1. Algebra 1
Section 8.4

2. Translations
The translation of the graph of a function is a sliding of the graph without any rotating or resizing.

3. Definition
A power function is a function of the form y = axn, where the base x is a variable while the exponent n and the coefficient a are constants.

4. Example 1 Graph the function y = x2.
First, make a table of ordered pairs. x x2 -3 -2 -1 1 2 3 9 4 1 1 4 9

5. Example 1
Plot the points and connect them with a smooth curve. y x

6. Example 1
This curve is called a parabola. y
Vertex
(0, 0) x

7. Example 2 y
y = x2
y = x2 + 2
y = x2 – 3 x

8. Translations
A translation of a function results when a number is added to or subtracted from either the input or the output of the function.

9. Translations
In general, given y = x2 + k, the graph will shift k units vertically.

10. Example 3 y
y = x2
y = (x – 2)2
y = (x + 3)2 x

11. Translations
Given y = (x - h)2, the graph will shift h units horizontally.
The following chart applies to various types of functions.

12. Translations +k after performing the base function
+h before performing the base function
-h before performing the base function
k units up
k units down
h units left
h units right

13. Translations
Given y = (x - h)2 + k, the graph will shift h units horizontally and k units vertically.
The vertex (turning point) of the function will be located at (h, k).

14. y = (x – 2)2 + 4 Horizontal translation of 2 units to the right
Vertical translation of 4 units up
Vertex: (2, 4)

15. y = (x + 3)2 – 2 Horizontal translation of 3 units to the left
Vertical translation of 2 units down
Vertex: (-3, -2)

16. Translations
The graph of y = x3 can also be drawn using a table of ordered pairs.
The point of inflection of this graph is at (0, 0).

17. Translations In general, the point of inflection of the graph ofy
In general, the point of inflection of the graph of
y = (x – h)3 + k is located at (h, k). x

18. Example 4 Compared to the graph of y = x3,
y = (x – 4)3 + 5 is translated 4 units right and 5 units up.
Its point of inflection is located at (4, 5).

19. Example 4 Compared to the graph of y = x3,
y = (x + 5)3 – 2 is translated 5 units left and 2 units down.
Its point of inflection is located at (-5, -2).

20. Example 4
y = x3
y = (x – 4)3 + 5
y = (x + 5)3 – 2

21. Homework: pp