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=ax 2 +bx+c Vertex Form y=a(x-h) 2 +k Intercepts Form y=a(x-p)(x-q)

Example: Write a quadratic function 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)) =a(3) =9a 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=-2a 3=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: ax 2 +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=-4 2. a(-1) 2 +b(-1)+c=0 3. a(9) 2 +b(9)+c=-10 A -1 * B = X =a =b =c

Assignment 5.8 A (all)