Quadratic Project Modeling Algebra Section 11009

Quadratic ProjectModeling Algebra Section

11009

Juvenile Violent Crime Arrests

Source: http://wps.prenhall.com/esm_lehmann1/51/13153/3367376.cw/-/3367383/index.html

Scatter Plot of Data

Years x=number of years after 1988

# o

f Juve

nile

A

rrest

Why is this a Quadratic Function?

This data represents a quadratic function because the second differences of the “Juvenile Arrest” are very close to being equal which makes this a parabola

The Scatter Plot also forms almost a perfect parabola which makes the data quadratic

Y- VALUES FIRST DIFFERENCESECOND DIFFERENCE

327 97 33

424 64 25

488 39 28

527 -67 23

460 -90

370

How to Find Equation

To find equation use the Quadratic Regression Feature on calculatorAx2+Bx+C=0F(x)= -6.5x2+70.3x+320

F(x)= -6.5x2+70.3x+320This equation is a maximum because the leading coefficient is negativeFinding the Vertex:

Use formula: -b/2a -70.3/2(-6.5)= -70.3/-13 X=5.41 Plug in X to the original equation to find Y F(5.41)=-6.5(5.41)2+70.3(5.41)+320= -190.24265+380.323+320= 510.1=Y VERTEX= (5.41, 510.1)

This means in the year 1993 (5.41+1988) there were 510.1 juvenile crime arrests

Finding Horizontal Intercept

To find horizontal intercept the Y value must be 0(0)= -6.5x2+70.3x+320To find solution use Quadratic Formula -b+- 2ac -70.3+-

There are no juvenile arrests in the year 1984 and 2002. This works with our data

2(-6.5)X=14.26, -3.5

b2-4ac

42 (-6.5)(320)

Finding Vertical Intercept

To find vertical intercept X value must be 0F(x)= -6.5x2+70.3x+320F(0)= -6.5(0)2+70.3x+320Y= 320In the year 1988 the number of juvenile arrests is 320 which is seven off from the data, the original data states in the year 1988 there are 327 juvenile arrests

Quadratic Parabola of Data

Years x=number of years after 1988

# o

f Juve

nile

A

rrest

Predictions Made with Equation

The model can be used to find the number of juvenile arrests when no data was collected Year 1991= F(3)= -6.5(3)2+70.3(3)+320= 227.6 arrests Year 2000= F(12)= -6.5(12)2+70.3(12)+320= 472.4 arrests