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Transformations of Functions Graphs of Common Functions See Table 1.4, pg 184. Characteristics of Functions: 1. Domain 2. Range 3. Intervals where its increasing, decreasing, or constant 4. symmetry Vertical Shifts Let f be a function and c a positive real number. •The graph of y = f (x) + c is the graph of y = f (x) shifted c units vertically upward. •The graph of y = f (x) – c is the graph of y = f (x) shifted c units vertically downward. y = f (x) + c y = f (x) c c y = f (x) y = f (x) - c Example • Use the graph of y = x2 to obtain the graph of y = x2 + 4. 10 8 6 4 2 -10 -8 -6 -4 -2 2 -2 -4 -6 -8 -10 4 6 8 10 Example cont. • Use the graph of y = x2 to obtain the graph of y = x2 + 4. Step 1 Graph f (x) = x2. The graph of the standard quadratic function is shown. Step 2 Graph g(x) = x2+4. Because we add 4 to each value of x2 in the range, we shift the graph10of f vertically 4 units up. 8 6 4 2 -10 -8 -6 -4 -2 2 -2 -4 -6 -8 -10 4 6 8 10 Horizontal Shifts Let f be a function and c a positive real number. •The graph of y = f (x + c) is the graph of y = f (x) shifted to the left c units. •The graph of y = f (x - c) is the graph of y = f (x) shifted to the right c units. y = f (x) y = f (x + c) c y = f (x) c y = f (x - c) Text Example Use the graph of f (x) to obtain the graph of h(x) = (x + 1)2 – 3. Solution 5 x2. Step 1 Graph f (x) = The graph of the standard quadratic function is shown. 4 3 2 1 -5 -4 -3 -2 1)2. -1 -1 -2 1 Step 2 Graph g(x) = (x + Because we -3 -4 add 1 to each value of x in the -5 domain, we shift the graph of f horizontally one unit to the left. Step 3 Graph h(x) = (x + 1)2 – 3. Because we subtract 3, we shift the graph vertically down 3 units. 2 3 4 5 Reflection about the x-Axis • The graph of y = - f (x) is the graph of y = f (x) reflected about the x-axis. Reflection about the y-Axis • The graph of y = f (-x) is the graph of y = f (x) reflected about the y-axis. Stretching and Shrinking Graphs Let f be a function and c a positive real number. •If c > 1, the graph of y = c f (x) is the graph of y = f (x) vertically stretched by multiplying each of its y-coordinates by c. •If 0 < c < 1, the graph of y = c f (x) is the graph of y = f (x) vertically shrunk by multiplying each of its y-coordinates by c. g(x) = 2x2 f (x) = x2 10 9 8 7 6 5 4 3 2 1 -4 -3 -2 -1 h(x) =1/2x2 1 2 3 4 Sequence of Transformations A function involving more than one transformation can be graphed by performing transformations in the following order. 1. Horizontal shifting 2. Vertical stretching or shrinking 3. Reflecting 4. Vertical shifting Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 Solution: 10 8 6 Step 1: Because x is replaced with x+3, the graph is shifted 3 units to the left. 4 2 -10 -8 -6 -4 -2 2 -2 -4 -6 -8 -10 4 6 8 10 Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 Solution: 10 8 6 Step 2: Because the equation is not multiplied by a constant, no stretching or shrinking is involved. 4 2 -10 -8 -6 -4 -2 2 -2 -4 -6 -8 -10 4 6 8 10 Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 Solution: 10 8 6 Step 3: Because x remains as x, no reflecting is involved. 4 2 -10 -8 -6 -4 -2 2 -2 -4 -6 -8 -10 4 6 8 10 Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 10 Solution: 8 6 Step 4: Because 4 is subtracted, shift the graph down 4 units. 4 2 -10 -8 -6 -4 -2 2 -2 -4 -6 -8 -10 4 6 8 10