1. CHAPTER 4 : QUADRATICS. © John Wiley and Sons 2013 www.wiley.com/college/Bradley© John Wiley and Sons 2013 Essential Mathematics for Economics and Business, 4 th Edition

2. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.1 Solving quadratic equations ax 2 = 0; ax 2 + c = 0; ax 2 + bx = 0. Worked Example 4.2 Solving quadratic equations ax 2 + bx + c = 0. Worked Examples 4.3 Graphs of simple quadratic functions y = ± x 2 Worked Example 4.4 Comparing graphs of quadratic functions. Worked Examples 4.5 Horizontal and vertical translations Worked Examples 4.6 Graphs of quadratic functions y = ax 2 + bx + c Plus other examples on characteristics of quadratic functions.

3. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.1 Solving less general quadratic equations (a) Solve 5x 2 = 0 5x 2 = 0 x 2 = 0/5 = 0 x 2 = 0 x = ± 0 …..a repeated real root (or solution) ……x = 0 and x = 0

4. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.1 Solving less general quadratic equations (b) 2x 2 – 32 = 0 2x 2 = 32 x 2 = 16 x = ± 4 ….. two real roots (or solutions) x = - 4, x = 4

5. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.1 Solving less general quadratic equations (c) 2x 2 + 32 = 0 2x 2 = - 32 x 2 = √(-16) x = ± 4i …..two imaginary roots (or solutions) x = - 4i, x = 4i

6. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.1 Solving less general quadratic equations (d) 2x 2 - 32x = 0 x (2x -32) = 0 x = 0 and/or 2x – 32 = 0 x = 0 and/or 2x = 32 x = 0 and/or x = 16 …….two real roots (or solutions) x = 0 and x = 16 Factor the LHS You loose a solution if you divide both sides by x

7. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Find the roots of any quadratic equation using the Quadratic or ‘-b’ formula To find the roots of any quadratic equation, ax 2 + bx + c = 0 1. State the values of a, b and c 2. Substitute the values of a, b and c (signs included!) into the quadratic or ‘-b’ formula 3. Simplify and solve for x.

8. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.2 (a) Find the roots of x 2 + 6x + 5 = 0 The quadratic equation is x 2 + 6x + 5 = 0 1. state the values of a, b and c: a = 1, b = 6, c = 5 2. Substitute the values of a, b and c (signs included !) into the ‘- b’ formula 3. Simplify and solve for x, as shown in the next slide.

9. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.2 (a) Find the roots of x 2 + 6x + 5 = 0 continued …. a = 1, b = 6, c = 5 Two roots x = -5 x = -1 Two roots x = -5 x = -1

10. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.2 (b) Find the roots of x 2 + 6x + 9 = 0 The quadratic equation is x 2 + 6x + 9 = 0 1. state the values of a, b and c: a = 1, b = 6, c = 9 2. Substitute the values of a, b and c (signs included !) into the ‘- b’ formula 3. Simplify and solve for x, as shown in the next slide.

11. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.2 (b) Find the roots of x 2 + 6x + 9 = 0 continued …. a = 1, b = 6, c = 9 Repeated roots x = -3 Repeated roots x = -3

12. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.2 (c) Find the roots of x 2 + 6x + 10 = 0 The quadratic equation is x 2 + 6x + 10 = 0 1. state the values of a, b and c: a = 1, b = 6, c = 10 2. Substitute the values of a, b and c (signs included !) into the ‘- b’ formula 3. Simplify and solve for x, as shown in the next slide.

13. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.2 (c) Find the roots of x 2 + 6x + 9 = 0 continued …. a = 1, b = 6, c = 10 Complex roots

14. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Graphs of quadratics The basic quadratic is y = x 2 is U shaped The quadratic, y = ax 2 for a > 0. …. is wider than y = x 2 when a < 1: … is narrower than y = x 2 when a > 1:

15. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Calculate a table of points Plot the points. Join in a smooth curve 5 15 25 35 -4-3-21234 x y Curve symmetrical about y-axis Minimum at (0,0) Worked Example 4.3 (a)The basic quadratic, y = x 2 x- 4-3-201234 169410149

16. www.wiley.com/college/Bradley © John Wiley and Sons 2013 y= - x 2 is the mirror image of y = x 2 through the x -axis Worked Example 4.3: Sketch and Calculate the table of points for x = -3 to x = 3 x-3-20123 9410149 -9-40 -4-9 U-shaped curve Symmetrical about y-axis Minimum at (0,0)

17. www.wiley.com/college/Bradley © John Wiley and Sons 2013 y = 2x 2 narrower (steeper) than y = x 2 Worked Example 4.4, and Calculate the table of points for x = -3 to x = 3 -3-20123 y = x 2 9410149 y = 2x 2 1882028 y=0.5x 2 4.520.50 24.5 Symmetrical about y-axis Minimum at (0,0) y = 0.5x 2 wider than y = x 2

18. www.wiley.com/college/Bradley © John Wiley and Sons 2013 y = 2+x 2 is y = x 2 … translated up by 2 units Like Worked Example 4.5, and Calculate the table of points for x = -3 to x = 3 `-3-20123 y = x 2 9410149 y =x 2 +21163236 y=x 2 -3 61-2-3-216 Symmetrical about y-axis Minimum at (0,0) y = x 2 -3 is y = x 2 … translated down by 3 units

19. www.wiley.com/college/Bradley © John Wiley and Sons 2013 y = (x-1) 2 is y = x 2 translated to the right by 1 unit Worked Example 4.5, and Calculate the table of points for x = -3 to x = 3 -3-20123 y = x 2 9410149 y =(x+2) 2 101491625 y=(x-1) 2 16941014 Symmetrical about y-axis Minimum at (0,0) y = (x+2) 2 is y = x 2 translated to the left by 2 units

20. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Properties of quadratic functions, illustrated graphically TThe quadratic is a minimum type if a > 0 TThe quadratic is a maximum type if a < 0 TThe graph of any quadratic is symmetrical about the vertical line drawn through its maximum or minimum point TThe roots of the quadratic equation f(x) = 0 are at the points of intersection of the graph y = f(x) with the x-axis TThe roots are equidistant, from the x-coordinate of the maximum or minimum point TThese points are illustrated in Worked Example 4.6, for the quadratic: y = 2x 2 - 7x + 9

21. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Calculate: y for values of x from x = -2 to x = 6. See (Table 4.3) Plot the points and join them to give the graph in Figure 4.7. x y Worked Example 4.6: Plot the graph of y = 2x 2 - 7x - 9

22. www.wiley.com/college/Bradley © John Wiley and Sons 2013 The points where the graph y = 2x 2 - 7x - 9 crosses the x-axis are the roots of 2x 2 – 7x - 9 = 0. Hence, from the graph, the roots of at x = -1 and x = 4.5 Root at x = -1 Root at x = 4.5 Worked Example 4.6: Estimate the roots of 2x 2 - 7x–9 = 0 graphically

23. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.6 (b) Find the roots of 2x 2 - 7x - 9 = 0 algebraically The quadratic equation is 2x 2 - 7x - 9 = 0 a = 2, b = -7, c = -9 Substitute the values of a, b and c (signs included !) into the ‘-b’ formula Simplify and solve for x, as shown in the next slide.

24. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Find the roots of 2x 2 - 7x - 9 = 0 using the ‘-b’ formula 2x 2 - 7x - 9 = 0 Root at x = -1 and root at x = 4.5

25. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.6: Find the minimum point for y = 2x 2 - 7x - 9 = 0 The x-coordinate of the minimum point is mid-way between x = - 1 and x = 4.5 The distance (along the axis) between the roots is (4.5 – (-1) ) = 5.5. That is, (5.5/2) = 2.75 from root at x = -1 and a further 2.75 to the root at x = 4.5 Hence x = 1.75 at the minimum point (2.75 from either root) 2.75 (1.75, - 17) Minimum point x = -1 x = 4.5 x =1.75

26. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.1 revisited, with graph. Solving less general quadratic equations (a) Solve 5x 2 = 0 5x 2 = 0 x 2 = 0/5 = 0 x 2 = 0 x = ± 0 …..a repeated real root (or solution) ……x = 0 and x = 0

27. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.1 revisited with graph. Solving less general quadratic equations (b) 2x 2 – 32 = 0 2x 2 = 32 x 2 = 16 x = ± 4 … two real roots (or solutions) x = - 4, x = 4

28. www.wiley.com/college/Bradley © John Wiley and Sons 2013 Worked Example 4.1 revisited with graph. Solving less general quadratic equations (c) 2x 2 + 32 = 0 2x 2 = - 32 x 2 = √(-16) x = ± 4i …..two imaginary roots (or solutions) x = - 4i, x = 4i