3-1: Characteristics of Quadratic Functions

Unit 3: Quadratic Functions

MCR3U1: Functions

Introduction

You are currently raising money for this year’s food drive and would like to sell a 4GB iPod nano to get this money. These iPods were donated to you so you can sell them at any price. Your job is to determine a what price should you sell these iPods to generate the greatest revenue?


LESSON : Properties of Quadratic Functions, f(x)=ax2+bx + c

Unit : Quadratic Functions

PART A: Introduction

·  Quadratic functions have degree and produce when graphed.

·  Quadratic functions can be represented by quadratic equations in different forms.

·  Each form gives different information about the function:

(i)  Standard Form: f(x) = ax2+bx + c , a≠0

·  This form gives the y-intercept, c.

(ii) Factored Form: f(x) = a(x – s)(x – t) , a≠0

·  This form gives the zeros (roots or x-intercepts), x = s and x = t.

(iii)  Vertex Form: f(x) = a(x – h)2 + k , a≠0

·  This form gives the vertex (h, k) and the maximum or minimum value of the function, k, when x = h.

Sign of a

Property

/ Positive, a>0 / Negative, a<0
Vertex
Axis of Symmetry
Direction of Opening
Min/Max y-value

PART B: Writing a Quadratic Equation

Ex. 1: Micha owns a business selling snowboards. She collects the following profit data:

Profits from Snowboard Sales

Snowboards Sold (x1000) / Profit, P(x)
(x$10000) / 1st Diff. / 2nd Diff.
0 / -32 /
1 / -14
2 / 0
3 / 10
4 / 16
5 / 18
6 / 16
7 / 10
8 / 0
9 / -14

·  Quadratic functions represented

as a table of values have

constant differences.

·  When 2nd differences are:

+ve: parabola opens

-ve: parabola opens

1.  a) Write an algebraic equation to model Micha’s Profit using the vertex form of a quadratic function.

b) What information do you need to write the vertex form of the quadratic function?

2.  a) Write an algebraic equation to model Micha’s Profit using the factored form of a quadratic function.

b) What information do you need to write the factored form of the quadratic function?

Part C: Determining the Properties of a Quadratic Function

Ex. 2: A window washer tosses a tool to his partner across the street. The height of the tool above the ground is modeled by the quadratic function, h(t) = -5t2 + 20t + 25, where h(t) is the height in metres and t is the time in seconds after the toss.

a)  How high above the ground is the window?

b)  If his partner misses the tool, when will it hit the ground?

c)  If the path of the tool’s height were graphed, where would the axis of symmetry be?

d)  Determine the domain and range of this function.

Part D: Graphing a Quadratic Function Using the Vertex Form

Ex. 3: Given f(x) = 2(x – 1)2 – 5,

a) state the vertex, axis of symmetry, direction of opening, y-intercept,

domain and range.

b) Graph the function.

c) Find two other points on the parabola to help you graph more accurately.

Homefun: p. 145 # 3-4, 5bcd, 6ab, 8 (find equation in 3 different forms), 9ab, 10-12