«From straight line graph to polynomial inequalities»

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Rudolf Weaver
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1 «Fro straight line graph to polynoial inequalities» Bakground inforation: Solving linear and quadrati inequalities are the two topis over in Eleentary and Additional Matheatis at level. Solving polynoial inequality is required at A level. I about to present a graphial-aided way in fulfilling this task. This ethod an be readily served as an alternative to the standard approahes in any textbooks. The Straight line graph: For a non-vertial line, a straight line an be represented in an equation y x where is the gradient, is the y-axis interept. To find the x-axis( y = ) interept, siply solve the equation x. So the x-axis interept, x. We an ake use of these two points to draw a straight line. y x However, we an ake skethes of straight lines by siply aking use of the x-axis interept and its respetive gradient. y x y x Page of 9 Learning & Teahing KLAng De

2 To sketh y x, first find the x-axis interept line is a upward sloping line passes through the point x, and gradient is positive, so the,. To sketh y x, first find the x-axis interept x, and gradient is negative, so the line is a downward sloping line passes through the point,. Let s take a loser look at y x, y x y x, the blue part of the line. y x, the red part of the line. Fro the graph, we an draw the onlusion that: (i) (ii) (iii) x, x ; x, x ; x, x. This has given us a way to solve linear inequality. Consider y x, When y, x. The straight line has a positive gradient, so the line is upward sloping that passes through point,. Page of 9 Learning & Teahing KLAng De

3 Now, we an sketh the straight line: y x (i) (ii) (iii) x, x ; the blue part x, x ; the red part x, x. the x-axis interept To find the range of values of x for x, we just take a look at the graph, the blue part orrespond to x. f ourse, we have also learnt that, x x x You should be failiar with both approahes. Quadrati inequalities: a. Find the range of values of x for x x. For x, x x x Page of 9 Learning & Teahing KLAng De

4 For x, x Sketh the two lines, you an see that the x-axis y x has been divided into setions: x, x, x y x When x x x x x For x x, we will need or. Fro the table or the graph, we get x or x. Drawing of the table is optional one you are used to observe fro the graphs. In addition, if the question is to be finding the range of values of x for whih x x. For x x, we will need or. Fro the table or the graph, we get x only. b. Find the range of values of x for 4x 4x. x x For x, x For x, x Sketh the two lines. Page 4 of 9 Learning & Teahing KLAng De

5 y y x y x For x x, we will need or. Fro the graph, we get It is that easy!! Hooray! x or x. Now, ore good news, you don t need so uh details in your sketh. This will be deonstrated in the next exaple.. Find the range of values of x for 46x x 6. x 8 x For x, x For 8x, 8 x Sketh the two lines. y x 8 y 8 x For x 8 x, we will need or. we get x or 8 x. Page of 9 Learning & Teahing KLAng De

6 For x 8 x, we will need or. Fro the graph, we get 8 x only. Cobine the equality and the inequality, we get x 8 We an now extend this ethod to ubi polynoial. d. Find the range of values of x for x x 6x. x x x For x, x For x, x Sketh the three lines. y x y x For x x x, we will need or. Fro the graph, we get x or x. y x e. Find the range of values of x for whih. x x where x For x, x Sketh the line. Page 6 of 9 Learning & Teahing KLAng De

7 y x For x, we will need. Fro the graph, we get x. That siple! x f. Find the range of values of x for whih. x x x where x For x, x For x, x Sketh the two lines. y x y x x For x, we will need. or Fro the graph, we get x only. Let us opliate the question further Page 7 of 9 Learning & Teahing KLAng De

8 x g. Find the range of values of x for whih. x x x where x For x, x. y x x For x, we will need. Fro the graph, we get x For x x., we will need. For x, x. Cobine the equality and the inequality, we get x x x x h. Find the range of values of x for whih. For x x x x x x, we will need where x,. For x, x. y x y x y x Page 8 of 9 Learning & Teahing KLAng De

9 For x x x, we will need or or. Fro the graph, we get x or x. Cobine the equality and the inequality, we get An Extension to the Appliation: x or x We an atually ake use of this ethod to produe a sketh of a polynoial. For exaple, y x x 6x y x x x y x y x y x In suary: This ethod is workable for all polynoials that an be fatorised in real nuber set. It ertainly an be extended to Quarti Polynoials. For quadrati fator that annot be fatorised into linear fators, we will just have to deterine if it is a always positive or always negative urve. This lesson is suitable for se and above. Page 9 of 9 Learning & Teahing KLAng De

12 Rational Functions

12 Rational Functions Funtions Conepts: The Definition of a Funtion Identifing Funtions Finding the Domain of a Funtion The Big-Little Priniple Vertial and Horizontal Asmptotes The Graphs of Funtions (Setion.) Definition. A