5.8 Modeling with Quadratic Functions

By: L. Keali’i Alicea

Goals

Write quadratic functions given characteristics of their graphs.

Use technology to find quadratic models for data.

Remember the 3 forms of a quadratic equation!

Standard Form y=ax2+bx+c

Vertex Form y=a(x-h)2+k

Intercepts Form y=a(x-p)(x-q)

Example: Write a quadraticfunction for a parabola with a vertex of (-2,1) that passes through the point (1,-1).

Since you know the vertex, use vertex form! y=a(x-h)2+k

Plug the vertex in for (h,k) and the other point in for (x,y). Then, solve for a.

-1=a(1-(-2))2+1

-1=a(3)2+1

-2=9a

a9

21)2(

9

2 2

xy

Now plug in a, h, & k!

Example: Write a quadratic function in intercept form for a parabola with x-intercepts (1,0) & (4,0) that passes through the point (2,-6).

Intercept Form: y=a(x-p)(x-q) Plug the intercepts in for p & q and the point in

for x & y. -6=a(2-1)(2-4)

-6=a(1)(-2)-6=-2a3=a

y=3(x-1)(x-4)

Now plug in a, p, & q!

Example: Write a quadratic equation in standard form whose graph passes through the points (-3,-4), (-1,0), & (9,-10).

Standard Form: ax2+bx+c=y Since you are given three points that could be

plugged in for x & y, write three eqns. with three variables (a,b,& c), then solve using your method of choice such as linear combo, inverse matrices, or Cramer’s rule.

1. a(-3)2+b(-3)+c=-42. a(-1)2+b(-1)+c=03. a(9)2+b(9)+c=-10

A-1 * B = X =a

=b

=c

1 9 81

1 1- 1

1 3- 9

A

10-

0

4-

B

45

1 4

1

Xyxx

4

5

4

1 2

Assignment

5.8 A (all)