Quadratic Functions Unit Objectives: Solve a quadratic equation. Graph/Transform quadratic functions with/without a calculator Identify function attributes: domain, range, vertex, line of symmetry, number and nature of roots, maximum/minimum values. Model situations with quadratic functions. Today’s Objective: Identify attributes and graph quadratic functions

Quadratic Function: 𝑓 𝑥 =𝑎 𝑥 2 +𝑏𝑥+𝑐, where 𝑎≠0 𝑦= 𝑥 2 Graph: Parabola Parent function/equation: Vertex Point where graph changes direction Minimum or maximum Vertex Form: 𝑦=±𝑎 (𝑥−ℎ) 2 +𝑘 Vertex: (h, k) Axis of Symmetry (line) Divides the graph into 2 mirror images x = h

Transformation of 𝑓 𝑥 = 𝑥 2 Translation: Vertical Translation: Horizontal Up k units Right h units 𝑦= 𝑥 2 +𝑘 𝑦= (𝑥−ℎ) 2 Down k units Left h units 𝑦= 𝑥 2 −𝑘 𝑦= (𝑥+ℎ) 2 Reflection Dilation: 𝑦=𝑎 𝑥 2 Stretch: Across x-axis 𝑦=− 𝑥 2 𝑎>1 Vertex Form: 𝑦=±𝑎( 𝑥−ℎ) 2 +𝑘 Compression: 0<𝑎<1

Graphing a Quadratic Function in vertex form 𝑦= (𝑥−3) 2 +2 Vertex: (3, 2) Plot the vertex Find and plot two points to the right of vertex. Plot the point across axis of symmetry. Sketch the curve. Units right of vertex x Units up from vertex 1 2 𝑥 2 1 Axis of Symmetry: Domain: Range: 4 𝑥=3 3 key-points All Real Numbers 𝑦≥2

Graphing a Quadratic Function in vertex form 𝑦= 2𝑥 2 Vertex: (0, 0) Plot the vertex Find and plot two points to the right of vertex. Plot the point across axis of symmetry. Sketch the curve. Units right of vertex x Units up from vertex 1 2 2𝑥 2 2 Axis of Symmetry: Domain: Range: 𝑥=0 8 3 key-points All Real Numbers 𝑦≥0

Graphing a Quadratic Function in vertex form 𝑦= − 1 2 (𝑥+4) 2 −3 Vertex: (−4, −3) Plot the vertex Find and plot two points to the right of vertex. Plot the point across axis of symmetry. Sketch the curve. Units right of vertex x Units up from vertex 1 2 − 1 2 𝑥 2 − 1 2 Axis of Symmetry: Domain: Range: −2 𝑥=−4 All Real Numbers 𝑦≤−3

Writing a Quadratic function: vertex form 𝑦=±𝑎 (𝑥−ℎ) 2 +𝑘 Identify the Vertex: (−2, −7) 𝑦=𝑎 (𝑥+2) 2 −7 Finding dilation factor: Choose another known point and solve for a. −5=𝑎 (−1+2) 2 −7 (-1, -5) 2=𝑎 𝑦=2 (𝑥+2) 2 −7 (-2, -7)

Writing a Quadratic function: vertex form 𝑦=±𝑎 (𝑥−ℎ) 2 +𝑘 (3, 9) Identify the Vertex: (3, 9) 𝑦=𝑎 (𝑥−3) 2 +9 (5, 7) Finding dilation factor: Choose another known point and solve for a. 7=𝑎 (5−3) 2 +9 −2=4𝑎 Practice W.S.: Graphing Quadratic Functions in Vertex Form − 1 2 =𝑎 𝑦= − 1 2 (𝑥−3) 2 +9