![The vertex form of a quadratic equation is used to
easily identify the vertex of the parabola. The general
vertex form is defined as .
Example:
Transform into .
𝑦 = 𝑥 2
− 6 𝑥 − 6
𝑦 =( 𝑥2
− 6 𝑥 ) − 6
𝑦 =( 𝑥
2
− 6 𝑥 + 9 ) − 6 − 9
𝑎
[ − 𝑏
2 ]
2
=1[− (− 6)
2 ]
2
= 9
𝑦 =( 𝑥
2
− 6 𝑥 + 9 ) − 15
𝒚 =( 𝒙 − 𝟑 )𝟐
− 𝟏𝟓](https://image.slidesharecdn.com/15-vertexform-241011232627-72921dc4/85/15-Vertex-Form-of-a-Quadratic-Function-12-320.jpg)
![Example:
Transform into .
𝑦 = 𝑥 2
− 6 𝑥 + 3
𝑦 =( 𝑥 2
− 6 𝑥 ) + 3
𝑦 =( 𝑥
2
− 6 𝑥 + 9 ) + 3 − 9
𝑎
[ − 𝑏
2 ]
2
=1[− (− 6)
2 ]
2
= 9
𝑦 =( 𝑥
2
− 6 𝑥 + 9 ) − 6
𝒚 =( 𝒙 − 𝟑 )𝟐
− 𝟔](https://image.slidesharecdn.com/15-vertexform-241011232627-72921dc4/85/15-Vertex-Form-of-a-Quadratic-Function-13-320.jpg)
![Example:
Transform into .
𝑦 =− 4 𝑥 2
− 24 𝑥 + 15
𝑦 =− 4 ( 𝑥 2
+ 6 𝑥 ) + 15
𝑦 =− 4 ( 𝑥
2
+6 𝑥 +9 )+3 − 36
𝑎
[−𝑏
2 ]
2
=− 4 [−(6)
2 ]
2
=− 4 (9 )=− 3 6
𝑦 =− 4 ( 𝑥
2
+6 𝑥 +9 ) − 33
𝒚 =− 𝟒 ( 𝒙 + 𝟑)𝟐
− 𝟑𝟑](https://image.slidesharecdn.com/15-vertexform-241011232627-72921dc4/85/15-Vertex-Form-of-a-Quadratic-Function-14-320.jpg)
![Example:
Transform into .
𝑦 = 𝑥 2
+ 8 𝑥 + 18
𝑦 =( 𝑥 2
+ 8 𝑥 ) + 18
𝑦 =( 𝑥
2
+8 𝑥 +16 ) +18 − 16
𝑎
[ − 𝑏
2 ]
2
=[ − (8 )
2 ]
2
= 16
𝑦 =( 𝑥
2
+ 8 𝑥 + 16 ) + 2
𝒚 =( 𝒙 + 𝟒 )𝟐
+ 𝟐](https://image.slidesharecdn.com/15-vertexform-241011232627-72921dc4/85/15-Vertex-Form-of-a-Quadratic-Function-15-320.jpg)
![Example:
Transform into .
𝑦 = 𝑥 2
− 2 𝑥 + 5
𝑦 =( 𝑥 2
− 2 𝑥 ) + 5
𝑦 =( 𝑥
2
− 2 𝑥 +1 ) + 5 − 1
𝑎
[ − 𝑏
2 ]
2
=[ − (− 2 )
2 ]
2
=1
𝑦 =( 𝑥
2
− 2 𝑥 + 1 ) + 4
𝒚 =( 𝒙 − 𝟏 )𝟐
+ 𝟒](https://image.slidesharecdn.com/15-vertexform-241011232627-72921dc4/85/15-Vertex-Form-of-a-Quadratic-Function-16-320.jpg)
15 - Vertex Form of a Quadratic Function
- 2. Pre-Activity: Tell whether each picture models a quadratic function or not. Justify your answer. QUADRATIC FUNCTION
- 3. Pre-Activity: Tell whether each picture models a quadratic function or not. Justify your answer. QUADRATIC FUNCTION
- 4. Pre-Activity: Tell whether each picture models a quadratic function or not. Justify your answer. QUADRATIC FUNCTION
- 5. The concepts of quadratic function is very useful in our life if you know further about it. You can solve different problems that involves quadratic function in real-life situations such as building structures, computing the maximum height or minimum point of an object my reach, analyzing the movement of an object, and etc. Here are some examples of situations that models quadratic functions in real-life situations: 1. Targets an object in upward direction 2. Throwing an object downward 3. Shooting ball vertically upward 4. Minimum point submarine to submerge 5. Launching rocket to its maximum point
- 6. A quadratic function is a second degree polynomial represented as or , where and a, b and c are real numbers. The graph of a quadratic function is a Parabola. It has different properties including the vertex, axis of symmetry, opening of the parabola, and the intercepts.
- 8. Is a quadratic function or not? is a quadratic function since its highest degree is 2 and all numerical coefficients are real numbers. Is a quadratic function or not? is a not quadratic function since its highest degree is 1. Is a quadratic function or not? is a not quadratic function because if you will expand the right side, its highest degree will be 4.
- 9. Consider the graph of the quadratic function .
- 11. Quadratic Function’s Vertex Form General Form to Vertex Form
- 12. The vertex form of a quadratic equation is used to easily identify the vertex of the parabola. The general vertex form is defined as . Example: Transform into . 𝑦 = 𝑥 2 − 6 𝑥 − 6 𝑦 =( 𝑥2 − 6 𝑥 ) − 6 𝑦 =( 𝑥 2 − 6 𝑥 + 9 ) − 6 − 9 𝑎 [ − 𝑏 2 ] 2 =1[− (− 6) 2 ] 2 = 9 𝑦 =( 𝑥 2 − 6 𝑥 + 9 ) − 15 𝒚 =( 𝒙 − 𝟑 )𝟐 − 𝟏𝟓
- 13. Example: Transform into . 𝑦 = 𝑥 2 − 6 𝑥 + 3 𝑦 =( 𝑥 2 − 6 𝑥 ) + 3 𝑦 =( 𝑥 2 − 6 𝑥 + 9 ) + 3 − 9 𝑎 [ − 𝑏 2 ] 2 =1[− (− 6) 2 ] 2 = 9 𝑦 =( 𝑥 2 − 6 𝑥 + 9 ) − 6 𝒚 =( 𝒙 − 𝟑 )𝟐 − 𝟔
- 14. Example: Transform into . 𝑦 =− 4 𝑥 2 − 24 𝑥 + 15 𝑦 =− 4 ( 𝑥 2 + 6 𝑥 ) + 15 𝑦 =− 4 ( 𝑥 2 +6 𝑥 +9 )+3 − 36 𝑎 [−𝑏 2 ] 2 =− 4 [−(6) 2 ] 2 =− 4 (9 )=− 3 6 𝑦 =− 4 ( 𝑥 2 +6 𝑥 +9 ) − 33 𝒚 =− 𝟒 ( 𝒙 + 𝟑)𝟐 − 𝟑𝟑
- 15. Example: Transform into . 𝑦 = 𝑥 2 + 8 𝑥 + 18 𝑦 =( 𝑥 2 + 8 𝑥 ) + 18 𝑦 =( 𝑥 2 +8 𝑥 +16 ) +18 − 16 𝑎 [ − 𝑏 2 ] 2 =[ − (8 ) 2 ] 2 = 16 𝑦 =( 𝑥 2 + 8 𝑥 + 16 ) + 2 𝒚 =( 𝒙 + 𝟒 )𝟐 + 𝟐
- 16. Example: Transform into . 𝑦 = 𝑥 2 − 2 𝑥 + 5 𝑦 =( 𝑥 2 − 2 𝑥 ) + 5 𝑦 =( 𝑥 2 − 2 𝑥 +1 ) + 5 − 1 𝑎 [ − 𝑏 2 ] 2 =[ − (− 2 ) 2 ] 2 =1 𝑦 =( 𝑥 2 − 2 𝑥 + 1 ) + 4 𝒚 =( 𝒙 − 𝟏 )𝟐 + 𝟒
- 17. Quadratic Function’s Vertex Form Vertex Form to General Form
- 18. Example: Transform into . 𝑦 =( 𝑥 − 1 )2 − 1 𝑦 =( 𝑥 2 − 2 𝑥 + 1 ) − 1 𝒚 = 𝒙 𝟐 − 𝟐 𝒙 Example: Transform into . 𝑦 =2 ( 𝑥 − 2 )2 − 3 𝑦 =2 ( 𝑥 2 − 4 𝑥 + 4 ) − 3 𝑦 =(2 𝑥 2 − 4 𝑥 + 4 ) − 3 𝒚 =𝟐 𝒙𝟐 − 𝟒 𝒙 + 𝟏
- 19. Example: Transform into . 𝑦 =( 𝑥 + 2 )2 − 3 𝑦 =( 𝑥 2 + 4 𝑥 + 4 ) − 3 𝒚 = 𝒙𝟐 + 𝟒 𝒙 + 𝟏 Example: Transform into . 𝑦 =2 ( 𝑥 + 7 )2 − 4 𝑦 =2 ( 𝑥 2 +14 𝑥 + 49 ) − 4 𝑦 =(2 𝑥 2 +28 𝑥 + 98 ) − 4 𝒚 =𝟐 𝒙𝟐 +𝟐𝟖 𝒙 +𝟗𝟒
- 20. Example: Transform into . 𝑦 =( 𝑥 − 3 )2 + 2 𝑦 =( 𝑥 2 − 6 𝑥 + 9 ) + 2 𝒚 = 𝒙𝟐 − 𝟔 𝒙 + 𝟏𝟏 Example: Transform into . 𝑦 =− 3 ( 𝑥 − 4 )2 + 2 𝑦 =− 3 ( 𝑥 2 − 8 𝑥 +16 ) +2 𝑦 =(− 3 𝑥2 + 24 𝑥 − 48 )+ 2 𝒚 =− 𝟑 𝒙 𝟐 +𝟐𝟒 𝒙 − 𝟒𝟔