Completing the square Tuesday, 23 April 2019

Minimum value here would be –14 Revision of completing the square Example Complete the square for Minimum value here would be –14

Complete the square for the following (i) (ii) (iii) Example Complete the square for the following (i) (ii) (iii) (i) (i) (i) This one shows that the quadratic is always positive. i.e. Always 10 or more.

Example Express in the form

Example Express in the form

From this the minimum value is 3 A Level Question Express in the form , where a, b and c are constants whose values are to be found. [3] Use your answer to part (a) to find the greatest value of [2] (a) From this the minimum value is 3 (b) Therefore maximum value =

Here the minimum value is 5 when x = 2 A Level Question Express in the form , where the values of the constants a, b and c are to be found. Hence, sketch the graph of , indicating the coordinates of its stationary point. [5] Here the minimum value is 5 when x = 2 x y (2, 5)

The Circle Centre the origin

The Circle Centre (0, 0) radius r

Angles between a tangent and its radius is always 90º To find the gradient of any tangent at a specific point on a circle find the gradient of the radius, then the gradient of the tangent using knowledge of perpendicular lines

Find the gradient of the tangent to the circle at the point Example Find the gradient of the tangent to the circle at the point Gradient of the radius = Gradient of the tangent = (2, –3)

Completing the square and Circles Introduction In questions 1 – 5 complete the square 1. 2. 3. 4. 5.

9. Find the gradient of the tangent to the circle at the point In questions 6 – 8 express in the form , where a, b and c are constants and state the minimum value. 6. 7. 8. 9. Find the gradient of the tangent to the circle at the point 10. Find the gradient of the tangent to the circle at the point Minimum = Minimum = Minimum = Gradient = Gradient =