5.3 Translating Parabolas

GRAPHING Vertex Form y = a(x-h)² + k We still look for the same things in vertex form: HAPPY or SAD ? 2Where is the VERTEX = ( h, k )? 3T- Chart 4Axis of Symmetry Standard form: y = ax² + bx + c Vertex form: y = a(x-h)² + k

From Last Time: GRAPHING - - Vertex Form ( y = a(x-h)² + k ) y = (x+5)² + 2 1) It is happy because a>0 2) FIND VERTEX a =1 h = -5 k = 2 So V = (h, k) = (-5,2) 3) T-CHART X Y = (x+5)² Y = (-4+5)² + 2 = 3 -3Y = (-3+5)² + 2 = 6

GRAPHING - - Vertex Form ( y = a(x-h)² + k ) y = -.5(x-1)² + 3 1) It is sad because a<0 2) FIND VERTEX a = -.5 h = 1 k = 3 So V = (h, k) = (1,3) 3) T-CHART X Y = -.5(x - 1)² + 3 3Y = -.5(3 - 1)² + 3 = 1 5Y = -.5(5 -1)² + 3 = -5

Practice: Graph -1/2(x-2) 2 +3 Hint: find the vertex and axis of symmetry; then find where it hits the y axis (let x=0) Vertex: (2,3) Y intercept: (0,1) Mirror point to (0,1) = (4, 1)

Working Backwards: How to write the equation of a parabola 1. Find the vertex: (3, 4) 2. Substitute into the vertex form equation Y = a(x – h) 2 + k h = 3, k = 4 Y = a(x – 3) Choose any point on the graph to plug in for x an y (4, 2) 2 = a(4 – 3) 2 + 4

Working Backwards: How to write the equation of a parabola 4. Solve for a: 2 = a(4 – 3) = a (1) a = -2 Final Equation: Y = -2(x-3) 2 + 4

Practice: Find the equation of the parabola in vertex form Vertex: (-1, 0) Solve for a: 2 Final Equation: Y = 2(x+1) 2

Completing the square Easy case x² + 8x – 5 = 9 x² + 8x – 5 = 9 x² + 8x = 14 x² + 8x + 16 = (x+4)² = 30 (x+4) = ±√30 x = -4 ±√30 Add