Algebra 1 Section 8.4.

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Presentation transcript:

Algebra 1 Section 8.4

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

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.

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

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

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

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

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.

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

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

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

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

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).

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

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

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).

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

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).

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).

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

Homework: pp. 348-349