the graph of y=f(x) is transformed into y=-f(1/3x+9)+8.
a.Describe the transformations in correct sequence.
b.Is there any other sequence of transformations that could produce the final graph? If so, describe all possible ones.
-f(1/3 x+9)+8 -f(1/3 (x+3)) + 8
so, shift left 3, scale horizontally by 3, reflect through the x-axis, and shift up 8.
a. To describe the transformations in the correct sequence, we can break down the given equation:
1. Horizontal Reflection: The negative sign in front of f(x) (y = -f(x)) indicates a reflection across the x-axis, resulting in the graph being vertically flipped.
2. Horizontal Compression: The expression inside the function, 1/3x, indicates a horizontal compression by a factor of 3. This squeezes the graph horizontally.
3. Horizontal Translation: The "+9" in the expression 1/3x + 9 indicates a horizontal translation of 9 units to the left. This shifts the graph to the left.
4. Vertical Translation: The "+8" at the end of the equation indicates a vertical translation of 8 units upward. This moves the entire graph upward.
So, in correct sequence, the transformations are:
1. Horizontal Reflection
2. Horizontal Compression
3. Horizontal Translation
4. Vertical Translation
b. Yes, there are other sequences of transformations that could produce the final graph. Different sequences of transformations yield the same final graph. Here are some possible sequences:
1. Vertical Translation (+8)
2. Horizontal Translation (-9)
3. Horizontal Compression (by a factor of 3)
4. Horizontal Reflection
Or
1. Horizontal Reflection
2. Horizontal Translation (-9)
3. Horizontal Compression (by a factor of 3)
4. Vertical Translation (+8)
The order of transformations can vary as long as the final transformations applied result in the same graph.
a. To describe the transformations in the correct sequence, we need to break down the given transformation equation:
1. Horizontal Compression/Expansion: We have a horizontal transformation of 1/3x. Since the coefficient is less than 1 (1/3), it corresponds to a horizontal compression. If the coefficient was greater than 1, it would indicate a horizontal expansion.
2. Horizontal Translation: Following the horizontal compression, we have a horizontal translation of +9. The "+9" implies a shift of the graph to the left by 9 units. If the value was negative ("-9"), it would represent a shift to the right.
3. Vertical Reflection: Next, we have a vertical reflection of -f(x), indicated by the negative sign (-) in front of the function f(x). This transformation results in a mirror image of the original graph across the x-axis.
4. Vertical Translation: After the vertical reflection, we have a vertical translation of +8. The "+8" indicates a shift of the graph upwards by 8 units. If the value was negative ("-8"), it would represent a shift downwards.
Therefore, in correct sequence, the transformations are:
1. Horizontal Compression of 1/3
2. Horizontal Translation left by 9 units
3. Vertical Reflection across the x-axis
4. Vertical Translation upwards by 8 units
b. It is important to note that the sequence of transformations is unique and cannot be changed. It is not possible to produce the final graph with any other sequence of transformations. The given sequence of transformations is the only correct sequence to obtain the specific graph y=-f(1/3x+9)+8.