3. Functions and Their Graphs
Picture is from Microsoft Clip Art

4. 5 basic graphs Formula: y = x y = x² y = x³ y = √x y = |x| Graph:
Linear— “line”
Quadratic— “parabola”
Cubic— “squiggly”
Square Root– “half of a parabola
Absolute Value– “V-shaped”

5. Shifts y = a(x - h)²+k
h is the horizontal shift. The graph will move in the opposite direction.
K is the vertical shift. The graph will move in the same direction.
If a is positive, then the graph will go up.
If a is negative, then the graph will go down.

6. If a is positive, then the graph will go up.
If a is positive, then the graph will go up.
If a is negative, then the graph will go down.

7. More info about parabolas y = a(x – h)² + k

8. The Vertex of a Parabola y = a(x – h)² + k
The vertex is (h, k).
In other words, the vertex is (H.S., V.S).
For example, y = (x – 2)² – 9.
H.S. = Right 2
V.S. = Down 9
Therefore, the vertex is denoted by
V(2, -9).

9. For example: y = x² + 6
y = (x – 0)² + 6, so the V.S. is up 6 and the H.S. is none. Therefore, the vertex is
V (0, 6).
Since a is positive, the direction of the parabola is up.
Since a is 1, then the parabola is neither fat or skinny. It is a standard parabola.

10. Another example: y = 2(x + 2)² + 6
H.S. = left 2
V.S. = up 6 V (–2, 6)
Direction is up
Since a = 2, then the parabola is skinny

11. Axis of Symmetry of a Parabola y = a(x – h)² + k
x = H.S. or x = h
For example, y = (x – 2)²
Axis of symmetry is x = 2

12. Practice up or down? fat or skinny? V.S.? H.S.? Axis Symmetry:
Graph it

13. Practice

14. Practice y = -|x – 6| + 3 What does this graph look like?
Horizontal shift?
Vertical shift?
Vertex?
Fat or skinny?
Up or down?

15. Practice y = -|x – 6| + 3

16. Practice y = (x + 1)³ - 5 What does this graph look like? H.S.? V.S.?
There is no vertex.
Right or Left?
Fat or skinny?

17. Practice y = (x + 1)³ - 5

18. Practice What does this graph look like? H.S.? V.S.?
There is no vertex.
Up or down?
Fat or skinny?

19. Practice

20. Domain and Range
Domain is the set of all x-values. You will look at the graph from left to right (like you’re reading a book). Ask yourself: where does the graph begin? Where does it end?
Range is the set of all y-values. You will look at the graph from bottom to top. Ask yourself: where does the graph begin? Where does it end?

22. Find the domain and range.
From left to right, while following the x-axis, where does the graph begin? Where does it end?
From bottom to top, while following the y-axis, where does the graph begin? Where does it end?

23. Answer:

24. Symmetry

25. Symmetry

26. Symmetry

27. Symmetry

28. Any questions?