2. Parabolas show up in the architecture of bridges. The parabolic shape is used when constructing mirrors for huge telescopes, satellite dishes and highly sensitive listening devices.

3. focus If this parabola is a satellite dish or a telescopic mirror, the receiver is located at the focus. Parallel rays strike the parabola and all reflect to the focus If it is the reflective surface of a lamp, then the light-bulb is located at the focus of the mirror and the arrows point in the other direction.

4. Quadratic Functions

5. y = ax 2 + bx + c The parabola will open down when the a value is negative. The parabola will open up when the a value is positive. Standard Form The standard form of a quadratic function is a > 0 a < 0 Vertex NOTE: if the parabola opened left or right it would not be a function!

6. y x Line of Symmetry If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the line of symmetry. The line of symmetry ALWAYS passes through the vertex.

7. Find the line of symmetry of y = 3x 2 – 18x + 7 Finding the Line of Symmetry When a quadratic function is in standard form The equation of the line of symmetry is y = ax 2 + bx + c, For example… Using the formula… Thus, the line of symmetry is x = 3.

8. Finding the Vertex We know the line of symmetry always goes through the vertex. Thus, the line of symmetry gives us the x-coordinate of the vertex. To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. STEP 1: Find the line of symmetry STEP 2: Plug the x – value into the original equation to find the y value. y = –2x 2 + 8x –3 y = –2(2) 2 + 8(2) –3 y = –2(4)+ 8(2) –3 y = –8+ 16 –3 y = 5 Therefore, the vertex is (2, 5)

9. Graphing a Quadratic Function in Standard Form The standard form of a quadratic function is given by y = ax 2 + bx + c STEP 1: Find the vertex. STEP 2: MAKE A TABLE using x–values close to the line of symmetry. STEP 3: Find two other points and reflect them across the line of symmetry and connect the points with a smooth curve.

10. Graphing a Quadratic Function in Standard Form Thus the vertex is (1,–3). STEP 1: Find the vertex Let's Graph ONE! Try … y = 2x 2 – 4x – 1

11. 5 –1 STEP 2: MAKE A TABLE using x–values close to the line of symmetry. STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. A Quadratic Function in Standard Form 3 2 yx

12. You Try… Graph y = 2x 2 +8x + 7

13. x = ay 2 + by + c opens left for a < 0 opens right for a > 0 standard form: a > 0 a < 0 Analyzing a Sideways Parabola Line of Symmetry equation of the line of symmetry is vertex: plug in y for line of symmetry to get x

14. Graph (1/8)y 2 – (1/4)y – (15/8) = x by finding the coordinates of the vertex, the axis of symmetry, and the direction of opening. Graphing a Sideways Parabola STEP 1: Find the line of symmetry. STEP 2: Find the vertex. The y – value is given by the line of symmetry, y = 1. (1/8)(1) 2 – (1/4)(1)– (15/8) = x -2 = x STEP 3: Find two other points and reflect them across the line of symmetry. (0, 5), (0, -3)

15. You Try… Graph x = 2y 2 + 8y + 7 by finding the coordinates of the vertex, the axis of symmetry, and the direction of opening.

16. A quadratic function can be defined by a formula in one of the following forms: Vertex form allows you to be able to see the transformations the parabola has undergone.

17. Vertex Forms of the Parabola To convert a quadratic function from vertex form to standard form, simply multiply out the squared term.

18. Writing Quadratic Equations in Vertex Form by Completing the Square GOAL: Turn  Step 1: First factor out the coefficient of x 2 if not 1.  Step 2: Divide the coefficient of x by 2.  Step 3: Square the result of Step 2.  Step 4: Add the result of Step 3 after the x term, then subtract it.  Step 5: Factor the perfect square and simplify the rest.  Step 6: Finally, distribute the coefficient from Step 1.

19. Example: Put the following quadratic function into vertex form by completing the square: perfect square  Step 1: First factor out -4.  Step 2: Divide the coefficient of x by 2, giving 3/2.  Step 3: Square the result: (3/2) 2 = 9/4.  Step 4: Add the result after the x term, then subtract it.  Step 5: Factor the perfect square and simplify the rest.  Step 6: Finally, multiply through by the – 4.

20. **Remember that whatever you add in the grouping must be subtracted from the c-value** Group x-terms Factor out the coefficient Complete the Square Rewrite as subtraction

21. You Try… Write the following equation in vertex form. State the vertex, axis of symmetry, direction of opening, and transformations.

22. Example: Find the vertex form of the equation, vertex, axis of symmetry, direction of opening of and transformations x = 2y² + 4y + 5. What about a sideways parabola? x = 2y² + 4y + 5 = 2(y² + 2y) + 5 = 2(y² + 2y + 1) –2 + 5 = 2(y + 1)² +3

23. You Try… Write the following equation in vertex form. State the vertex, axis of symmetry, direction of opening, and transformations. x = 3y² + 6y + 8 = 3(y² + 2y) + 8 = 3(y² + 2y + 1) − 3 + 8 = 3(y + 1)² + 5

24. Closure Explain the difference between the standard and vertex form of a parabola. Include the formulas for each.

25. Solving Quadratic Functions (Finding x-intercepts of a quadratic equation.) 1) factoring 2) completing the square 3) using the quadratic formula 4) taking the square root 5) using a calculator

26. 1) Factoring Ex: Solve y = 2x 2 – 4x – 6

27. 2) Completing the Square (Sketchpad Animation) (Sketchpad Animation) Completing the square is useful when the quadratic equation does not factor easily. Ex: Find the zeros of f(x) = 2x 2 + 6x – 7.

28. 3) Quadratic Formula When the quadratic equation does not factor easily, you may choose to use the quadratic formula instead of completing the square.

29. Ex: Find the zeros of g(x) = 2x 2 – 5x – 9. Let a = 2, b =  5, and c =  9.

30. Quadratic Equations and the Discriminant The discriminant tells you the number and types of answers (roots) you will get. Since the discriminant is under a radical, think about what it means if you have a +, -, or 0 under the radical.

31. Example Find the value of the discriminant and describe the types of roots (real, imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula. a=2, b=7, c=-11 Discriminant = 2 Reals - Irrational

32. Example (continued) Solve using the Quadratic Formula

33. 4) Taking the Square Root (no x term) Example: Find the solutions of y = -16t 2 + 68.

34. You Try… Try the following examples. Do your work on your paper and then check your answers using a calculator. Solve:

35. Closure Explain what it means to solve a quadratic equation. Describe the ways for solving a quadratic equation.

36. Warm Up 1.Find two positive real numbers whose sum is 110 and whose product is a maximum. 2.Find two positive real numbers such that the sum of the first and twice the second is 24 and whose product is a maximum.

37. Applications of Quadratic Functions Ball

38. Back to Warm Up (using parabolas) 1.Find two positive real numbers whose sum is 110 and whose product is a maximum. 2.Find two positive real numbers such that the sum of the first and twice the second is 24 and whose product is a maximum.

39. Projectile Motion When a projectile is launched, several factors determine its height: 1. The force of gravity pulls the projectile back to Earth. By Galileo’s formula, gravity pulls the projectile towards Earth 16t 2 feet in t seconds. 2. The initial upward velocity with which the projectile is thrown. 3. The initial height of the projectile. Since 16 feet ≈ 4.9 meters, if the units are in meters in the formula above, then h = −4.9t 2 + vt + s.

42. 4.3 s

44. Example You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What are the dimensions needed to enclose the maximum area?

45. Closure Explain how you can use quadratic formulas to find maximum or minimum values in real life applications.