What is the transformation that transforms the graph of the function [tex]f(x)=x^2[/tex] to the graph of the function [tex]g(x)=(x+9)^2[/tex]?
A: Vertical shift 9 units upward
B: Vertical shift 9 units downward
C: Horizontal shift 9 units to the left
D: Horizontal shift 9 units to the right
Answer (2)
The transformation from f ( x ) = x 2 to g ( x ) = ( x + 9 ) 2 involves replacing x with x + 9 , resulting in a horizontal shift of the graph 9 units to the left. This means the correct option is C : Horizontal shift 9 units to the left.
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The function g ( x ) = ( x + 9 ) 2 is obtained from f ( x ) = x 2 by replacing x with x + 9 .
Replacing x with x + a in f ( x ) results in a horizontal shift of ∣ a ∣ units.
Since we replaced x with x + 9 , we have a horizontal shift of 9 units to the left.
Therefore, the transformation is a horizontal shift 9 units to the left: C .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 2 and g ( x ) = ( x + 9 ) 2 . We want to determine the transformation that maps the graph of f ( x ) to the graph of g ( x ) . The options are vertical and horizontal shifts.
Analyzing the Transformation Notice that g ( x ) = ( x + 9 ) 2 = f ( x + 9 ) . This means that the input to the function f is being changed from x to x + 9 .
Recalling Horizontal Shifts Recall that replacing x with x + a in a function f ( x ) results in a horizontal shift. If 0"> a > 0 , the shift is to the left by a units. If a < 0 , the shift is to the right by ∣ a ∣ units.
Determining the Shift In our case, we are replacing x with x + 9 , so a = 9 . Since 0"> 9 > 0 , the transformation is a horizontal shift to the left by 9 units.
Conclusion Therefore, the transformation that transforms the graph of f ( x ) = x 2 to the graph of g ( x ) = ( x + 9 ) 2 is a horizontal shift 9 units to the left.
Examples
Imagine you're drawing a parabola on a graph. The function f ( x ) = x 2 gives you a basic parabola centered at the origin. Now, if you want to shift this parabola 9 units to the left, you would use the function g ( x ) = ( x + 9 ) 2 . This kind of transformation is useful in physics to model projectile motion or in engineering to design parabolic reflectors, where the position of the parabola needs to be adjusted.
