Select the correct answer.
In which direction must the graph of [tex]f(x)=7^x[/tex] be shifted to produce the graph of [tex]g(x)=7^x+7[/tex]?
A. left
B. down
C. up
D. right
Answer (1)
Recognize that the function g ( x ) is obtained by adding a constant to f ( x ) .
Understand that adding a positive constant to a function shifts its graph vertically upwards.
Conclude that the graph of f ( x ) is shifted upwards by 7 units to obtain the graph of g ( x ) .
The correct answer is C: up. C
Explanation
Understanding the Problem We are given two functions: f ( x ) = 7 x and g ( x ) = 7 x + 7 . We want to determine how the graph of f ( x ) must be shifted to obtain the graph of g ( x ) .
Identifying the Transformation Notice that g ( x ) is obtained by adding a constant to f ( x ) : g ( x ) = f ( x ) + 7 . Adding a positive constant to a function shifts its graph vertically upwards.
Determining the Direction of the Shift Since we are adding a positive constant (7) to f ( x ) to get g ( x ) , the graph of f ( x ) is shifted upwards by 7 units to obtain the graph of g ( x ) . Therefore, the correct answer is C.
Examples
Imagine you're adjusting the brightness on a screen. The original brightness is represented by f ( x ) = 7 x . If you increase the brightness by a fixed amount, say 7 units, the new brightness can be represented by g ( x ) = 7 x + 7 . This is a vertical shift upwards. Similarly, in music, adding a constant value to each note in a melody shifts the entire melody up in pitch. Understanding function shifts helps in various fields, from image processing to audio engineering, where adjustments are made by adding or subtracting constant values.
