1. 1 § 1 - 2 Quadratic Functions The student will learn about: Quadratic function equations, quadratic graphs, and applications.

2. 2 Introduction A quadratic is an equation of the form f (x) = a x 2 + b x + c, and graphs as a parabola.

3. 3 Life has many paths! There are several ways in which we may study functions. Algebraically – as we have been doing. Calculus – throughout the upcoming weeks. Graphing Calculator – as we have been doing. We need to be careful of the last method. Many instructors do not permit the use of graphing calculators. For that reason I expect you to be conversant in all three methods!

4. 4 Definition The graph of a quadratic function is called a parabola. Def: A function f is a quadratic function if f (x) = a x 2 + b x + c, where a  0, and a, b, and c are real numbers. The domain of a quadratic function is the set of all real numbers.

5. 5 Example 1 f (x) = x 2 + x – 2. 1. Find the x intercepts algebraically Sketch the graph of f (x) in a Cartesian coordinate system. To find the x intercept, let f (x) = 0 and solve for x. Factor or use the quadratic formula.

6. 66 Solving a Quadratic Equation f (x) = x 2 + x – 2. Solutions may be found algebraically. Let f (x) = 0 and solve for x : The solutions of a quadratic (the x-intercepts) are also called roots or zeros. a.Factor : 0 = (x + 2)(x – 1) and x = 1 and – 2. OR b. Use the quadratic formula – Next slide please!

7. 77 Solving a Quadratic Equation f (x) = x 2 + x – 2. b. Use the quadratic formula : a = 1, b = 1, c = -2.

8. 8 Example 1 - Continued f (x) = x 2 + x – 2. 2. Find the y-intercept algebraically Sketch the graph of f (x) in a Cartesian coordinate system. To find the y-intercept let x = 0 and solve for f (x). The solution will be c. 3. Using the x-intercepts of 1 and -2 and the y- intercept of -2 we can now sketch this parabola. A graphing calculator might help! Knowing the vertex might help!

9. 99 2. Enter the function Y 1 = x 2 + x – 2 and then press ZOOM and 6 for a standard window. x-intercepts using CALC and zero giving 1 and -2 Graphing a Quadratic Function: Calculator 1. Turn the calculator on and press the y = button. If something is there press clear. 3. Find the x and y intercepts using the calculator. y-intercept using CALC and value giving -2 OR use table!

10. 10 Properties of Quadratics Many useful properties of the quadratic function can be found by transforming the general equation f (x) = a x 2 + b x + c into the form f (x) = a (x – h) 2 + k. This second form is called the standard form (or vertex form) and can be derived from the general form by completing the square. There is a simple (non algebraic) way to do this.

11. 11 Example 2 * f (x) = -2x 2 + 4x + 6 f (x) = ax 2 + bx + c = a(x – h) 2 + k 2. Solve f(x) by completing the square: 1. The intercepts can be calculated as previously shown. y = -2x 2 + 4x + 6 Hence a = - 2, h = 1 and k = 8. (-1, 0), (3, 0), and (0, 6) y = -2 (x 2 - 2x) + 6 y = -2 (x 2 - 2x + 1) + 6 y = -2 (x –1) 2 + 8 + 2 There is a simple way to do this. ? ? ?

12. 12 3. The graph is a parabola with: With intercepts: (-1, 0), (3, 0), and (0, 6) and vertex at (h, k), and a = -2, h = 1 and k = 8. axis of symmetry: x = h, and a minimum if a > 0 or a maximum if a < 0. Example 2 Continued * f (x) = -2x 2 + 4x + 6 f (x) = ax 2 + bx + c = a(x – h) 2 + k

13. 13 With intercepts: (-1, 0), (3, 0), and (0, 6) and a = -2, h = 1 and k = 8. 4. The graph of f is a transformation of the graph of g (x) = x 2. reflected in the x axis since a is negative stretched double since |a| = 2 > 1 shifted one to the right since h = 1 shifted eight up since k = 8. Example 2 Continued * f (x) = -2x 2 + 4x + 6 f (x) = ax 2 + bx + c = a(x – h) 2 + k

14. 14 Example 2 Continued f (x) = -2x 2 + 4x + 6 With intercepts: (-1, 0), (3, 0), and (0, 6) and a = -2, h = 1 and k = 8 and reflected in the x axis stretched double shifted one to the right shifted eight up. There is a simple way to do this.

15. 15 Review f (x) = ax 2 + bx + c OR y = a (x – h) 2 + k x intercepts(x, 0) opens up ifa > 0 y intercept(0, y) stretch if|a| > 1 squish if|a| < 1 opens down ifa < 0

16. 16 Review f (x) = ax 2 + bx + c OR y = a (x – h) 2 + k Has a minimum ifa > 0 Domain is all Reals Has a maximum if a < 0 Range is [k,  ) if a > 0 Range is (- , k] if a < 0

17. 17 Review f (x) = ax 2 + bx + c OR y = a (x – h) 2 + k vertex at(h, k) MAX or min of y = k symmetry axisx = h noteh = -b/2a note k = c – b 2 /4a MAX or min at x = h There is a simple way to do this. So, what is it?

18. 18 The simple way! f (x) = ax 2 + bx + c OR y = a (x – h) 2 + k From example 2, f (x) = - 2x 2 + 4x + 6 we need to get y = a (x – h) 2 + k. But we already know from the standard form that a = - 2, The equation must be: y = - 2 (x – 1) 2 + 8. TA DA! But we already know from the standard form that a = - 2. Press CALC and 3 for a minimum. Giving h = 1 and k = 8. I love my calculator! Graph it!

19. 19 Summary. We learned about quadratics and the different forms for quadratic equations.

20. 20 ASSIGNMENT §1.2; Page 7; 1 – 11, odd.