1. 2.5 Stretching, Shrinking, and Reflecting Graphs
Vertical Stretching of the Graph of a Function
If c > 1, the graph of is obtained by vertically stretching the graph of by a factor of c. In general, the larger the value of c, the greater the stretch.

2. 2.5 Vertical Shrinking Vertical Shrinking of the Graph of a Function
If the graph of is obtained by vertically shrinking the graph of by a factor of c. In general, the smaller the value of c, the greater the shrink.

3. 2.5 Reflecting Across an Axis
Reflecting the Graph of a Function Across an Axis
For a function
(a) the graph of is a reflection of the graph of f across the x-axis.
(b) the graph of is a reflection of the graph of f across the y-axis.

4. 2.5 Example of Reflection Given the graph of sketch the graph of
(a) (b)
Solution
(a) (b)

5. 2.5 Reflection with the Graphing Calculator

6. 2.5 Combining Transformations of Graphs
Example
Describe how the graph of can be obtained by transforming the graph of Sketch its graph.
Solution
Since the basic graph is the vertex of the parabola is shifted right 4 units. Since the coefficient of is –3, the graph is stretched vertically by a factor of 3 and then reflected across the x-axis. The constant +5 indicates the vertex shifts up 5 units.
shift 4 units right
shift 5 units up
vertical stretch by a factor of 3
reflect across the x-axis

7. Graphs:

8. 2.5 Caution in Translations of Graphs
The order in which transformations are made is important. If they are made in a different order, a different equation can result.
For example, the graph of is obtained by first stretching the graph of by a factor of 2, and then translating 3 units upward.
The graph of is obtained by first translating horizontally 3 units to the left, and then stretching by a factor of 2.

9. 2.5 Transformations on a Calculator- Generated Graph
Example
The figures show two views of the graph and another graph illustrating a combination of transformations. Find the equation of the transformed graph.
Solution
The first view indicates the lowest point is (3,–2), a shift 3 units to the right and 2 units down. The second view shows the point (4,1) on the graph of the transformation. Thus, the slope of the ray is
Thus, the equation of the transformed graph is
First View
Second View