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

Posted on April 18, 2020 by user

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

Posted on April 18, 2020 by user

The graph of quadratic function is parabola.
When the coefficient of x² is positive the graph is a parabola with ∪ shape.
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

Posted on April 18, 2020 by user

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.

$f (x) = (5 x - 3) (3 x + 8)$
$f (x) = 2 (3 x + 8)$
$f (x) = \frac{5}{2 x^{2}}$

Answer:
(a)

$\begin{array}{l} f (x) = (5 x - 3) (3 x + 8) \\ f (x) = 15 x^{2} + 40 x - 9 x - 24 \\ f (x) = 15 x^{2} + 31 x - 24 \end{array}$
Quadratic function

(b)

$\begin{array}{l} f (x) = 2 (3 x + 8) \\ f (x) = 6 x + 16 \end{array}$
Not quadratic function

(c) Not quadratic function