graphing. (approximate roots)

TermPicture/FormulaIn your own words: Quadratic Equation Zeros Roots Vocabulary Cases: two real roots one real root no real roots, Ø x- intercepts of the graph solutions of the quadratic equation

Example: Not a point but the 2 places where the graph crosses the x-axis. This is called a double root. -3 is the answer twice.

Part 1 : Exact roots Example 1: Solve x 2 + x – 6 = 0 by graphing. Vertex: ________ x f(x) Exact Roots of the equation (or zeros of the function): ________ & _______ Old School or Vintage style:

Part 1 : Exact roots Your Turn 1: Solve x 2 - 4x – 5 = 0 by graphing. Vertex: ________ x f(x) Exact Roots of the equation (or zeros of the function): ________ & _______

Part 2 : Approximate roots Example 2: Solve x 2 - 2x – 2 = 0 by graphing. Vertex: ________ x f(x) Approximate roots: _____________________________

Part 2 : Approximate roots Your Turn 2: Solve x 2 - 4x + 4 = 0 by graphing. Vertex: ________ x f(x) Approximate roots: _____________________________

Use your Graphing Calculator to solve the following problems: Word Problem 1: An object is propelled upward from the top of a 500 foot building. The path that the object takes as it falls to the ground can be modeled by h = -16t t where t is the time (in seconds) and h is the corresponding height of the object. The velocity of the object is v = -32t +100 where t is seconds and v is velocity of the object y = How high does the object go? ___________________ When is the object 550 ft high? __________________ With what velocity does the object hit the ground? __________________

Word problem 2: An astronaut standing on the surface of the moon throws a rock into the air with an initial velocity of 27 feet per second. The astronaut’s hand is 6 feet above the surface of the moon. The height of the rock is given by h = -2.7t2 + 27t + 6 y = How many seconds is the rock in the air? How high did the rock go?

Lesson 6.2: Assessment Check for Understanding: Closure 6.2 & “The quadratic family activity” Homework: Practice 6.2

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