Connecting Everything

Warm Up Factor the following 𝑓 𝑥 = 𝑥 2 −10𝑥+16 𝑓 𝑥 = 𝑥 2 −𝑥−30 State the vertex of each parabola: 𝑓 𝑥 = (𝑥−12) 2 −10 𝑓 𝑥 = −(𝑥+17) 2 𝑓 𝑥 = 3𝑥 2 −28

What we know How to represent a quadratic in: Vertex form:𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘 Standard form: 𝑓 𝑥 =𝑎𝑥 2 +𝑏𝑥+𝑐 Factored form:𝑓 𝑥 = 𝑥+𝑎 (𝑥+𝑏) How to graph each of these using a table or graphing calculator

Learning Targets Want to be able to represent one function in each of its different forms: Vertex Standard Factored Describe the unique features of the graph that each form tells us.

How it figures into our ‘web’ Vertex Form Standard Form Equation Factored Form Table Graph Context

Who cares which one I use?! *Each quadratic representation has specific characteristics that are unique to its graph, table or equation. Let’s use the example of: 𝑓 𝑥 =𝑥 2 +4𝑥+3

Standard Form 𝑓(𝑥)=𝑥 2 +4𝑥+3 The Standard Form includes the y-intercept, i.e. where the graph intersects the y-axis x f(x) -4 3 -3 -2 -1

Factored Form 𝑓 𝑥 =𝑥 2 +4𝑥+3=(𝑥+1)(𝑥+3) x f(x) -4 3 -3 -2 -1 The Factored Form tells us where our roots are going to be located by using the zero product property (roots are where the output is zero). x f(x) -4 3 -3 -2 -1

Factored Form 𝑓(−2)=(−2+1)(−2+3) 𝑓 −2 =−1 The Factored Form of the quadratic can also tell us what the Vertex Form is. We can substitute this X value and evaluate what our Y value should be. 𝑓 −2 =−1

Vertex Form 𝑥 2 +4𝑥+3= 1 𝑥+2 2 −1 x f(x) -4 3 -3 -2 -1 𝑥 2 +4𝑥+3= 1 𝑥+2 2 −1 The Vertex Form tells us where the vertex is going to be located, if it is compressed/stretched and which direction it is pointing. x f(x) -4 3 -3 -2 -1

Summary Each form tells us something different about our equation Standard Form: y-intercept Factored Form: roots (where the graph crosses the x-axis) Vertex Form: vertex point, direction and compression/stretch factor

You try… Given the graph below, state the vertex, factored and standard forms of the quadratic 𝑥−1 𝑥−2 = 𝑥 2 −3x+2= 𝑥−1.5 2 +.25

You try… Given the table below, state the vertex, factored and standard forms of the quadratic x f(x) -2 -1 -3 -4 1 2 x f(x) -2 -1 -3 -4 1 2 𝑥+2 𝑥−2 = 𝑥 2 −4= 𝑥 2 −4

COMPLETE THE SQUARE Wait a second But how do I go from Standard Form to Vertex Form if I don’t have a table or a graph? We can use a technique called… COMPLETE THE SQUARE

Complete The Square This technique allows us to make a square out of the Standard form so that we can find our ℎ and 𝑘 in the vertex form The vertex form always contains a perfect square: 𝑥−ℎ 2 This means it is two of the same quantities being multiplied by each other

𝑥−ℎ 2 in Algebra Tiles Not a Perfect Square 𝑥 ℎ ℎ 𝑥 𝑥 2 ℎ𝑥 ℎ𝑥 ℎ ℎ𝑥 ℎ 2

𝑥−ℎ 2 in Algebra Tiles Perfect Square 𝑥 ℎ 𝑥 𝑥 2 ℎ𝑥 ℎ ℎ𝑥 ℎ 2 Our goal in Completing the Square is to take any quadratic and find out what its “ 𝑥−ℎ 2 ” is and then write it using our vertex form.

For Next Class How many of you have used Algebra Tiles to factor or multiply polynomials? Here is a brief demo: Video on Algebra Tiles We only need to watch the first 2:54 to understand how they work

Homework Worksheet #2