Quadratic Functions and Parabolas Linear or Not? Month May June July Aug Sep Oct Nov Dec Avg Temp
64 67 71 72 71 67 62 58 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) = ax2 + 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 xintercepts, 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) = x2 f(x) = x2 + 2 f(x) = x2 + 5 Graphing Quadratics: TI 84 Graph the following f(x) = x2 f(x) = x2 - 2
f(x) = x2 - 5 Graphing Quadratics in Vertex form f(x) = x2 f(x) = (x + 2) 2
f(x) = (x + 5) 2 f(x) = x2 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)2 + 5 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)2 + 81 Where A(t) represents the number of
applicants for asylum in thousands t years since 1990. 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) g(x)
8.6 = 2(x + 7)2 + 4 = -0.3(x 2.7)2 Quadratic Model Avg temp in Charlotte NC. Month Temp (oF) 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
53oF. How well does your model predict this value?
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