Quadratic Equations, Quadratic Functions and Absolute Values:

Quadratic Equations, Quadratic Functions and Absolute Values

Solving a Quadratic Equation:

Solving a Quadratic Equation by factorization by graphical method by taking square roots by quadratic equation by using completing square

By factorization:

By factorization roots (solutions)

Slide 4:

By graphical method x y O roots

By taking square roots:

By taking square roots A quadratic equation must contain two roots. ?

By taking square roots:

By taking square roots

Solving a Quadratic Equation by the quadratic Formula:

Solving a Quadratic Equation by the quadratic Formula

By quadratic equation:

By quadratic equation

Slide 9:

a = b = c = 1 10 -7

In general, a quadratic equation may have ::

In general, a quadratic equation may have : (1) two distinct (unequal) real roots (2) one double (repeated) real root (3) no real roots

Slide 11:

Two distinct (unequal) real roots x -intercepts

Slide 12:

One double (repeated) real roots x -intercept

Slide 13:

No real roots no x -intercept

Exercise 1.1:

Exercise 1.1 P.8

Nature of Roots:

Nature of Roots

Slide 16:

△ = b 2 - 4 ac Since the expression b 2 - 4 ac can be used to determine the nature of the roots of a quadratic equation in the form ax 2 – bx + c = 0, it is called the discriminant of the quadratic equation.

Slide 17:

Two distinct (unequal) real roots x-intercepts △ = b 2 - 4 ac > 0

Slide 18:

One double (repeated) real roots x-intercept △ = b 2 - 4 ac = 0

Slide 19:

No real roots no x -intercept △ = b 2 - 4 ac < 0

Solving a Quadratic Equation by Completing the Square:

Solving a Quadratic Equation by Completing the Square

Solving a Quadratic Equation by Completing the Square:

Solving a Quadratic Equation by Completing the Square

Exercise 1.2:

Exercise 1.2 P.13

Relations between the Roots and the Coefficients:

Relations between the Roots and the Coefficients

Slide 24:

If α and β(p and q, x 1 and x 2 ) are the roots of ax 2 + bx + c = 0, then sum of roots = α + β and product of roots = αβ

Slide 25:

Forming Quadratic Equations with Given Roots

Slide 26:

x 2 – (sum of the roots) x + (product of roots) = 0 Forming Quadratic Equations with Given Roots In S.3, when α = 2 and β = -3 x = 2 or x = -3 x – 2 = 0 or x + 3 = 0 ( x – 2)( x + 3) = 0 x 2 + x – 6 = 0

Exercise 1.3:

Exercise 1.3 P.21

Slide 28:

Linear Functions and Their Graphs

Slide 30:

c ＞ 0 x y O

Slide 31:

c ＜ 0 x y O

Slide 32:

Linear Functions

Slide 34:

c ＞ 0 x y O m ＞ 0 c

Slide 35:

c ＜ 0 x y O m ＞ 0 c

Slide 36:

c ＞ 0 x y O m ＜ 0 c

Slide 37:

c ＜ 0 x y O m ＜ 0 c

Slide 38:

c ＝ 0 x y O m ＜ 0 c

Slide 40:

Open upwards Vertex Open upwards Line of symmetry ( a ＞ 0)

Slide 41:

Open downwards Vertex Line of symmetry ( a ＞ 0)

Vertex (Turning point):

Vertex (Turning point) Local (Relative) Maximum point (max. pt.) Local (Relative) Minimum point point (mini. pt.)

Slide 43:

y = ax 2

Slide 44:

x y O y = ax 2 ( a ＞ 0)

Slide 45:

x y O y = ax 2 + bx + c ( a ＞ 0) b 2 - 4 ac ＞ 0 roots 2 real roots ( c ＜ 0)

Slide 46:

x y O ( a ＞ 0) b 2 - 4 ac ＝ 0 repeated roots root y = ax 2 + bx + c ( c ＞ 0)

Slide 47:

x y O ( a ＞ 0) B 2 - 4 ac ＜ 0 No real roots y = ax 2 + bx + c ( c ＞ 0) No intercept

Finding the turning point of a Quadratic Function by Completing the Square:

Finding the turning point of a Quadratic Function by Completing the Square mini value of the function = -1 mini point = (-2, -1) Because a = +ve, there exists a minimum point.

Exercise 1.4:

Exercise 1.4 P.28

Absolute Values:

Absolute Values

Let x be any real number. The absolute value of x, denoted by | x |, is defined as:

Let x be any real number. The absolute value of x , denoted by | x |, is defined as x if x ≧ 0. - x if x < 0. eg. | 5 | = 5, | 0 | = 0, | -5 | = 5

For all real numbers x and y,:

For all real numbers x and y , ( y ≠ 0)

Generalization:

Generalization If | x | = a , where a ≧0, then x = a or x = - a

THE END:

THE END

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