Which of the following order of transformations best describes how to sketch the graph of h(x) = (1 - x)^3?
The correct order of transformations to sketch the graph of h(x) = (1 - x)^3 is as follows:
1. Reflect the graph about the y-axis: y = (-1)(1 - x)^3
2. Stretch the graph vertically by a factor of 1: y = 1(-1)(1 - x)^3
3. Move the graph 1 unit to the right: y = 1(-1)(1 - (x - 1))^3
So the correct order of transformations is: Reflect, Stretch, Move.
To sketch the graph of h(x) = (1 - x)^3, you can follow these steps:
1. Start with the parent function f(x) = x^3, which has a basic cubic shape.
2. Begin by reflecting the graph of f(x) across the y-axis to account for the negative coefficient in front of the x term. This gives us g(x) = (-x)^3, which is the same as g(x) = -x^3.
3. Next, shift the graph of g(x) one unit to the right to obtain h(x) = g(x - 1) = -(x - 1)^3.
4. Finally, shift the graph of h(x) one unit upwards by adding 1 to the function, resulting in h(x) = (1 - x)^3.
Therefore, the order of transformations to sketch the graph of h(x) = (1 - x)^3 is: reflection across the y-axis, horizontal shift one unit to the right, and vertical shift one unit upwards.
To sketch the graph of the function h(x) = (1 - x)^3, you need to follow a series of transformations. Let's break it down step-by-step:
1. Start with the parent function f(x) = x^3. This is a simple cubic function with a graph that passes through the origin and has a positive slope.
2. Begin with the innermost transformation, which is the reflection across the y-axis. This means that every point (x, y) on the graph of f(x) will be transformed to (-x, y) on the graph of f(-x). In other words, the positive x-values become negative and vice versa. The new function becomes f(-x).
3. Next, apply the horizontal shift. The function h(x) = f(-x + a) represents a horizontal shift of a units to the right if a is positive or to the left if a is negative. In this case, there is no horizontal shift, so a = 0. The function becomes h(x) = f(-x + 0) = f(-x).
4. Finally, apply the vertical shift. The function h(x) = f(-x) + k represents a vertical shift of k units upward if k is positive or downward if k is negative. In this case, there is no vertical shift, so k = 0. The final function is h(x) = f(-x) + 0 = f(-x).
So, the order of transformations for h(x) = (1 - x)^3 is as follows:
1. Reflection across the y-axis
2. Horizontal shift (no shift in this case)
3. Vertical shift (no shift in this case)