Topics 8: Quadratics. Introduction to Solving Quadratics. Table of Contents. Two Formats of Quadratics. Convert from Vertex to

Transcription

1 Topics 8: Quadratics 3 Methods to Solving Quadratics Factoring Completing the Square Box Method Perfect Squares Graphing Quadratic Formula Table of Contents 1. Introduction to Solving Quadratics 2. Solving Quadratic Functions by Graphing 3. Transformation of Quadratic Functions 4. Solving Quadratics using the Quadratic Formula 5. Graphing Polynomials Two Formats of Quadratics Standard Form y = ax 2 + bx + c Vertex Form y= a(x h) 2 + k We can re arrange standard form to get vertex form. Vertex form is related to completing the square. Introduction to Solving Quadratics Convert from Vertex to Standard Form 1. Expand squared factor and multiply. 2. Distribute a 3. Combine Like Terms. 1

2 1. y= 2(x + 3) 2 5 Let s Practice 1. y = x 2 8x + 15 Let s Practice 2. y = 3 (x 4) y = 2x 2 + 8x y = 4 (x + 2) y = 3x 2 6x + 15 Convert from Standard to Vertex Form Solving Quadratic Functions by Graphing Convert from Standard to Vertex Form 1. Move c to left. 2. Complete the square on the right using ( )2 3. Add value from step 2 to both sides. 4. Divide by GCF and factor right. 5. Simplify to get y alone. Parts of Quadratic Graph The b o tto m (or top) of U is calle d v erte x, or tu rn i ng point.th e vert ex of a para bola open ing upw ard is also calle d mini mum point. The vert ex of a para bola open ing dow nwa rd is also calle d maxi mum point. The x- in ter cept s are calle d r oots, or zero s. To find x - in ter cept s, set a x 2 + bx+ c = 0. The ends of grap hco n tin u e to posit iv e in fin ity (or nega tiv e in fin ity ) unle ss dom ain (the x's to be grap hed) is othe rwis e spec ified. 2

3 Up or Down?! Standard Form: Calculate Vertex 1. Find vertex x value using formula: 2. Plug in x into equation to solve for y. Methods of Graphing Quadratics 1. Plug in to create XY Table of Values 2. Standard Form: Calculate Vertex & Zeros 3. Vertex From: Identify Vertex & Zeros Calculate the: Graph: y = x 2 6x + 5 Zeros: Vertex: Is the vertex a min or max? Standard Form: Calculate Zeros 1. Factor 2. Set factors equal to zero and solve. Calculate the: Graph: y = x 2 2x 8 Zeros: Vertex: Is the vertex a min or max? 3

4 Calculate the: Graph: y = x 2 + 5x + 4 Zeros: Graph: y = (x 2) Identify the: Vertex: Axis of Symmetry: Table: X Y Vertex: Is the vertex a min or max? Graphing in Vertex From 1. Identify and graph vertex. Remember: y = a (x h) 2 + k Coordinate = (h, k) 2. Determine axis of symmetry. (Hint: it s h!) 3. Pick 4 values to plug in. Two must be less than h and two must be greater than h. Zero is a great option! Graph: y = (x + 5) Identify the: Vertex: Axis of Symmetry: Table: X Y Graph: y = (x + 3) Identify the: Vertex: Axis of Symmetry: Table: X Y Transformations of Quadratic Functions 4

5 Parent Functions The general family/group that the basic graph belongs to. A function/graph s last name. Compression If a is greater than 1, the graph is narrower. Graphical Transformations 5 major types of graphical transformations: Reflection Compression Stretches Horizontal Shift Vertical Shift Stretches If a is less than 1 (a fraction) the graph is wider. Reflection If a is negative the graph flips over the x axis Horizontal Shift If b is negative the graph shifts right. If b is positive the graph shifts left. 5

6 Vertical Shift If c is negative the graph shifts down. If c is positive the graph shifts up. Identify and Graph y = (x + 3) 2 6 Reflection: Compression: Stretch: Vertical: Horizontal: Identify and Graph y = (x + 1) Reflection: Compression: Stretch: Vertical: Horizontal: Identify and Graph y = (x 2) 2 Reflection: Compression: Stretch: Vertical: Horizontal: Solving Quadratics using the Quadratic Formula 6

7 The Quadratic Formula Let s Practice Solve using the Quadratic Equation. 1. 2x 2 + 5x + 3 = 0 1. Identify a, b and c. 2. Plug in. 3. Simplify under radical. 4. Separate positive and negative solutions. 5. Solve. 2. 5x x 84 = 0 Solve: Let s Practice The Discriminant You can use the discriminant to determine the number and type of solutions to the equation. Solve: Let s Practice 7

8 Find the discriminant and give the number of solutions of the equation. Review: Standard Form of a Polynomial Function The standard form of a polynomial function arranges the terms by degree in descending numerical order. Graphing Polynomials Even vs. Odd Function Even function:highest degree is an even number Odd function: highest degree is an odd number Even or Odd? 1. 4x 3 + 2x x 9 11x 2 + 8x x 4 + 3x 3 2x 2 Essential Understanding A polynomial function has distinguishing behaviors. You can look at its algebraic form and know something about its graph. You can look at its graph and know something about its algebraic form. Even vs. Odd & End Behavior The degree of a polynomial function effects the end behavior, or the directions of the graph to the far left and to the far right. Up and Up Down and Up Down and Down Up and Down 8

9 What s the Pattern?! What is the end behavior of the graph? All of these functions are EVEN. What s the Pattern?! All of these functions are ODD. Odd vs. Even & Turning Points The graph of a polynomial function of degree n has at most n 1 turning points. The graph of a polynomial function of odd degree has an even number of turning points. The graph of a polynomial function of even degree has an odd number of turning points. End Behavior of a Polynomial Function You can determine the end behavior of a polynomial function of degree n from the leading coefficient in the standard form. Leading Coefficient Positive Negative Leading Degree= Even Leading Degree =Odd Odd vs. Even & Turning Points Degree: Turning Points: Degree: Turning Points: Degree: Turning Points: Degree: Turning Points: 9

10 Zeros Like quadratic functions, polynomial functions have zeros. Zeros occur where the function crosses or touches the x axis. What function best represents the graph? 10