x 2 = 16 ax 2 + bx + c

The parabola is used to make lights that have a focused beam as those used in motor vehicles. It is also used in parabolic louvres and as light fixtures inside buildings.

1. To factorize…. 2. To solve …. 3. To draw the graph … 4. To locate the roots of the equation. 5. To locate the axis of symmetry … 6. To state the maximum or minimum value … 7. To locate and state the turning point … 8. To use the graph to solve simple inequalities

To factorize a x 2 + bx + c 1. Find the product ac 2. Find two factors of ac that will add to give the value of b 3. Replace bx by the two factors found in Factorize in pairs.

Factorize 2x 2 – x – 3 a = 2 b = -1 c = -3 ac = 2 x -3 = -6 Two factors of -6 are -2 and 3 but = 1 which is not equal to b. Use the two factors 2 and -3 ( check that 2 + (-3 ) = -1 which is equal to b) The expression is now re-written as 2x 2 + 2x – 3x -3 (factorize by pairing) 2x(x + 1) -3(x + 1) (x + 1)(2x – 3)

1. Take all terms to one side of the equation. 2. Factorize the equation. 3. Put each factor equal to 0 and solve for x. 4. ALL QUADRATC EQUATIONS YIELD TWO ANSWERS! Perfect squares give two identical values.

To solve 2x 2 – x = 3 ( Take all terms to one side of the equation.) 2x 2 – x – 3 = 0 ( Factorize the equation.) (x+1)(2x – 3) = 0 (Put each factor equal to 0 and solve for x.) x + 1 = 0and2x – 3 = 0 x = -1 x = 1.5 The two solutions are x = -1 and 1.5

1. Choose at least 5 values of x from a given range. 2. Substitute each value into the equation and find the corresponding values of y. 3. Plot each point on a graph page. 4. Draw a smooth curve connecting the points.

The first two steps can be represented in the table as shown below. Working for the values of y do not need to be shown but if shown you will not lose any marks. x y

The roots are the values of x for which the function is equal to zero. i.e. the answer you get when you solve: y = 2x 2 – x – 3 = 0 Remember when we solved this equation we got: x = -1 or 1.5. The axis of symmetry is the average of the roots. x = ½ ( ) = 0.25

 In this case the turning point is a minimum.  The minimum point is the coordinates of the point where the axis of symmetry crosses the curve.  The minimum point for this graph is (0.25, -3.1)  The minimum value is -3.1

 A graph that is already drawn can be used to determine the solution to other equations or inequalities by drawing a suitable line on the curve that was already drawn.  For example to solve 2x 2 – x – 3 = 2 we simply need to 1. draw the line y = 2 on the graph of 2x 2 – x – 3 = 0 and 2. state the values of x where the curve and the line intersect.

 The solutions are the x coordinates of the points where the curve and the line intersect. QUESTION:  Do you think this method can be used to solve 2x 2 – x – 3 = x 2 + 4x – 5?  How would you do this?

To solve 2x 2 – x – 3 < 3  Draw the curve y = 2x 2 – x – 3 and the line y = 3 on the same graph.  Draw vertical lines from the points of intersection to the x-axis.  Shade the region on the graph that makes the inequality true.  State the range of values of x for all points on the curve within the shaded area.