4.2 Quadratic Functions –Vertex and Intercept Form

𝑎𝑥2 + 𝑏𝑥 + 𝑐1. Opens up if 𝑎 > 0, Opens down if 𝑎 < 0

2. The axis of symmetry is 𝒙 = −𝒃

𝟐𝒂

3. The vertex has the x-coordinate −𝒃

𝟐𝒂: −

𝒃

𝟐𝒂, 𝑓(−

𝒃

𝟐𝒂)

4. The y-intercept is 𝑐, 𝟎, 𝒄

5. How might we determine the Max and Min of a quadratic equation?

WARM UP! Hot Potato with your partner.

Graph the following functions

𝒚 = 𝒙𝟐 − 𝟔𝒙 + 𝟒 𝒇 𝒙 = 𝒙𝟐 − 𝟑𝒙 + 𝟑

𝒚 = 𝒙 − 𝟒 − 𝟐 𝒇 𝒙 = −𝟐 𝒙 + 𝟐 + 𝟑

Graph the following functions

Graph the following functions

𝒚 = 𝒙𝟐 − 𝟔𝒙 + 𝟒𝒇 𝒙 = 𝒙𝟐 − 𝟑𝒙 + 𝟑𝒚 = 𝒙 − 𝟒 − 𝟐𝒇 𝒙 = −𝟐 𝒙 + 𝟐 + 𝟑

Vertex Form 𝒚 = 𝒂(𝒙 − 𝒉)𝟐+𝒌

Using what you already know, how might you graph the function -

𝒚 = −𝟐(𝒙 − 𝟒)𝟐+𝟐

Vertex Form 𝑦 = 𝑎(𝑥 − ℎ)2+𝑘

Parent Equation 𝑦 = 𝑎𝑥2𝑦 = (𝑥 − 2)2+1𝑦 = (𝒙 − 𝟐)2+1𝑦 = (𝒙 − 𝟐)2+𝟏

Vertex (𝟐 , 𝟏)

Steps to graph from vertex form

1. Identify the constants 𝒚 = 𝒂(𝒙 − 𝒉)𝟐+𝒌

2. Does it open up or down? Is 𝑎 < 0 𝑜𝑟 𝑎 > 0

3. Plot the vertex (𝒉 , 𝒌)

4. Evaluate a two points symmetric about 𝒉

5. Plot the graph

You Try!

𝑦 = (𝑥 + 4)2 𝑦 = 2(𝑥 + 1)2 − 3

Intercept Form 𝑦 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞)

• x-intercepts are 𝑝 𝑎𝑛𝑑 𝑞

• 𝑝 𝑎𝑛𝑑 𝑞 are symmetric about the vertex

𝑥 =𝑝+𝑞

2

• The vertex is at𝒑+𝒒

𝟐, 𝒚

𝑝+𝑞

2, 𝑓

𝑝+𝑞

2

𝑦 =1

2(𝑥 − 1)(𝑥 − 5)𝑦 =

1

2(𝑥 − 1)(𝑥 − 5)

Steps to graph from intercept form

1. Does it open up or down? Is 𝑎 < 0 𝑜𝑟 𝑎 > 0

2. Identify the x-intercepts (𝑝 , 0) and 𝑞 , 0

3. Axis of Symmetry is 𝑥 =𝑝+𝑞

2

4. Plot the vertex 𝑝+𝑞

2, f(

𝑝+𝑞

2)

5. Plot the graph

You Try!

𝑦 = 2(𝑥 − 1)(𝑥 − 4) 𝑦 = −2(𝑥 + 2)(𝑥 − 4)

Use the FOIL method to multiply the binomials

First

Outside

Inside

Last

+𝒙𝟐 +𝟏𝟐+𝟑𝒙+𝟒𝒙

𝒙𝟐 + 𝟕𝒙 + 𝟏𝟐

(𝒙 + 𝟑)(𝒙 + 𝟒)

𝑦 = 2(𝑥 − 2)(𝑥 + 1) 𝑦 = 2(𝑥 − 2)2+2

Converting to Standard Form

Homework

Section 4.2 1 – 53 odd