Quadratic Functions and Parabolas

Linear or Not?Month Avg Temp

May 64June 67July 71Aug 72Sep 71Oct 67Nov 62Dec 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 enterThe 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 buttonAdjust 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 waysStandard: f(x) = ax2 + bx + cVertex: f(x) = a(x – h) 2 + k

Parabolas

A ‘U’ shape or an upside down ‘U’

VertexPoint where the lowest or minimum

value occursPoint 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 formGraph the following on

the calculator:f(x) = x2

f(x) = x2 + 2f(x) = x2 + 5

Graphing Quadratics: TI 84

Graph the followingf(x) = x2

f(x) = x2 - 2f(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 + 5Increasing/Decreasing?Find the vertexFind the axis of symmetryWhat is the vertical interceptSketch the graph

Application

Number of applicants for asylum in the UK in thousands can be modeled

by A(t) = -3.5(t - 11)2 + 81Where 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 modelc) 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)2 + 4

g(x) = -0.3(x – 2.7)2 – 8.6

Quadratic Model

Avg temp in Charlotte NC.Month Temp (oF)

Mar 62Apr 72May 80Jun 86July 89Aug 88Sep 82Oct 72Nov 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?