Quadratic Functions and Parabolas

Linear or Not? MonthAvg Temp May64 June67 July71 Aug72 Sep71 Oct67 Nov62 Dec58

Avg Temp Define the variables: T(m) = Avg monthly temp, in degrees F, in San Diego, California. m = month of the year; ex: m = 5 represents may

TI – 84 STAT button then 1:Edit enter The screen will be L1. Enter the values for x (Independent values) for L1. In the L2 column enter the values for y (Dependent Values) for L2. Then WINDOW button Adjust the x min to be lower than the given domain and the max is higher than the max value. Adjust the y min to be lower than min given value and higher than max value. Graph

Quadratic Functions Non linear functions that have a squared term and cannot be written in the general form of a line. Represented 2 ways Standard: f(x) = ax 2 + bx + c Vertex: f(x) = a(x – h) 2 + k

Parabolas  A ‘ U ’ shape or an upside down ‘ U ’ Vertex Point where the lowest or minimum value occurs Point where the maximum value occurs. Quadratic Function models this pattern.

Reading a quadratic

Reading a Quadratic  For what x-values is the curve increasing?  Decreasing  Vertex?  Y-intercept  X-intercept  f(1)

Intercepts  Quadratic can have two x- intercepts, one x-intercept or no x- intercept  Can have one y- intercept or no y- intercept.

Graphing Quadratics in Vertex form  Graph the following on the calculator:  f(x) = x 2  f(x) = x  f(x) = x 2 + 5

Graphing Quadratics: TI 84  Graph the following  f(x) = x 2  f(x) = x  f(x) = x 2 - 5

Graphing Quadratics in Vertex form  f(x) = x 2  f(x) = (x + 2) 2  f(x) = (x + 5) 2  f(x) = x 2  f(x) = (x - 2) 2  f(x) = (x - 5) 2

Vertex  The vertex in Vertex form: F(x) = a(x – h) 2 + k Vertex: (h, k) Symmetry about the vertical line through the vertex: Axis of Symmetry x = h

Practice f(x) = (x + 3) Increasing/Decreasing? Find the vertex Find the axis of symmetry What is the vertical intercept Sketch the graph

Application  Number of applicants for asylum in the UK in thousands can be modeled by A(t) = -3.5(t - 11) Where A(t) represents the number of applicants for asylum in thousands t years since a) How many applicants applied in 2003? b) Sketch a graph of this model c) Give a reasonable domain and range

Domain and range  Domain and range for Quadratics application are restricted.  Model breakdown occurs if the spread is too wide.  Therefore to determine the domain and range rely on the information to make an informed decision.

Quadratic Function  Domain for quad functions that are not applications has no restriction  Range is dependent on the vertex of the graph. facing upward [k, 00) facing downward (-oo, k] Where the vertex is (h, k)

What is the domain and range?  f(x) = 2(x + 7)  g(x) = -0.3(x – 2.7) 2 – 8.6

Quadratic Model Avg temp in Charlotte NC. Month Temp ( o F) Mar 62 Apr 72 May 80 Jun 86 July 89 Aug 88 Sep 82 Oct 72 Nov 62

Model  Find an equation for a model of these data  Using the model estimate the temp in Dec  The actual avg high temp in Dec for Charolette is 53 o F. How well does your model predict this value?