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F. Everything Quadratics

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F. Everything Quadratics
Math 20: Foundations FM20.9 Demonstrate an understanding of the characteristics of quadratic functions of the form y = a(x - p)² + q , including: vertex intercepts domain and range axis of symmetry.

Getting Started String Art p.356

What DO YOU Think? P.357

1. What is a Quadratic? FM20.9 Demonstrate an understanding of the characteristics of quadratic functions of the form y = a(x - p)² + q , including: vertex intercepts domain and range axis of symmetry.

1. What is a Quadratic? Quadratic Relation – A relation that can be written in the standard form𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐, where 𝑎≠0; for example, 𝑦=4 𝑥 2 +2𝑥+1 Parabola – The shape of the graph of any quadratic relation.

Explore the Math p.359 Grab your graphing Calculators!! How does changing the coefficients and constant in a relation that is written in the form 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐affect the graph of the relation?

Summary p.359

Practice Ex. 7.1 (p.360) #1-6

2. Properties of Quadratic Graphs
FM20.9 Demonstrate an understanding of the characteristics of quadratic functions of the form y = a(x - p)² + q , including: vertex intercepts domain and range axis of symmetry.

Reflections p.362

Vertex - The point at which the quadratic function reaches its maximum or minimum value.
Axis of Symmetry - A line that separates a 2-D figure into two identical parts. For example, a parabola has a vertical axis of symmetry passing through its vertex.

Example 1

Example 2

Does this last Function have a max or min value?

Example 3

Summary p.368

Practice Ex. 7.2 (p.368) #1-16 #4-19

3. Graphing to Solve Quadratic Equations
FM20.9 Demonstrate an understanding of the characteristics of quadratic functions of the form y = a(x - p)² + q , including: vertex intercepts domain and range axis of symmetry.

3. Graphing to Solve Quadratic Equations
A zero is a number that when subbed in for the x variable it makes the equation equal to zero A zero is another name for an x-intercept

Investigate the Math p.373

Example 1

Example 2

Is it possible for a Quad Equation to have more than 2 roots?

Example 3

Summary p.379

Practice Ex. 6.3 (p.379) #1-13 #5-15

4. Quadratics in Factored Form
FM20.9 Demonstrate an understanding of the characteristics of quadratic functions of the form y = a(x - p)² + q , including: vertex intercepts domain and range axis of symmetry.

Investigate the Math p.382

To find your x-intercepts for your quadratic you can factor the function then set each part equal to zero and solve for x. You can then also average your x-intercepts together to get your axis of symmetry

Example 1

Example 2

Example 3

Example 4

Summary p.390

Practice Ex. 7.4 (p.391) #1-16 #4-20

5. Solving Quadratics by Factoring
FM20.9 Demonstrate an understanding of the characteristics of quadratic functions of the form y = a(x - p)² + q , including: vertex intercepts domain and range axis of symmetry.

5. Solving Quadratics by Factoring

When you rewrite your Quadratic equation in Standard form (=0) you can factor the equation to easily find your zeros (x- intercepts)

How can you verify that your answers are correct?
If I gave you the roots (x-intercepts) 2 and 6 can you give me the quadratic equation in standard form? Can all quadratic equations be solved by factoring?

Example 1

Example 2

When your quadratic function is in factored from how can you tell how many roots there will be 2 or 1?

Example 3

Example 4

Summary p.405

Practice Ex. 7.5 (p.405) #1-15 #3-18

6. Vertex Form FM20.9 Demonstrate an understanding of the characteristics of quadratic functions of the form y = a(x - p)² + q , including: vertex intercepts domain and range axis of symmetry.

6. Vertex Form Vertex Form of a Quadratic Function 𝑦=𝑎 (𝑥−ℎ) 2 +𝑘

Investigate the Math p.408 Read problem then go straight to the questions

Example 1

Example 2

Example 3

Example 4

Summary p.416

Practice Ex. 7.6 (p.417) #1-14 #4-19

7. The Quadratic Formula FM20.9
Demonstrate an understanding of the characteristics of quadratic functions of the form y = a(x - p)² + q , including: vertex intercepts domain and range axis of symmetry.

7. The Quadratic Formula When we want to solve a quadratic function, find the roots, zeros or x-intercepts but we can not factor the function we use the Quadratic Formula The Quadratic Formula solve for x when we can not factor

Quadratic Formula: 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 When 𝑎 𝑥 2 +𝑏𝑥+𝑐=0

Example 1

Example 2

Example 3

Example 4

Summary p.427

Practice Ex. 7.7 (p.427) #1-11 #4-13

8. Solving Problems FM20.9 Demonstrate an understanding of the characteristics of quadratic functions of the form y = a(x - p)² + q , including: vertex intercepts domain and range axis of symmetry.

8. Solving Problems

Example 2

Example 3

Example 4

Summary p.436

Practice Ex. 7.8 (p.436) #1-8 #2-12

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