Graphing Quadratic Functions: Vertex and Standard Forms
Today's objective is to graph quadratic functions in vertex or standard form. Learn about the vertex form, standard form, transformations, and how to plot parabolas with given equations. Understand how to find the vertex, axis of symmetry, and other key attributes of quadratic functions.
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Presentation Transcript
4-1 & 4-2 Quadratic Functions: Vertex Form & Standard Form Today s Objective: I can graph a quadratic function in vertex or standard form
Quadratic Function: ? ? = ??2+ ?? + ?, where ? 0 Graph: Parabola ? = ?2 Parent function/equation: Vertex Point where graph changes direction Minimum or maximum Vertex Form: ? = ?(? )2+? Vertex: (h, k) Axis of Symmetry (line) Divides the graph into 2 mirror images x = h
Transformation of ? ? = ?2 Translation: Vertical Up k units Translation: Horizontal Right h units ? = ?2+ ? Down k units ? = ?2 ? ? = (? )2 Left h units ? = (? + )2 Reflection Across x-axis ? = ??2 Dilation: Stretch: ? = ?2 Vertex Form: ? = ?(? )2+ ? ? > 1 Compression: 0 < ? < 1
Quadratic Function: Vertex Form ?(?) = ?(? )2+? Attributes: Opens up (a > 0) or down (a < 0) Vertex is maximum or minimum Vertex: (h, k) Axis of symmetry: ? = ( ,?) ? =
Quadratic Function: Standard Form Attributes: Opens up (a > 0) or down (a < 0) Vertex is maximum or minimum y-intercept: (0, c) ?(?) = ??2+ ?? + ? (0, c) Can be determined with a little work Axis of symmetry: ? = ? 2? ? 2?,? ? 2? ? 2?,? ? 2? Vertex: ? = ? Evaluate f(x) at ? 2? 2?
Graphing a Quadratic Function: Standard form ? = ?2+ 2? + 3 Vertex: ? = ? ( ?,?) 1. Plot the vertex 2. Find and plot two points to the right of vertex. 3. Plot the point across axis of symmetry. 4. Sketch the curve. 2 2(1)= 1 2?= ? = ( 1)2+2 1 + 3 = 2 Units up from vertex ?2 Units right of vertex x 1 2 Axis of Symmetry: Domain: All Real Numbers ? = 1 Range: 1 ? 2 4
Graphing a Quadratic Function: Standard form ? = 2?2 4? 5 Vertex: ? = ? (?, ?) 1. Plot the vertex 2. Find and plot two points to the right of vertex. 3. Plot the point across axis of symmetry. 4. Sketch the curve. 4 2?= 2(2)= 1 ? = 2(1)2 4 1 5 = 7 Units up from vertex 2?2 Units right of vertex x 1 2 Axis of Symmetry: Domain: All Real Numbers ? = 1 Range: 2 ? 7 8
Graphing a Quadratic Function: Standard form ? = 0.5?2+ 2? 3 ? = 2 Vertex: ? = ? (?, ?) Axis of Symmetry: Domain: All Real Numbers 2 2?= 2( .5)= 2 ? = 0.5 22+ 2 2 3 Range: ? 1 = 1 Vertex on Calculator: [2nd], [trace] Choose minimum or maximum Move curser left of vertex, [enter] Move curser right of vertex, [enter] [enter] Units up from vertex -0.5?2 Units right of vertex x 1 2 0.5 2
Standard form to Vertex form a value is the same Find the vertex ? 2?,? ? = ?(? )2+? ? = ??2+ ?? + ? ? 2? ? = ?2+ 4? 5 4 2( 1) ? = 22+ 4 2 5 ? = 2?2+ 10? + 7 ? = 10 2(2)= 2.5 ? = 2( 2.5)2+10 2.5 + 7 ? = = 2 = 1 = 5.5 ? = (? 2)2 1 ? = 2(? + 2.5)2 5.5
Bungee Jumping You can model the arch of this bridge with the function ? = 0.000498?2+ 0.847? How high above the river is the arch? 0.847 2( 0.000498)= 850 ? = ? = 0.000498 8502+ 0.847(850) = 360 Maximum Arch height: 516 + 360 = 876 ?? p.206:9-29 odds (850,360)
