Quadratic Functions (Graphing)

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Presentation transcript:

Quadratic Functions (Graphing) Algebra 2A – Unit 2 Quadratic Functions (Graphing)

2-1: Graphing Quadratic Functions Learning Targets: I can graph quadratics using five points. I can find and interpret maximum and minimum values.

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

Using the TI to Graph Know these function keys on your calculator: y = set window table max and min Choose some values around the axis of symmetry, so around the x=2 point

Example 1: Graph f(x) = 3x2 - 6x + 7 Axis of symmetry:     y-intercept:   Direction of opening: Up / down? “a” value?   Maximum / Minimum? Value: _____________ Minimum of 4 @ x=1 Now plot the points:       x f(x) -1 1 2 3 Vertex: (1, 4) 16 7 4 7 16

Example 2: Graph f(x) = - x 2 + 6x - 4 Now plot the points: Axis of symmetry:     y-intercept:   Direction of opening: Up / down? “a” value?   Maximum / Minimum? Value: _____________ Maximum of 5 @ x=3 x f(x) 1 2 3 4 5 Vertex: (3, 5) 1 4 5 4 1      

Lesson 2.1: Closing H.W. Practice 2.1 Can you graph quadratics using five points and/or your calculator? Can you find max. and min. values? H.W. Practice 2.1

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Analyzing graphs of Quadratic Functions Algebra 2A – Unit 2 Lesson 2.2 Analyzing graphs of Quadratic Functions

Lesson 2.2 Learning Targets: I can graph quadratic functions in vertex form. I can write quadratic functions in standard and/or vertex form.

Picture/Formula In your own words: Vertex Form Standard Form Term Picture/Formula In your own words: Quadratic Function Vertex Form: Vertex Max/Min x-coordinate of vertex Axis of symmetry           Vertex Form Standard Form

Exploration Activity

Example 1: Graph f(x) = (x – 3)2 – 2 Now plot the points: Direction of opening: Up / down ? Vertex: ________ Max / Min ? Axis of symmetry: ___________         x f(x)                    

Example 2: Graph f(x) = ½ (x + 5)2 + 3 Direction of opening: Up / down ? Vertex: ________ Max / Min ? Axis of symmetry: ___________         x f(x)                    

Your Turn 1: Graph f(x) = 4(x + 3)2 – 2 Direction of opening: Up / down ? Vertex: ________ Max / Min ? Axis of symmetry: ___________         x f(x)                    

Your Turn 2: Graph f(x) = -2 (x + 1)2 + 4 Direction of opening: Up / down ? Vertex: ________ Max / Min ? Axis of symmetry: ___________         x f(x)                    

Example 3: Plug in values for h, k and x, y to find a.     Plug in values for h, k and x, y to find a. Rewriting the equation in standard form. Now look at this: -20 = a(0 – 4)2 - 36 Next, solve for a. -20 = 16a - 36 y = 1(x – 4)2 - 36 16a = 16 y = 1(x– 4)(x – 4) - 36 a = 1 y = 1(x2 -8x +16)- 36 Finally, rewrite the equation using a, h and k. y = x2 -8x - 20 y = 1(x – 4)2 - 36 Same parabola in different forms.

Example 4: Plug in values for h, k and x, y to find a. 2 = a(0+3)2 - 1       Plug in values for h, k and x, y to find a. 2 = a(0+3)2 - 1 Next, solve for a. 2 = 9a - 1 3 = 9a   Finally, rewrite the equation using a, h and k.  

Your Turn 3: Plug in values for h, k and x, y to find a.       Plug in values for h, k and x, y to find a. 0 = a(2 - 3)2 +- 1 Next, solve for a. 0= a - 1 1 = a Finally, rewrite the equation using a, h and k. y = 1(x - 3)2 - 1

How high does the object go? ___________________ 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 = -16t2 +100t + 500 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? __________________                

How many seconds is the rock in the air? Use your Graphing Calculator to solve the following problems: 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 2.2: Assignment Check for Understanding: Closure Assignment: Practice 2.2

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Solving Quadratic Equations by graphing Algebra 2A – Unit 2 Lesson 2.3 Solving Quadratic Equations by graphing

Lesson 2.3 Learning Targets: I can solve quadratic equations by graphing. (exact roots) I can estimate solutions of quadratic equations by graphing. (approximate roots)

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

Example: Not a point but the 2 places where       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: Old School or Vintage style:       Solve x2 + x – 6 = 0 by graphing.         Vertex: ________ -3 x f(x) -2 -1   1 2 -4 -6   -6 -4   Exact Roots of the equation (or zeros of the function): ________ & _______ -3 2  

Part 1 : Exact roots Your Turn 1: Old School or Vintage style:         Solve x2 - 4x – 5 = 0 by graphing. Vertex: ________   -1 x f(x) 1 2 3 4 5 -5 -8 -9 -8 -5   Exact Roots of the equation (or zeros of the function): ________ & _______ -1 5  

Part 2 : Approximate roots Example 2: Use your Graphing Calculator to solve the following problems: Part 2 : Approximate roots Example 2:     Solve x2 - 2x – 2 = 0 by graphing. Look under the table in the calculator. Vertex: ________   x f(x) 2 -2 -1 3 1 -3   Approximate roots: _____________________________    

Part 2 : Approximate roots Your Turn 2: Use your Graphing Calculator to solve the following problems: Part 2 : Approximate roots Your Turn 2: Solve x2 - 4x + 4 = 0 by graphing. Vertex: ________   x f(x) 1 3 4 2 Double Root of x=2   Approximate roots: _____________________________

Their sum is 4, and their product is -12.                     Use your Graphing Calculator to graph and find the roots:  

Lesson 2.3: Assignment Check for Understanding: Closure Assignment: Practice 2.3

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