2.2 Key Features in Vertex Form

2.2 Key Features in Vertex Form

Quadratic Function in Vertex Form $y\ =\ -3\ \left(x-1\right)^2+4$  

• Does this parabola open up or down?

• Concave down / a = - 3

• What is the vertex? *Hint: (h, k) 

• Vertex: (1 , 4)
  
  

Absolute Value Function in Vertex Form $y\ =\ \left|x+3\right|-4$  

• Does this graph open up or down?

• Up / a = 1

• What is the vertex? 
  

• ( - 3 , - 4)

Multiple Choice

Does this parabola open up or down?  $y\ =\ 2\left(x-3\right)^2+1$  

• Opens Up

• Opens Down

Multiple Choice

What is the vertex of this parabola? $y\ =\ 2\left(x-3\right)^2+1$   

• ( - 3 , 1 )

• ( 3 , 1 )

• ( - 3 , - 1 )

• ( 3 , - 1 )

Multiple Choice

Does the graph of this function open up or down? $y\ =\ -\left|x+4\right|-2$  

• Opens Up

• Opens Down

Multiple Choice

What is the vertex of this absolute value? $y\ =\ -\left|x+4\right|-2$   

• ( 4 , 2 )

• ( - 4 , - 2 )

• ( - 4 , 2 )

• ( 4 , - 2 )

Line of Symmetry

• x = h

• Example:  $y\ =\ \left(x-7\right)^2+1$ 

• What is the line of symmetry?

• x = 7

Multiple Choice

What is the line of symmetry for $y\ =\ -2\left|x+4\right|-4$ ? 

• x = 4

• x = - 4

• y = 4

• y = - 4

Maximum

 $y\ =\ -5\left(x-3\right)^2+6$  

• If a<0, meaning the parabola opens down, the vertex is a maximum.

• Where is the maximum point?

• ( 3 , 6 )

Minimum $y\ =\ 4\left|x+2\right|-9$  

• If a>0, meaning the parabola opens up, the vertex is a minimum.

• Where is the minimum point?

• ( - 2 , - 9 )

Multiple Choice

Does this function have a maximum or a minimum? What is the vertex? $y\ =\ 2\ \left(x-4\right)^2+3$   

• Maximum
  Vertex: ( 4, 3)

• Minimum
  Vertex: (4,3)

• Maximum
  Vertex: (-4 , 3)

• Minimum
  Vertex: (-4 , 3)

Multiple Choice

Does this function have a maximum or a minimum? What is the vertex? $y\ =\ -\left(x+2\right)^2+4$   

• Maximum
  Vertex: (2 , 4)

• Minimum
  Vertex: (-2 , 4)

• Maximum
  Vertex: (-2, 4)

• Minimum
  Vertex: (2 , 4)

Domain/Range

• The DOMAIN of a parabola or absolute value, that opens up or down, will always be $\left(-\infty,\infty\right)$   unless restrictions are given.

• RANGE:  

• Step 1 - Determine if the vertex is a maximum or minimum, based off the value of 'a'

• Step 2 - If the vertex is a MAXIMUM: Your range will be  $\left(-\infty,\ k\right]$  

• Step 3 - If the vertex is a MINIMUM: Your range will be  $\left[k,\infty\right)$  

Domain/Range

• EXAMPLE 1: $y\ =\ -3\ \left|x-2\right|+5$   

• What is the domain?

•  $\left(-\infty,\infty\right)$  

• What is the range?

•  $\left(-\infty,5\right]$  
  

• EXAMPLE 2:  $y\ =\ 4\ \left(x+1\right)^2-2$  

• What is the domain?
  
  What is the range?

Multiple Choice

What is the range of this function? $\ y\ =\ -2\ \left(x-3\right)^2+4$  

•  $\left[4,\ \infty\right)$  
  

•  $\left(-\infty,\ 4\right]$  

• [3, 4]

•  $\left(-\infty,\infty\right)$  
  

Multiple Choice

What is the range of this function? $\ y\ =\ 2\ \left(x-3\right)^2-1$  

•  $\left[-1,\ \infty\right)$  
  

•  $\left(-\infty,\ 1\right]$  

• [3, -1]

•  $\left(-\infty,\infty\right)$  
  

Multiple Choice

What is the domain of this function? $\ y\ =\ 2\ \left(x-3\right)^2-1$  

•  $\left[-1,\ \infty\right)$  
  

•  $\left(-\infty,\ 1\right]$  

• [3, -1]

•  $\left(-\infty,\infty\right)$  
  

SUMMARY: Given $y\ =\ -2\left(x+3\right)^2-4$   

• Is this concave up (a>0) or concave down (a <0)? 

• What is the vertex (h , k)?

• What is the line of symmetry (x = h)?

• Does this function have a maximum or a minimum?
  

• What is the domain and range?

• ANSWERS: Concave down, (-3 , 4), x = -3, Maximum, D: $\left(\infty,\ \infty\right)$ R:  $\left(-\infty,4\right]$   

Accomplished Objectives

• I am able to identify key features of a quadratic function and an absolute value function in vertex form. 

  Key Features Include:

• Vertex

• Maximum/Minimum

• Domain/Range

• Line of Symmetry

EXIT TICKET

Multiple Choice

Does this parabola open up or down? $y\ =\ -3\ \left(x+4\right)^2+2$   

• Opens Up

• Opens Down

Multiple Choice

Does this parabola have a maximum or minimum? $y\ =\ -3\ \left(x+4\right)^2+2$   

• Maximum

• Minimum

Multiple Choice

Where is the vertex? $y\ =\ -3\ \left(x+4\right)^2+2$   

• (-4 , 2)

• (4 , 2)

• (-4 , -2)

Multiple Choice

What is the line of symmetry? $y\ =\ -3\ \left(x+4\right)^2+2$   

• x = 4

• y = 4

• x = - 4

• y = - 4

Multiple Choice

What is the line of symmetry? $y\ =\ -3\ \left(x+4\right)^2+2$   

• x = 4

• y = 4

• x = - 4

• y = - 4

Multiple Choice

What is the domain? $y\ =\ -3\ \left(x+4\right)^2+2$   

• ( -3 , 2)

•  $\left(-\infty,\infty\right)$  
  

•  $\left[2,\ \infty\right)$  

•  $\left(-\infty,\ 2\right]$  

Multiple Choice

What is the range? $y\ =\ -3\ \left(x+4\right)^2+2$   

• ( -3 , 2)

•  $\left(-\infty,\infty\right)$  
  

•  $\left[2,\ \infty\right)$  

•  $\left(-\infty,\ 2\right]$