3.2a Finding the Maximum/minimum and Axis of Symmetry of a Quadratic Function

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Example 1
State the maximum or minimum value of   for each of the following quadratic function and state the value of x at which the function is maximum or minimum.  Find the maximum or minimum point and finally state axis of symmetry for each case.

(a) f ( x ) = − 2 ( x − 3 ) 2 + 4
(b) f ( x ) = 3 ( x − 4 ) 2 + 10
(c) f ( x ) = − 3 ( x + 2 ) 2 − 9
(d) f ( x ) = − 8 + 2 ( x + 5 ) 2






Correction for part (d) of the question,
when x + 5 = 0, x = -5
minimum point is (-5, -8)
Axis of symmetry, x = -5




3.1 Graph of Quadratic Functions

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  1. The graph of quadratic function is parabola.
  2. When the coefficient of x2 is positive the graph is a parabola with ∪ shape.
  3. When the coefficient of x2 is negative the graph is a parabola with ∩ shape.

(A) Axis of Symmetry

The axis of symmetry is a vertical line passing through the maximum or minimum point of the parabola.

 

 

 

(A) General Form of Quadratic Function

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3.1 General Form of Quadratic Function
General form of a quadratic function is f ( x ) = a x 2 + b x + c where a, b, and c are constants and a ≠ 0, and x as a variable.

Example:
Determine which of the following is a quadratic function.
  1. f ( x ) = ( 5 x − 3 ) ( 3 x + 8 )
  2. f ( x ) = 2 ( 3 x + 8 )
  3. f ( x ) = 5 2 x 2
Answer:
(a)
  f ( x ) = ( 5 x − 3 ) ( 3 x + 8 ) f ( x ) = 15 x 2 + 40 x − 9 x − 24 f ( x ) = 15 x 2 + 31 x − 24
Quadratic function

(b)
f ( x ) = 2 ( 3 x + 8 ) f ( x ) = 6 x + 16
Not quadratic function


(c) Not quadratic function

SPM Practice (Paper 1)

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Question 11:
The quadratic equation x 2 −4x−1=2p(x−5) , where p is a constant, has two equal roots. Calculate the possible values of p.

Solution:




Question 12:
Find the range of values of k for which the equation x 2 −2kx+ k 2 +5k−6=0 has no real roots.

Solution:




Question 13:
Find the range of values of p for which the equation 5 x 2 +7x−3p=6 has no real roots.

Solution:

SPM Practice (Paper 1)

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Question 6:
Write and simplify the equation whose roots are the reciprocals of the roots of 3 x 2 +2x−1=0 , without solving the given equation.

Solution:





Question 7:
Find the value of p if one root of x 2 +px+8=0 is the square of the other.

Solution:





Question 8:
If one root of 2 x 2 +px+9=0 is twice the other, find the values of p.

Solution:


SPM Practice (Paper 1)

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Question 1:
Solve the following quadratic equations by factorisation.
(a)  x 2 −5x−10=−4 (b) 3−x−2 x 2 =0 (c) 11a=2 a 2 +12 (d)  2x+7 3x−2 =x

Solution:








Question 2:
Solve the following quadratic equations by completing the square.
(a) 5 x 2 +10x−3=0 (b) 2 x 2 −5x−6=0

Solution:





Finding the Sum of Roots (SoR) and Product of Roots (PoR)

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2.6 Finding the Sum of Roots (SoR) and Product of Roots (PoR)



Example
Find the sums and products of the roots of the following equations.
a. x 2 + 7 x − 3 = 0
b. x ( x − 1 ) = 5 ( 1 − x )

Answer:
(a)
x 2 + 7 x − 3 = 0 a = 1 ,   b = 7 ,   c = − 3 Sum of Roots α + β = − b a = − 7 1 = − 7 Product of Roots α β = c a = − 3 1 = − 3

(b)
x ( x − 1 ) = 5 ( 1 − x ) x 2 − x = 5 − 5 x x 2 − x + 5 x − 5 = 0 x 2 + 4 x − 5 = 0 a = 1 ,   b = 4 ,   c = − 5 Sum of Roots α + β = − b a = − 4 1 = − 4 Product of Roots α β = c a = − 5 1 − 5