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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