1.3 Maximum Or Minimum Of A Quadratic Function Worksheet - Mcgraw-Hill Education Canada Page 4
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1
2
3
4
5
6
7
8
9_
To find the y-coordinate of the vertex, substitute x 5 3
Conn ec ti on s
into
2
y 5 4x
12x 3.
2
By partial factoring,
_
_
(
)
(
)
3
3
2
f (x) = ax
+ bx + k can
2
y 5 4
12
3
2
2
be expressed as
_
(
)
b
_
(
)
9
f (x) = ax
x +
a
+ k.
5 4
18 3
4
This is a family of
5 9 18 3
quadratic functions with
axis of symmetry
5 6
_
b
_
x = −
.
(
)
3
2a
The vertex of the function y 5 4x
12x 3 is at
, 6
. It is a
2
2
minimum, because the value of a is positive.
y
f(x) = ax
+ bx + k
2
0
x
example 3
Solve a Problem Involving a Minimum or a Maximum
Rachel and Ken are knitting scarves to sell at the craft show. The wool
for each scarf costs $6. They were planning to sell the scarves for $10
each, the same as last year when they sold 40 scarves. However, they
know that if they raise the price, they will be able to make more profit,
even if they end up selling fewer scarves. They have been told that for
every 50¢ increase in the price, they can expect to sell four fewer scarves.
What selling price will maximize their profit and what will the profit be?
Solution
Let x represent the number of 50¢ price changes.
Since each scarf cost $6 and was sold for $10, the profit was $4 per scarf.
As they raise the price, their profit per scarf will be (4 0.5x) for x
changes to the price. They will sell 40 4x scarves when they make the
price change.
Profit 5 profit per scarf × number sold
P(x) 5 (4 0.5x)(40 4x)
4x 160
2
5 2x
Method 1: Complete the Square to Determine the Vertex
P(x) 5 2(x
2x) 160
2
2x 1 1) 160
2
5 2(x
5 2(x 1)
2
2 160
5 2(x 1)
162
2
The maximum value of this quadratic function is 162 when x 5 1. This
means that they will make a maximum profit of $162 if they increase the
price once. The selling price is 10 0.5(1) or $10.50.
8 MHR • Functions 11 • Chapter 1
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