Suppose the graph of f(x)=\sqrt{x} is shifted down 4 units and to the left 5 units. What is the equation of the new graph? Verify the result graphically.
The equation of the new graph is g(x)=\sqrt{x+5}-4
Verify the result graphically. Choose the graph showing f(x)=\sqrt{x} as a dashed blue curve and the transformed curve as a solid red curve below.
A.
B.
C.
D.
Answer (1)
The original function is f ( x ) = x .
Shifting the graph left by 5 units results in x + 5 .
Shifting the graph down by 4 units results in g ( x ) = x + 5 − 4 .
The correct graph representing this transformation is option A. g ( x ) = x + 5 − 4 , Graph A
Explanation
Understanding the transformations We are given the function f ( x ) = x . We need to shift this graph down 4 units and to the left 5 units.
Horizontal shift To shift the graph to the left by 5 units, we replace x with ( x + 5 ) in the original function. This gives us x + 5 .
Vertical shift To shift the graph down by 4 units, we subtract 4 from the function. This gives us g ( x ) = x + 5 − 4 .
Finding the new equation Therefore, the equation of the new graph is g ( x ) = x + 5 − 4 . Now we need to choose the correct graph that shows f ( x ) = x as a dashed blue curve and g ( x ) = x + 5 − 4 as a solid red curve.
Choosing the correct graph The correct graph is the one where the original function f ( x ) = x (dashed blue) is shifted 5 units to the left and 4 units down to obtain the transformed function g ( x ) = x + 5 − 4 (solid red). By visually inspecting the options, we can see that option A correctly represents this transformation.
Final Answer The equation of the new graph is g ( x ) = x + 5 − 4 , and the correct graph is option A.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs of motion change with different initial conditions or forces is essential. Similarly, in economics, shifts in supply and demand curves can be modeled using function transformations. In computer graphics, transformations are used to manipulate objects in 2D and 3D space. Knowing how to shift, stretch, and reflect functions allows us to model and analyze a wide range of real-world phenomena effectively.
