Describe the change in the graph of

Question

f
(
x
)
=
x
.


g
(
x
)
=
f
(
2
x
)
+
2


A number outside the function notation shows a horizontal stretch or shrink. Since the number is > 1, it represents a horizontal compression by a factor of 2. There is also a translation down the
y
-axis by 2 units.

A number inside the function notation shows a vertical stretch or shrink. Since the number is > 1, it represents a vertical compression by a factor of 2. There is also a translation up the
y
-axis by 2 units.

A number outside the function notation shows a vertical stretch or shrink. Since the number is > 1, it represents a vertical compression by a factor of 2. There is also a translation up the
y
-axis by 2 units.

A number inside the function notation shows a horizontal stretch or shrink. Since the number is > 1, it represents a horizontal compression by a factor of 2. There is also a translation up the
y
-axis by 2 units.

Answer 1

To analyze the transformation of the function \( g(x) = f(2x) + 2 \) where \( f(x) = x \), we need to understand how the individual components of the transformation affect the graph.

1. Base Function: The starting function is \( f(x) = x \), which is a straight line passing through the origin with a slope of 1.

2. Transformations:

   • The function \( g(x) = f(2x) + 2 \) can be broken down into two transformations:
     • Horizontal Compression: The term \( f(2x) \) means that we are taking the base function and compressing it horizontally by a factor of 2. For any x-value, the output of \( f(2x) \) is achieved at half the x-value of the original. This means that it will reach the same y-values more quickly as x increases. The new points will be reached at twice the speed along the x-axis.
     • Vertical Shift: The addition of \( +2 \) after \( f(2x) \) indicates a vertical translation upward by 2 units. This shifts the entire graph up without changing the shape of the line.

3. Final Description:

   • The graph of \( g(x) \) will be a straight line that is a horizontally compressed version of the graph of \( f(x) \) and is then shifted upwards by 2 units. Specifically:
     • The line will still pass through the y-axis at the point \( (0, 2) \) and will have a slope of 1, meaning that for every 1 unit it goes up, it goes 1 unit to the right. However, due to the horizontal compression, it will have a steeper appearance as it approaches the y-axis.

In summary, the correct interpretation of the transformations for \( g(x) = f(2x) + 2 \) is:

• A horizontal compression by a factor of 2.
• A vertical shift up by 2 units.