Method of Graph sketching

Solve the quadratic inequality x 2 – 5x + 6 > 0 graphically.

Procedures: Step (2): we have y = (x – 2)(x – 3), i.e. y = 0, when x = 2 or x = 3. Factorize x 2 – 5x + 6, The corresponding quadratic function is y = x 2 – 5x + 6 Sketch the graph of y = x 2 – 5x + 6. Step (1): Step (3): Step (4): Find the solution from the graph.

Sketch the graph y = x 2 – 5x + 6. x y 0 What is the solution of x 2 – 5x + 6 > 0 ? y = (x – 2)(x – 3),  y = 0, when x = 2 or x = 3.  23

above the x-axis.so we choose the portion x y 0 We need to solve x 2 – 5x + 6 > 0, The portion of the graph above the x-axis represents y > 0 (i.e. x 2 – 5x + 6 > 0) The portion of the graph below the x-axis represents y < 0 (i.e. x 2 – 5x + 6 < 0) 23

x y 0 x < 2 When x < 2, the curve is above the x-axis i.e., y > 0 x 2 – 5x + 6 > 0 x > 3 When x > 3, the curve is above the x-axis i.e., y > 0 x 2 – 5x + 6 > 0 23

From the sketch, we obtain the solution or

Graphical Solution: 0 2 3

Solve the quadratic inequality x 2 – 5x + 6 < 0 graphically. Same method as example 1 !!!

x y 0 2 < x < 3 When 2 < x < 3, the curve is below the x-axis i.e., y < 0 x 2 – 5x + 6 < 0 23

From the sketch, we obtain the solution 2 < x < 3

0 2 3 Graphical Solution:

Solve Exercise 1: x 1 Answer: x y 0 0–21 Find the x-intercepts of the curve: (x + 2)(x – 1)=0 x = –2 or x = 1 –2 1

Solve Exercise 2: –3 < x < 4 Answer: x y 0 0–34 Find the x-intercepts of the curve: x 2 – x – 12 = 0 (x + 3)(x – 4)=0 x = –3 or x = 4 –3 4

Solve Exercise 3: –7 < x < 5 Solution: x y 0 0–75 Find the x-intercepts of the curve: (x + 7)(x – 5)=0 x = –7 or x = 5 –7 5

Solve Exercise 4: Solution: x y 0 Find the x-intercepts of the curve: (x + 3)(3x – 2)=0 x = –3 or x = 2/3 – x  –3 or x  2/3