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Section 4.9. Graph – Standardization Theorem. Graph-Standardization Theorem (286). In a relation described by a sentence in x and y, the following processes yield the same graph: 1. replacing x by and y by in the sentence;
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Section 4.9 Graph – Standardization Theorem
Graph-Standardization Theorem (286) • In a relation described by a sentence in x and y, the following processes yield the same graph: • 1. replacing x by and y by in the sentence; • 2. Applying the scale change (x,y) (ax,by) where a, b≠ 0, followed by the applying the translation (x,y) (x + h, y + k) to the graph of the original relation.
Theorem • has • Amp: |b| • Period: 2π *|a| • Phase shift: h • Vertical shift: k Notice : to use this formula, x must be by itself!
Example 1 • Describe the graph of y = sin (2x – π/4) y = sin (2(x – π/8)) Amp: Period: Phase shift: Vertical shift: 1 2π/2 = π π/8 to the right 0
Example 2 • The composite of what scale change and translation maps y = cos x onto y = 3 cos 4(x + π) – 10 S(x,y) (x/4, 3y) T(x,y) (x – π, y – 10)
Example 3: S(x,y) T(x,y) Amp Period Vertical shift Phase shift (4x, 2y) (x + 4, y) 2 2π* 4 = 8π 0 4 to the right
Example 4: y = 2 sin (3x + π) + 2 y = 2 sin 3(x + π/3) + 2 S(x,y) T(x,y) Amp Period Vertical shift Phase shift (x/3, 2y) (x - π, y + 2) 2 2π* 1/3 = 2π/3 2 up π/3 to the left
Homework Pages 290 1 , 3 – 6, 9 – 11 *Don’t need to graph
