Unit 7 Quadratics! Section 9.2 Characteristics of Quadratic Functions

Guided Notes “Zeros,” or solutions, to a quadratic are the points where the graph touches the x-axis, aka x-intercepts. (i.e. where y = 0). Axis of symmetry: A vertical line that divides a parabola into two equal halves. Vertex: The tip of the parabola. Either the Minimum or Maximum point depending on if the graph opens upwards or downwards, respectively. What is the domain of this function? All x values What is the range? ! ≥ −4 What is the increasing interval? ! > −2 What is the decreasing interval?

! < −2

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Unit 7 Quadratics! Section 9.2 Characteristics of Quadratic Functions

Guided Notes How to find the Axis of Symmetry without looking at the graph:

Formula Example For a quadratic function ! = !!! + !" + ! ,

the axis of symmetry is the vertical line

! = !!!!

! = 2!! + 4! + 5

! = −!2!

! = −4

2(2) = −1

Theaxisofsymmetryis! = −1

Examples: Find the Axis of Symmetry for each of the following.

1. ! = 4!! + 16! − 12

2. ! = −3!! − 12! + 2

3. ! = −!! + 6! + 13 How to find the vertex without looking at the graph:

• Notice that the vertex will always lie on the axis of symmetry. Steps:

1. Find the axis of symmetry. 2. Plug that value back into the function and solve.

Examples: Find the vertex for each of the following.

1. ! = 4!! + 16! − 12

! = −162(4) =

−168 = −2

! = −(−12)2(−3) = 12

−6 = −2

! = −62(−1) =

−6−2 = 3

! = −162(4) =

−168 = −2 → ! = 4(−2)! + 16(−2) − 12 = −28

Vertex:(−2,−28)

Unit 7 Quadratics! Section 9.2 Characteristics of Quadratic Functions

Guided Notes

2. ! = !! + 10! + 1

3. ! = −2!! − 12!

! = −102(1) =

−102 = −5 → ! = (−5)! + 10(−5) + 1 = −24

! = −(−12)2(−2) = 12

−4 = −3 → ! = −2(−3)! − 12(−3) = 18

Vertex:(−5,−24)

Vertex:(−3,18)