Learning objectives

• Students are able to understand the characteristics of parabola y=ax²+bx+c from the given graphs

Review: y=2x²-8 (blue)

• Concave up

• Narrower than y=x²

• x-intercepts=-2 and 2, y-intercept=-8

• minimum=-8

• Domain: all real x, range y≥-8

• Vertex: (0,-8)

Parabola y=ax²+bx+c

• Given parabola y=x²-2x-8

• Concave up

• x-intercepts=-2 and 4, y-intercept=-8

• Domain: all real x, range y≥-9

• Vertex (1,-9)

Parabola y=ax²+bx+c

• x-intercept: ax²+bx+c=0, y-intercept=c

• axis of symmetry X_a= -b/2a

• maximum/minimum: Y_m=a(X_a)²+b(X_a)+c

• Vertex (X_a, Y_m)

Example 1

• x-intercepts=1 and 7, y-intercept=5

• axis of symmetry, X_a=4

• minimum: Y_m=-10

• Vertex: (4,-10)

Example 3: y=x²-2x-8

• a=1, b=-2,c=-8

• x-intercept: x²-2x-8=0, (x-4)(x+2)=0, so, x-intercepts=-2 and 4; y-intercept=-8

• axis of symmetry: X_a=-b/2a=-(-2)/2(1)=1

• minimum: Y_m=(1)²-2(1)-8=-9

• Vertex=(1,-9)

Example 4: y=x²-6x-16

• a=1, b=-6,c=-16

• x-intercept: x²-6x-16=0, (x-8)(x+2)=0, so, x-intercepts=-2 and 8; y-intercept=-16

• axis of symmetry: X_a=-b/2a=-(-6)/2(1)=3

• minimum: Y_m=(3)²-6(3)-16=-25

• Vertex=(3,-25)