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Vertical and Horizontal Shifts of Graphs

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Identify the basic function with a graph as below:

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Vertical Shift of graphs Discussion 1 x y f(x) = x 2 f(x) = x 2 +1 f(x) = x 2 -2 f(x) = x 2 -5 ↑ 1 unit ↓ 2 unit ↓ 5 unit What about shift f(x) up by 10 unit? shift f(x) down by 10 unit?

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Vertical Shift of Graphs Discussion 2 x y f(x) = x 3 f(x) = x 3 +2 f(x) = x 3 -3 ↑ 2 unit ↓ 3 unit

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Vertical Shift of Graphs The graph of y = f(x) + c is obtained by shifting the graph of y = f(x) upward a distance of c units. The graph of y = f(x) – c is obtained by shifting the graph of y = f(x) downward a distance of c units. ↑ f(x) + c ↓ f(x) - c

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Horizontal Shift of graphs Discussion 1 x y f(x) = x 2 f(x) = (x+1) 2 f(x) = (x-2) 2 f(x) = (x-5) 2 ← 1 unit → 2 unit → 5 unit What about shift f(x) left by 10 unit? shift f(x) right by 10 unit?

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Horizontal Shift of Graphs Discussion 2 x y f(x) = |x| f(x) = |x + 2| f(x) = |x - 3| ← 2 unit → 3 unit

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Horizontal Shift of Graphs The graph of y = f(x + c) is obtained by shifting the graph of y = f(x) to the left a distance of c units. The graph of y = f(x - c) is obtained by shifting the graph of y = f(x) to the right a distance of c units. f(x + c) ← → f(x - c)

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Combinations of vertical and horizontal shifts Equation write a description y 1 = |x - 4|+ 3. Describe the transformation of f(x) = |x|. Identify the domain / range for both.

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Combinations of vertical and horizontal shifts Description equation Write the function that shifts y = x 2 two units left and one unit up. answer: y1 = (x+2) 2 +1

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Combinations of vertical and horizontal shifts Graph equation Write the equation for the graph below. Assume each grid mark is a single unit. Answer: f(x) = (x-1) 3 -2 x y

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Combinations of vertical and horizontal shifts Equation graph Sketch the graph of y = f(x) = √x-2 -1. How does the transformation affect the domain and range? x y Step 1: f(x) = √x Step 2: f(x) = √x-2 Step 3: f(x) = √x-2 -1

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