MCR3U1
U3L2
DETERMINING THE MAXIMUM/MINIMUM VALUES
(of a Quadratic Function)
PART A ~ INTRODUCTION
Recall that a quadratic function can be expressed in three different algebraic forms:
1.
2.
3.
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑓(𝑥) = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠)
𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘
standard form:
factored form:
vertex form:
If a > 0, then the parabola opens upward and has a minimum value.
If a < 0, then the parabola opens downward and has a maximum value.
The maximum/minimum value of a quadratic function is the y–coordinate
of the vertex occurring at the x–coordinate of the vertex.
Example 
a)
Consider each of the following quadratic functions:
𝑦 = 2(𝑥 + 1)2 + 3
b)
𝑦 = −2(𝑥 − 5)2 + 4
y
y
x
x
There is a ________________________ value
There is a ________________________ value
of ____________ when x = ____________.
of ____________ when x = ____________.
Example 
a)
b)
Recall perfect square trinomials:
Expand each of the following in one step:
i)
(𝑥 − 5)2 =
ii)
(𝑥 + 3)2 =
Calculate the missing value so that the trinomial
is a perfect square and then factor:
i)
𝑥 2 + 8𝑥 +
=
ii)
𝑥 2 − 12𝑥 +
=
iii)
𝑥 2 + 3𝑥 +
=
MCR3U1
U3L2
PART B ~ DETERMINING THE MAX/MIN VALUE
There are primarily two methods to discover the vertex of a quadratic function:
 COMPLETING THE SQUARE ~ express the function in vertex form 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
𝑦 = −5𝑥 2 + 40𝑥 + 100
factor the a value from the first 2 terms
𝑏 2
complete the square using ( )
2
(add & subtract value inside bracket)
factor the perfect square trinomial;
distribute the a value
simplify
 the vertex is ____________ & there is a ________________________ value of ________ when x = ________
 FACTORING (USE ZEROS) ~ express the function in factored form 𝑦 = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠)
𝑦 = −5𝑥 2 + 40𝑥 + 100
factor the a value from all the terms
factor the trinomial
identify the x–intercepts/zeros
average the zeros to determine the
x–coordinate of the vertex
determine the y–coordinate of the
vertex by substitution
 the vertex is ____________ & there is a ________________________ value of ________ when x = ________
MCR3U1
U3L2
PART C ~ APPLICATIONS
One common application is revenue problems.
revenue = price x quantity
Example 
profit = revenue – cost
= (price x quantity) – cost
The revenue function for a new video game is 𝑅(𝑥) = −6𝑥 2 + 40𝑥 and
its cost function is 𝐶(𝑥) = 4𝑥 + 48, where x is the number of games sold
in hundreds.
a)
b)
Determine the profit function (in thousands of dollars).
Determine the maximum profit and the number of games that must
be sold in order achieve this profit.
HOMEWORK: p.153–154 #1–3, 4abcd (use different methods), 7ac, 8, 11ab, 15