Topics 8: Quadratics

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

Introduction to Solving Quadratics

3 Methods to Solving Quadratics Factoring – Completing the Square – Box Method – Perfect Squares Graphing Quadratic Formula

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.

Convert from Vertex to Standard Form 1.Expand squared factor and multiply. 2.Distribute “a” 3.Combine Like Terms.

Let’s Practice… 1. y= 2(x + 3) 2 – 5 2. y = -3 (x – 4) y = 4 (x + 2) 2 + 5

Convert from Standard to Vertex Form

Let’s Practice… 1. y = x 2 – 8x y = 2x 2 + 8x y = 3x 2 – 6x + 15

Solving Quadratic Functions by Graphing

Parts of Quadratic Graph The botto m (or top) of the U is calle d the v erte x, or the turni ng point. Th e vert ex of a para bola open ing upw ard is also calle d the mini mum point. The vert ex of a para bola open ing dow nwa rd is also calle d the maxi mum point. The x- inter cept s are calle d the r oots, or the zero s. To find thex - inter cept s, set a x 2 + bx + c = 0. The ends of the grap h co ntinu e to posit ive infin ity (or nega tive infin ity) unle ss the dom ain (the x's to be grap hed) is othe rwis e spec ified.

Up or Down?!

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

Standard Form: Calculate Zeros 1.Factor 2.Set factors equal to zero and solve.

Standard Form: Calculate Vertex

Graph: y = x 2 - 6x + 5 Calculate the: Zeros: _________________ Vertex:________________ Is the vertex a min or max?

Graph: y = x 2 - 2x - 8 Calculate the: Zeros: _________________ Vertex:________________ Is the vertex a min or max?

Graph: y = x 2 + 5x + 4 Calculate the: Zeros: _________________ 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 + 3) Identify the: Vertex: _______________ Axis of Symmetry:________ Table: XY

Graph: y = (x – 2) Identify the: Vertex: _______________ Axis of Symmetry:________ Table: XY

Graph: y = -(x + 5) Identify the: Vertex: _______________ Axis of Symmetry:________ Table: XY

Transformations of Quadratic Functions

Parent Functions The general family/group that the basic graph belongs to. A function/graph’s last name.

Graphical Transformations 5 major types of graphical transformations: Reflection Compression Stretches Horizontal Shift Vertical Shift

Reflection If “a” is negative the graph flips over the x-axis

Compression If “a” is greater than 1, the graph is narrower.

Stretches If “a” is less than 1 (a fraction) the graph is wider.

Horizontal Shift If “b” is negative the graph shifts right. If “b” is positive the graph shifts left.

Vertical Shift If “c” is negative the graph shifts down. If “c” is positive the graph shifts up.

Identify and Graph y = (x + 1) Reflection: __________ Compression:___________ Stretch: ______________ Vertical: _____________ Horizontal: ___________

Identify and Graph y = -(x - 2) 2 Reflection: __________ Compression:___________ Stretch: ______________ Vertical: _____________ Horizontal: ___________

Identify and Graph y = -(x + 3) Reflection: __________ Compression:___________ Stretch: ______________ Vertical: _____________ Horizontal: ___________

Solving Quadratics using the Quadratic Formula

The Quadratic Formula 1. Identify a, b and c. 2. Plug in. 3. Simplify under radical. 4. Separate positive and negative solutions. 5. Solve.

Solve: Let’s Practice…

Solve: Let’s Practice…

Solve using the Quadratic Equation. 1. 2x 2 + 5x + 3 = x x – 84 = 0

You can use the discriminant to determine the number and type of solutions to the equation. The Discriminant

Find the discriminant and give the number of solutions of the equation.

Graphing Polynomials

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.

Review: Standard Form of a Polynomial Function The standard form of a polynomial function arranges the terms by degree in descending numerical order.

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 7 – 5 2.5x 9 – 11x 2 + 8x x 4 + 3x 3 -2x 2

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. Down and Down Up and Up Down and Up Up and Down Even vs. Odd & End Behavior

What’s the Pattern?! All of these functions are EVEN.

What’s the Pattern?! All of these functions are ODD.

You can determine the end behavior of a polynomial function of degree n from the leading coefficient in the standard form. End Behavior of a Polynomial Function Leading Coefficient Leading Degree= Even Leading Degree =Odd Positive Negative

What is the end behavior of the graph?

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. Odd vs. Even & Turning Points

Degree: _______ Turning Points: ________ Odd vs. Even & Turning Points Degree: _______ Turning Points: ________ Degree: _______ Turning Points: ________ Degree: _______ Turning Points: ________

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?