Unit 3: Quadratic Functions

1|Page
MCR 3U
Unit 3:
Quadratic Functions
Quadratic Functions Unit
Big Idea:
-
Different representations of a quadratic function show different key features about the function more easily; the representation that
is most useful depends on the situation.
Learning Target
Text Questions
#1: Represent and interpret quadratics in a number
of different algebraic and graphical forms.
p. 145 #1-7
#2: Determine the maximum/minimum value of a
quadratic with the most efficient strategy
p. 153 # 1,3,4,8,9,10
#3: Simplify Radicals
p. 167 #1-4, 5ab, 6abc, 7
#4: Solve quadratic equations using all three forms
including simplifying radicals in the quadratic formula
p. 177 # 1, 2, 4, 7, 8, 13
#5: Use the discriminant to predict the number of
solutions to a quadratic equation
p. 185 # 1, 2, 3, 6 - 11
#6: Create an algebraic model given key features
and a point
p. 192 #1, 2, 4ac, 5ac, 6,
10
#7: Create an algebraic model to solve a problem
involving quadratics
Handout #1-9
Solve problems involving the
intersection of a linear and a quadratic
function
p. 198 # 3, 4ac, 5, 6, 8, 12
#8:
Review: p. 202 #1-5,9, 12-23
Practice Test: p. 204 #1-4, 6-9
I got it!
Not there yet! Action Scheduled to get there:
2|Page
MCR 3U
Unit 3:
Quadratic Functions
3.1 Properties of Quadratic Functions
Learning Target #1: Represent and interpret quadratics in a number of different algebraic and graphical
forms.
Warm-up Activity: List everything that you can remember about Quadratics
Parts of a Parabola
y
x
Quadratic Functions in Three Forms
Vertex Form
Big Idea:
Standard Form
Factored Form
Different representations of a quadratic function show different key features about the
function more easily; the representation that is most useful depends on the situation.
3|Page
MCR 3U
Unit 3:
Quadratic Functions
3.2 Determining the Max and Min Points
Learning Target #2: Determine the maximum/minimum value of a quadratic with the most efficient
strategy
Warm-up: Determine the Vertex for each of the following equations:
Standard Form to Vertex Form:
State the vertex:
How do you get the last term?
4|Page
MCR 3U
Unit 3:
Quadratic Functions
EX1: A golfer attempts to hit a golf ball over a gorge from a platform above the ground. The
2
height of the ball is modelled by the function h(t )  5t  40t  100 , where h is the height in
meters, and t is the time in seconds. There are powers lines 185 m above the ground directly in
the path of the ball. Will the golf ball hit the power lines?
2
EX2: Find the vertex of the function f ( x)  3x  9 x  2 .
5|Page
MCR 3U
Unit 3:
Quadratic Functions
3.3 Simplifying Radicals
(Section 3.4 in text)
Learning Target #3: Students will simplify and perform operations on mixed and entire radicals
Investigate
vs.
25  4 
25  4 
16  9 
16  9 
4  36 
4  36 
100  9 
100  9 
A radical is a square, cube, or higher root.
is the radical symbol.
In general,
a  b  ab
A mixed radical is a radical with a coefficient.
a
b

a
b
EX1: Express as a mixed radical in simplest form.
a)
72
b) 5 27
EX2: Simplify.
a)
8  50


b) 3  6 2  24

6|Page
MCR 3U
Unit 3:
Quadratic Functions
3.4 Solving Quadratic Equations
Learning Target #4: Solve quadratic equations using all three forms including simplifying radicals in the
quadratic formula.
Warm-up:
EX1: Remember the golfer…attempts to hit a golf ball over a gorge from a platform above the
2
ground. The height of the ball is modelled by the function, h(t )  5t  40t  100 , where h is
the height in meters, and t is the time in seconds. How long is the golf ball in the air?
EX2: Now, suppose there is a seagull hovering in the path of the ball, 125 m above the ground.
When will the ball hit the seagull?
Big Idea: Note that the form the quadratic is in tells you different characteristics.
7|Page
MCR 3U
Unit 3:
Quadratic Functions
3.5 The Discriminant
Learning Target #5: Use the discriminant to predict the number of solutions to a quadratic equation
How many roots can a quadratic function have?
How can we find the number of roots without solving?
The Discriminant:
If: b 2  4ac  0 , there are ___ roots.
b 2  4ac  0 , there are ___ roots.
b 2  4ac  0 , there are ___ roots.
EX1: A market researcher predicted that the profit function for the first year of a new business
2
would be P( x)  0.3x  3x  15 , where x is based on the number of items produced. Will it
be possible for the business to break even in the first year?
2
EX2: For what value(s) of k does the quadratic function f ( x)  2 x  kx  8 have one root?
8|Page
MCR 3U
Unit 3:
Quadratic Functions
3.6 Quadratic Function Models
Learning Target #6: Create an algebraic model given key features and a point
How many points do we need to define a quadratic function?
EX: Bridges often have a parabolic shape. If the anchors at either end of the
bridge are 42 m apart, and the maximum height of the bridge is 26 m, find the
equation that models the shape of the bridge.
Using Factored Form
Using Vertex Form
9|Page
MCR 3U
Unit 3:
Quadratic Functions
3. 7 Problem Solving with Quadratic Functions
Learning Target #7: Create an algebraic model to solve a problem involving quadratics
Warm-up:
EX1: The local transit company is trying to determine how much to raise fares in order to
maximize revenues. Currently there is an average of 56000 riders that pay an average of $2
per ride. Market research has determined that an increase of $0.10 in the fare will lead to a loss
of 2000 riders. Find the fare and the number of riders that will lead to the maximum revenue.
EX2: A rectangle has an area of 1800 m2. If the length is twice the width, what are the
dimensions of the rectangle?
10 | P a g e
MCR 3U
Unit 3:
Quadratic Functions
Practice: Problem Solving with Quadratic Functions
1. A ticket to a school dance is $8. Usually 300 students attend. The dance committee knows
that for every $1 increase in the price of the ticket, 30 fewer students attend the dance.
What ticket price will maximize the revenue?
2. A rectangle has an area of 330 m2. One side is 7 m longer than the other. What are the
dimensions of the rectangle?
3. The sum of squares of two consecutive integers is 685. What could the integers be?
4. A right triangle has a height 8 cm more than twice the length of the base. If the area of the
triangle is 96 cm2, what are the dimensions of the triangle?
5. A bus company has 4000 passengers daily, each paying a fare of $2. For each $0.15
increase, the company estimates that it will lose 40 passengers per day. If the company
needs to take in $10 450 per day to stay in business, what fare should they charge?
6. Vitaly and Jen have 24 m of fencing to enclose three sides of a vegetable garden at the back
of their house. What are the dimensions of the largest rectangular garden they could
enclose with this length of fencing?
House
Garden
House
7. A factory is to be built on a lot that measures 80 m by 60 m. A lawn of uniform width, equal
to the area of the factory, must surround it. How wide is the strip of lawn, and what are the
dimensions of the factory?
8. A magazine producer can sell 600 of her magazines at $6.00 each. A marketing survey
shows her that for every $0.50 she increases the price, she will lose 30 sales. What price
should she set to obtain the greatest income?
9. A rectangular solar-heat collecting panel is 2.5 m longer than it is wide. If its area is 21 m2,
what are its dimensions?
ANSWERS
1.
2.
3.
4.
5.
$9
15 m by 22 m
–19 & -18 or 18 & 19
base 8 cm, height 24 cm
between $2.75 and $14.25
6.
7.
8.
9.
6 m by 12 m
lawn: 10 m wide, factory: 60 m by 40 m
$8.00
3.5 m by 6.0 m
11 | P a g e
MCR 3U
Unit 3:
Quadratic Functions
3.8 Linear-Quadratic Systems
Learning Target #8: Solve problems involving the intersection of a linear and a quadratic function
Warm-up:
What happens when we graph a line and a parabola on the same set of axes?
2
EX: Find the point(s) of intersection of the functions: f ( x )  2 x  4 and g ( x)  x  6 x  4 .

Download Report