1. MATH 31 LESSONS PreCalculus 8. Sketching Functions

2. A. Relations and Functions Whenever one variable (y) is affected by another variable (x), we say they form a relation. A function is a special relation in which every x-value has at most one y-value. That is, if an x-value could have more than one y-value, it is not a function. All functions are relations, but not all relations are functions.

3. Vertical Line Test If a relation is a function, then any vertical line can cross the graph at most once. If the vertical line crosses the graph more than once, then it is not a function.

4. e.g. Which of the following relations are functions?

5. This is a function. The vertical lines never cross twice (or more).

6. This is not a function. A vertical line can cross twice.

7. This is not a function. A vertical line can cross twice.

8. This is a function. The vertical lines never cross twice (or more).

9. B. Parabolas A parabola is a function of the form

10. Basic form The basic (simplest) form of a parabola is

11. Vertex: (0, 0) Opens upward y x

12. Standard form The standard form (also called the completed-square form) of a parabola is or

13. Vertex: (h, k) y x k h V(h, k)

14. Vertex: (h, k) Axis of symmetry: x = h y x x = h V(h, k) The axis of symmetry cuts the graph “in half”

15. Vertex: (h, k) Axis of symmetry: x = h Max / Min Value: y = k y x y = k V(h, k) If the graph opens down, k is a maximum value. If the graph opens up, k is a minimum value.

16. If a < 0, then the parabola opens downward e.g.y = - x 2 y = -3 (x - 2) 2 + 7 y x

17. If a > 0, then the parabola opens upward e.g.y = x 2 + 11 y = 3 (x - 1) 2 - 4 y x

18. y x y = k x = h V(h, k)

19. Ex. 1Find the vertex of by completing the square. Try this example on your own first. Then, check out the solution.

20. Isolate the x-variables

21. Factor out the coefficient of the squared term

22. Determine the constant that is needed to make a perfect square, by squaring half of the middle (linear) term.

23. Add the coefficient to both sides. Don’t forget that the 16 is multiplied by 3 on the right side. You must do that on the left as well.

24. Complete the square

25. Put the parabola in standard form

26. Vertex: (-4, -28)

27. Ex. 2Fully sketch Identify key features of the graph, including vertex, max/min, axis of symmetry, and intercepts. Try this example on your own first. Then, check out the solution.

28.  Complete the square to put in standard form

29. Isolate the x-variables

30. Factor out the coefficient of the squared variable

31. Find the constant required to make a perfect square:

32. Add the constant to both sides. Don’t forget that the 25 is actually negative, due to the coefficient in front of the brackets.

33. Complete the square

34. Vertex: (5, 4) Opens downward (a < 0) Axis of symmetry: x = 5 Max Value: y = 4 Put in standard form.

35.  Find intercepts y-int: (x = 0)

36.  Find intercepts y-int: (x = 0) (0, -21)

37. x-int: (y = 0)

38. x-int: (y = 0)

39. x-int: (y = 0)

40. x-int: (y = 0) (3, 0) (7, 0)

41.  Sketch y x y = 4 x = 5 V(5, 4) 37