describe the transformations that produce the graph of g(x)=1/2(x-4)^3+5 from the graph of the parent function f(x)=x^3. Give the order in which they must be performed to obtain the correct graph
The transformations that produce the graph of g(x) = 1/2(x-4)^3 + 5 from the parent function f(x) = x^3 are:
1. Reflection: Multiply f(x) by -1, resulting in -f(x).
2. Horizontal translation: Shift f(x) 4 units to the right, resulting in f(x-4).
3. Vertical stretch/compression: Compress f(x-4) vertically by a factor of 1/2, resulting in (1/2)(f(x-4)).
4. Vertical translation: Shift (1/2)(f(x-4)) 5 units up, resulting in (1/2)(f(x-4)) + 5.
The correct order in which these transformations must be performed is:
1. Reflection: Multiply f(x) by -1, resulting in -f(x).
2. Horizontal translation: Shift -f(x) 4 units to the right, resulting in -f(x-4).
3. Vertical stretch/compression: Compress -f(x-4) vertically by a factor of 1/2, resulting in (1/2)(-f(x-4)).
4. Vertical translation: Shift (1/2)(-f(x-4)) 5 units up, resulting in (1/2)(-f(x-4)) + 5.
Therefore, the correct order is reflection, horizontal translation, vertical stretch/compression, and vertical translation.
To obtain the graph of g(x) = 1/2(x-4)^3 + 5 from the parent function f(x) = x^3, the following transformations must be performed in the given order:
1. Horizontal translation: The function g(x) is shifted 4 units to the right compared to the parent function f(x).
2. Vertical stretch: The function g(x) is vertically stretched by a factor of 1/2, compared to the parent function f(x).
3. Vertical translation: The function g(x) is shifted 5 units upward compared to the parent function f(x).
So, the correct order of transformations is:
1. Horizontal translation by 4 units to the right.
2. Vertical stretch by a factor of 1/2.
3. Vertical translation by 5 units upward.
To produce the graph of g(x) = 1/2(x-4)^3 + 5 from the parent function f(x) = x^3, the following transformations must be performed in the given order:
1. Horizontal Translation: The function g(x) is translated horizontally 4 units to the right compared to the parent function f(x). This horizontal translation is represented by (x - 4).
2. Vertical Stretch: The function g(x) is vertically stretched by a factor of 1/2 compared to f(x). This stretching is represented by the coefficient 1/2 in front of the function.
3. Vertical Translation: The function g(x) is translated vertically 5 units up compared to f(x). This vertical translation is represented by the + 5 at the end of the function.
Therefore, the correct order of transformations to obtain the graph of g(x) = 1/2(x-4)^3 + 5 from the graph of the parent function f(x) = x^3 is as follows:
1. Horizontal Translation of 4 units to the right
2. Vertical Stretch by a factor of 1/2
3. Vertical Translation of 5 units up
Performing these transformations in the given order will correctly produce the graph of g(x) from the graph of f(x).