The simplest quadratic function we can have is y = x 2, sketched below.

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1 Name: LESSON 6-8 COMPLETING THE SQUARE AND SHIFTING PARABOLAS COMMON CORE ALGEBRA II Date: Parabolas, and graphs more generall, can be moved horizontall and verticall b simple manipulations of their equations. This is known as shifting or translating a graph. You worked with this extensivel in Common Core Algebra I. The first exercise will review how to use a method known as completing the square to identif shifts and the turning point of a parabola. SHIFTING GRAPHS OF QUADRATIC FUNCTIONS The simplest quadratic function we can have is = x, sketched below. The vertex of this parabola is the origin (0, 0) We can easil shift the vertex, in an direction, b adding/subtracting a constant to the function = x. = (x h) + k h represents a shift : Vertex Form of a Quadratic h means shifted to the & +h means is shifted to the k represents a shift : k means shifted & +k means is shifted Most quadratics are stated in standard form, = ax + bx + c, and have to be manipulated to get it into vertex form. We will use a technique called Completing the Square to convert to vertex form. HOW TO COMPLETE THE SQUARE 1. Get all our x-terms on one side of the equation and everthing else on the other side.. Make sure ou have a 1x to get started. If not, GCF the leading coefficient of x ONLY! 3. Half the coefficient of the x term, Square it, and Add it to both sides of the equation. Be careful!! If ou had to GCF to get 1x, that multiplies the amount ou are adding to both sides of the equation. 4. Factor the x-side of our equation. It should alwas look like, + = (x + half) 5. Get alone.

2 Exercise #1: The function equation is x is shown alread graphed on the grid below. Consider the quadratic whose (a) Using the method of completing the square, write this equation in the form x h k. x (b) Describe how the graph of x would be shifted to produce the graph of (c) Sketch the graph of 8 18 b using its vertex form in (a). What are the coordinates of its turning point (vertex)? Exercise #: Using our calculator and the window shown below, sketch the graphs of the simple quadratics 1 x, 3 x, and x. Ever quadratic of the form ax has a turning point at: x

3 The algorithm of completing the square works best when a 1 and b is even in the form does work in ever case, even the mess ones. ax bx c. But, it Exercise #3: Place each of the following quadratic functions in vertex form and identif the turning point. (a) 3 1 (b) x x 6 1 Exercise #4: The method of completing the square can be performed on the standard quadratic equation ax bx c and after much manipulation can be placed in the form: (a) Based on this formula, what is the x-coordinate of the turning point of an parabola? Be careful. b b a x c a 4a (b) Use this formula to find the turning point of the parabola 10. (c) Verif our answer from part (a) b placing the quadratic 10 into vertex form. (d) Verif both answers b examining a table on our calculator using the original equation.

4 b Exercise #5: Use the formula x to find the turning points for each of the following quadratic functions. a 1 (a) f x x 1x 7 (b) g x x 5x 0 4

5 Name: FLUENCY HOMEWORK 6-8 COMPLETING THE SQUARE AND SHIFTING PARABOLAS COMMON CORE ALGEBRA II 1. Which of the following equations would result from shifting (1) x 5 4 (3) x 4 5 () x 5 4 (4) x 4 5 Date: x five units right and four units up?. Which of the following represents the turning point of the parabola whose equation is (1) 3, 7 (3) 7, 3 () 3, 7 (4) 3, 7 x 3 7? 3. Which of the following quadratic functions would have a turning point at 6,? (1) x 6 (3) x 5 6 () 3 x (4) x Which of the following is turning point of 1 4? (1) 1, 4 (3) 6,104 () 6, 40 (4) 4,1 5. In vertex form, the parabola 10 8 would be written as (1) x 5 33 (3) x 10 9 () x 5 17 (4) x

6 6. The turning point of the parabola 5 is (1).5,1.75 (3).5, 8.5 () 5, 10.5 (4).5, Write each of the following quadratic functions in its vertex form b completing the square. Then, identif its turning point. (a) 1 50 (b) x x b 8. Use the formula x to find the turning points of each of the following quadratic functions. Then, a place the function in vertex form to verif the turning points. (a) (b) x x 4 67

7 9. Consider the quadratic function whose equation is x 6x 40. (a) Determine the -intercept of this function algebraicall. (b) Write the function in its vertex form. State the coordinates of its turning point. (c) Algebraicall find the zeroes of the function using the zero product law. (d) Sketch a graph of the parabola, showing all relevant features found in parts (a) through (c).

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