CHAPTER 3: QUADRATIC FUNCTIONS
Chapter objectives:
• Find the minimum and maximum value of a quadratic function
• Sketch a quadratic graph determining its range for a given domain
• Know the conditions for 𝑓(𝑥 ) = 0, where 𝑓(𝑥) is a quadratic function,
to have
o Two real roots
o Two equal roots
o No real roots
• Know the related conditions for a line to
o Intersect a curve
o Be a tangent to a curve
o Not intersect a curve
• Solve quadratic equations for real roots
• Solve quadratic inequalities
What is a quadratic function?
A quadratic function is a function in which the highest power of the variable
(𝑥) is 2. It is a function in the form:
𝑓 (𝑥 ) = 𝑎𝑥 0 + 𝑏𝑥 + 𝑐 , where a, b and c are constants.
Completing the square
All quadratic functions can be expressed in the form:
𝑎(𝑥 + 𝑚)0 + 𝑘 , where 𝑎, 𝑚 and 𝑘 are constants with:
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, 𝑏
𝑚=
2𝑎
𝑏0
𝑘=𝑐−
4𝑎
The method of completing the square is as given in the following example.
Example 3.1
𝑓 (𝑥 ) = 2𝑥 0 + 8𝑥 + 4
Express the above function in the form (𝑥 + 𝑚)0 + 𝑘 , where 𝑎, 𝑚 and 𝑘
are constants.
SOLUTION
𝑓 (𝑥 ) = 2𝑥 0 + 8𝑥 + 4
Divide throughout by the coefficient of 𝑥 0 :
𝑓 (𝑥 )
= 𝑥 0 + 4𝑥 + 2
2
Add the square of half the coefficient of 𝑥 to both sides of the equation:
𝑓(𝑥) 4 0 0
4 0
+ ‡ ˆ = 𝑥 + 4𝑥 + ‡ ˆ + 2
2 2 2
𝑓(𝑥)
+ 20 = 𝑥 0 + 4𝑥 + 20 + 2
2
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,Using the fact (𝑥 + 𝑎)0 = 𝑥 0 + 2𝑎𝑥 + 𝑎0 :
𝑓(𝑥)
+ 20 = (𝑥 + 2)0 + 2
2
𝑓 (𝑥 )
= (𝑥 + 2)0 − 2
2
∴ 𝑓(𝑥 ) = 2(𝑥 + 2)0 − 4
Alternatively:
𝑎𝑥 0 + 𝑏𝑥 + 𝑐 ≡ 𝑎(𝑥 + 𝑚)0 + 𝑘
𝑏 𝑏0
𝑚= , 𝑘=𝑐−
2𝑎 4𝑎
Therefore:
𝑓 (𝑥 ) = 2𝑥 0 + 8𝑥 + 4
𝑎 = 2, 𝑏 = 8, 𝑐=4
𝑏 8
𝑚= = =2
2𝑎 2(2)
𝑏0 80
𝑘=𝑐− =4− = −4
4𝑎 4(2)
∴ 𝑓(𝑥 ) = 2(𝑥 + 2)0 − 4
42
, Because squares of real numbers are always positive or otherwise equal to
zero, the completed square can be used to find the maximum or minimum
value of a function. As for whether the quadratic function has a maximum or
minimum value, this depends on the sign of 𝑎 (the coefficient of 𝑥 0 ).
If 𝒂 is positive → the function has a minimum value
If 𝒂 is negative → the function has a maximum value
For 𝑓 (𝑥 ) = 𝑎(𝑥 + 𝑚)0 + 𝑘, the maximum or minimum value of 𝑓(𝑥 ) = 𝑘
7
and this value occurs at 𝑥 = −𝑚 = − .
06
Example 3.2
𝑓 (𝑥 ) = 2𝑥 0 + 8𝑥 + 4 for 𝑥 ∈ 𝑅
Does the graph of 𝑓 (𝑥 ) have a minimum or a maximum value?
Hence find this value and find the corresponding value of 𝑥.
SOLUTION
The graph has a minimum value because the coefficient of 𝑥 0 is positive.
Completing the square:
2𝑥 0 + 8𝑥 + 4 = 2(𝑥 + 2)0 − 4
→ 𝑘 = −4
and − 𝑚 = −2
∴The minimum value of 𝑓 (𝑥 ) is −4 and the corresponding value of 𝑥 is −2.
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Chapter objectives:
• Find the minimum and maximum value of a quadratic function
• Sketch a quadratic graph determining its range for a given domain
• Know the conditions for 𝑓(𝑥 ) = 0, where 𝑓(𝑥) is a quadratic function,
to have
o Two real roots
o Two equal roots
o No real roots
• Know the related conditions for a line to
o Intersect a curve
o Be a tangent to a curve
o Not intersect a curve
• Solve quadratic equations for real roots
• Solve quadratic inequalities
What is a quadratic function?
A quadratic function is a function in which the highest power of the variable
(𝑥) is 2. It is a function in the form:
𝑓 (𝑥 ) = 𝑎𝑥 0 + 𝑏𝑥 + 𝑐 , where a, b and c are constants.
Completing the square
All quadratic functions can be expressed in the form:
𝑎(𝑥 + 𝑚)0 + 𝑘 , where 𝑎, 𝑚 and 𝑘 are constants with:
40
, 𝑏
𝑚=
2𝑎
𝑏0
𝑘=𝑐−
4𝑎
The method of completing the square is as given in the following example.
Example 3.1
𝑓 (𝑥 ) = 2𝑥 0 + 8𝑥 + 4
Express the above function in the form (𝑥 + 𝑚)0 + 𝑘 , where 𝑎, 𝑚 and 𝑘
are constants.
SOLUTION
𝑓 (𝑥 ) = 2𝑥 0 + 8𝑥 + 4
Divide throughout by the coefficient of 𝑥 0 :
𝑓 (𝑥 )
= 𝑥 0 + 4𝑥 + 2
2
Add the square of half the coefficient of 𝑥 to both sides of the equation:
𝑓(𝑥) 4 0 0
4 0
+ ‡ ˆ = 𝑥 + 4𝑥 + ‡ ˆ + 2
2 2 2
𝑓(𝑥)
+ 20 = 𝑥 0 + 4𝑥 + 20 + 2
2
41
,Using the fact (𝑥 + 𝑎)0 = 𝑥 0 + 2𝑎𝑥 + 𝑎0 :
𝑓(𝑥)
+ 20 = (𝑥 + 2)0 + 2
2
𝑓 (𝑥 )
= (𝑥 + 2)0 − 2
2
∴ 𝑓(𝑥 ) = 2(𝑥 + 2)0 − 4
Alternatively:
𝑎𝑥 0 + 𝑏𝑥 + 𝑐 ≡ 𝑎(𝑥 + 𝑚)0 + 𝑘
𝑏 𝑏0
𝑚= , 𝑘=𝑐−
2𝑎 4𝑎
Therefore:
𝑓 (𝑥 ) = 2𝑥 0 + 8𝑥 + 4
𝑎 = 2, 𝑏 = 8, 𝑐=4
𝑏 8
𝑚= = =2
2𝑎 2(2)
𝑏0 80
𝑘=𝑐− =4− = −4
4𝑎 4(2)
∴ 𝑓(𝑥 ) = 2(𝑥 + 2)0 − 4
42
, Because squares of real numbers are always positive or otherwise equal to
zero, the completed square can be used to find the maximum or minimum
value of a function. As for whether the quadratic function has a maximum or
minimum value, this depends on the sign of 𝑎 (the coefficient of 𝑥 0 ).
If 𝒂 is positive → the function has a minimum value
If 𝒂 is negative → the function has a maximum value
For 𝑓 (𝑥 ) = 𝑎(𝑥 + 𝑚)0 + 𝑘, the maximum or minimum value of 𝑓(𝑥 ) = 𝑘
7
and this value occurs at 𝑥 = −𝑚 = − .
06
Example 3.2
𝑓 (𝑥 ) = 2𝑥 0 + 8𝑥 + 4 for 𝑥 ∈ 𝑅
Does the graph of 𝑓 (𝑥 ) have a minimum or a maximum value?
Hence find this value and find the corresponding value of 𝑥.
SOLUTION
The graph has a minimum value because the coefficient of 𝑥 0 is positive.
Completing the square:
2𝑥 0 + 8𝑥 + 4 = 2(𝑥 + 2)0 − 4
→ 𝑘 = −4
and − 𝑚 = −2
∴The minimum value of 𝑓 (𝑥 ) is −4 and the corresponding value of 𝑥 is −2.
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