Which of the following order of transformations best describes how to sketch the graph of h(x)=(1−x)^3?
A- Starting with the graph of the basic function f(x)=x^3, horizontally shift the graph to the left one unit, and then reflect the graph about the x-axis.
B- Starting with the graph of the basic function f(x)=x^3, horizontally shift the graph up one unit, and then reflect the graph about the y-axis.
C- Starting with the graph of the basic function f(x)=x^3, horizontally shift the graph to the left one unit, and then reflect the graph about the y-axis.
D- Starting with the graph of the basic function f(x)=x^3, horizontally shift the graph to the right one unit, and then reflect the graph about the y-axis.
The correct answer is C- Starting with the graph of the basic function f(x)=x^3, horizontally shift the graph to the left one unit, and then reflect the graph about the y-axis.
Explanation:
The given function is h(x) = (1-x)^3.
To sketch the graph of h(x), we can start with the basic function f(x) = x^3.
The function f(x) = x^3 represents a cubic function, which is V-shaped and opens upwards.
To obtain the graph of h(x) from f(x), we apply the following transformations step-by-step:
1. Horizontal shift to the left by one unit:
This shifts the entire graph of f(x) one unit to the left. As a result, the point (1, f(1)) on the graph of f(x) will be mapped to the point (0, h(0)) on the graph of h(x).
2. Reflect about the y-axis:
This reflection changes the sign of the x-coordinates of all the points on the graph. Therefore, the graph of h(x) will be a mirror image of the graph of f(x) with respect to the y-axis.
By following these step-by-step transformations, we obtain the graph of h(x) = (1-x)^3.
To determine the correct answer, we need to analyze the given transformation steps and compare them to the original function.
The function given is h(x) = (1 - x)^3.
First, let's consider the basic function f(x) = x^3. This is a cubic function, which is symmetric about the origin (0,0). It opens upwards and passes through the point (0,0).
Now, let's analyze the given transformations:
A- Starting with the graph of the basic function f(x) = x^3, horizontally shift the graph to the left one unit, and then reflect the graph about the x-axis.
Here, the given function h(x) is shifted to the left by one unit. To achieve this, we need to subtract 1 from the x-coordinate. However, the given function has the x-coordinate subtracted from 1 instead. Therefore, this transformation does not match the given function.
B- Starting with the graph of the basic function f(x) = x^3, horizontally shift the graph up one unit, and then reflect the graph about the y-axis.
This transformation does not involve any horizontal shifting, as it only shifts the graph vertically. Therefore, this transformation does not match the given function.
C- Starting with the graph of the basic function f(x) = x^3, horizontally shift the graph to the left one unit, and then reflect the graph about the y-axis.
This transformation matches the given function h(x) = (1 - x)^3. The function is shifted one unit to the left by subtracting 1 from the x-coordinate, and then reflected about the y-axis. This correctly describes the given function.
D- Starting with the graph of the basic function f(x) = x^3, horizontally shift the graph to the right one unit, and then reflect the graph about the y-axis.
This transformation involves shifting the graph to the right, which contradicts the given function's leftward shift. Therefore, this transformation does not match the given function.
Based on the analysis, the correct answer is C- Starting with the graph of the basic function f(x) = x^3, horizontally shift the graph to the left one unit, and then reflect the graph about the y-axis.