1. Connecting Everything

2. 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

3. 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

4. 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.

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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−

13. 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

14. 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

15. 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

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

17. 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.

18. 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

19. Homework
Worksheet #2