1. 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. 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.

3. 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

4. 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

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

6. 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

7. 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.

8. 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)

9. 2.6 – Transformations of Graphs
𝑦=−3 𝑥−
horizontal shift 4 units right
vertical stretch by a factor of 3
reflect across the x-axis
vertical shift 5 units up

10. 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