2.6 – Transformations of Graphs Vertical Shifting of the Graph of a Function If c > 0, then 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.

2.6 – Transformations of Graphs Horizontal Shifting of the Graph of a Function If c > 0, 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. If c > 0, 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.

2.6 – Transformations of Graphs Vertical and Horizontal Shifts Describe how the graph of y = |x + 2| − 6 would be obtained by translating the graph of y = |x|. Horizontal shift: 2 units left Vertical shift: 6 units down 𝑦= 𝑥 𝑦= 𝑥+2 −6

2.6 – Transformations of Graphs Write the equation of each graph using the appropriate transformations. 𝑦= 𝑥 𝑦= 𝑥 +4 𝑦= 𝑥−3 𝑦= 𝑥+7 −5

2.6 – Transformations of Graphs Vertical Stretching and Shrinking of the Graph of a Function If c > 1, then the graph of y = cf(x) is a vertical stretching of the graph of y = f(x) by applying a factor of c. If 0 < c < 1, then the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x) by applying a factor of c. If a point (x, y) lies on the graph of y = f(x) then the point (x, cy) lies on the graph of y = cf(x). 𝑦= 𝑥 2 −3 𝑦=2 𝑥 2 −3 𝑦=0.2 𝑥 2 −3

2.6 – Transformations of Graphs Horizontal Stretching and Shrinking of the Graph of a Function (a) If c > 1, then the graph of y = ƒ(cx) is a horizontal shrinking of the graph of y = ƒ(x). (b) If 0 < c < 1, then the graph of y = ƒ(cx) is a horizontal stretching of the graph of y = ƒ(x). If a point (x, y) lies on the graph of y = ƒ(x), then the point (x/c, y) lies on the graph of y = ƒ(cx). 𝑦= 𝑥 2 −3 𝑦= 3𝑥 2 −3 𝑦= 0.3𝑥 2 −3

2.6 – Transformations of Graphs Reflections Across the x and y Axes For a function, y = f(x), the following are true. (a) the graph of y = –f(x) is a reflection of the graph of f across the x-axis. (b) the graph of y = f(– x) is a reflection of the graph of f across the y-axis.

2.6 – Transformations of Graphs Reflections Across the x and y Axes Given the graph of a function y = f(x) sketch the graph of: f(x) f(x) y = –f(x) f(– x)

2.6 – Transformations of Graphs 𝑦=−3 𝑥−4 2 +5 horizontal shift 4 units right vertical stretch by a factor of 3 reflect across the x-axis vertical shift 5 units up

2.6 – Transformations of Graphs Summary of Transformations y = f(x) + C C > 0 moves it up C < 0 moves it down y = f(x + C) C > 0 moves it left C < 0 moves it right y = C·f(x) C > 1 stretches it in the y-direction 0 < C < 1 compresses it y = f(Cx) C > 1 compresses it in the x-direction 0 < C < 1 stretches it y = -f(x) Reflects it about x-axis y = f(-x) Reflects it about y-axis