Modelling Quadratic Functions

Terms The degree of a polynomial is given by the highest power (exponent) in any term of the polynomial A linear relation is a straight-line graph with a general form: y = mx + b. The algebraic expression involved is a polynomial of degree 1. Ex: y = 3x – 5 A quadratic function is a parabola with a general form: f(x)=a(x - h)2 + k. It is a polynomial with a degree of 2. Ex: f(x) = 2x2 + 3x – 5

Parabolas A parabola is a U-shaped graph that is symmetrical around a straight line (the axis of symmetry) The vertex is the highest or lowest point of a parabola that opens upwards or downwards. If a parabola opens upwards, the y-coordinate of the vertex is the minimum If a parabola opens downwards, the y-coordinate of the vertex is a maximum The x – intercepts are called the zeroes or roots.

1st and 2nd Differences When 1st differences are constant, (Or close to it with experimental data), then the polynomial has a degree of 1. IT IS LINEAR. f(x) = mx + b When 2nd differences are constant, then the polynomial has a degree of 2. IT IS A QUADRATIC. f(x) = a(x – h)2 + k Similarily, a cubic function has a degree of 3 and so the 3rd differences are constant, etc.

Data Example Time (s) Distance (m) 1 3 2 12 27 4 48 5 75 1 3 2 12 27 4 48 5 75 Calculate the 1st and 2nd differences to determine the degree and conclude re: graph type.

Data Example Distance (m) 1st Differences 2nd Differences -- 3 12 9 6 27 15 48 21 75 2nd differences are the same, so this motion can be represented by a polynomial of degree 2 and is a quadratic function. A plot of this data would yield a parabola, starting at the origin and opening upwards.

Questions What is the relationship between the degree of a function and the differences in a table of values? How can you tell if a function is linear or quadratic from a: Table of values? Graph? Equation?

Class/Home work Math Power 10 Page 275 #1, 2, 5, 7, 9, 13, 16 Principles of Math 10 Page 171 C1 Page 172 #1, 3, 5, 6, 8, 9