Suppose the graph of f is given describe how the graph of each function can be obtained from the graph of f?
2f(x+9)-1
1/4f(x-7)+9
To obtain the graph of 2f(x + 9) - 1 from the graph of f, you would shift the graph of f horizontally by 9 units to the left, then stretch the graph vertically by a factor of 2, and finally shift the entire graph downward by 1 unit.
To obtain the graph of 1/4f(x - 7) + 9 from the graph of f, you would shift the graph of f horizontally by 7 units to the right, then compress the graph vertically by a factor of 1/4, and finally shift the entire graph upward by 9 units.
To obtain the graph of each function from the graph of f, we can apply the given transformations step-by-step.
1. Function: 2f(x+9)-1
Start with the graph of f.
Step 1: Shift the graph 9 units to the left by replacing x with (x+9).
Step 2: Stretch the graph vertically by a factor of 2.
Step 3: Shift the graph down by 1 unit.
The resulting graph will be the graph of 2f(x+9)-1 obtained from the graph of f.
2. Function: 1/4f(x-7)+9
Start with the graph of f.
Step 1: Shift the graph 7 units to the right by replacing x with (x-7).
Step 2: Compress the graph vertically by a factor of 4.
Step 3: Shift the graph up by 9 units.
The resulting graph will be the graph of 1/4f(x-7)+9 obtained from the graph of f.
To understand how the graph of each function can be obtained from the graph of f, we need to analyze the transformations applied to f in each case.
1. Function: 2f(x+9) - 1
This function involves two transformations: a horizontal shift and a vertical shift, followed by a vertical stretch.
a) Horizontal shift: f(x+9)
The graph of f is shifted 9 units to the left. This means that every point on the graph of f will be shifted 9 units to the left to obtain the corresponding point on the new graph.
b) Vertical shift: 2f(x+9)
The graph of f is multiplied by 2, which vertically stretches the graph. Every y-coordinate of the graph of f will be multiplied by 2 to obtain the corresponding y-coordinate on the new graph.
c) Vertical shift: 2f(x+9) - 1
Finally, the entire graph is shifted 1 unit downward. This means that every y-coordinate of the graph obtained in the previous step will be decreased by 1.
To summarize, to obtain the graph of 2f(x+9) - 1 from the graph of f, you need to shift the original graph 9 units to the left, vertically stretch it by a factor of 2, and then shift it 1 unit downward.
2. Function: 1/4f(x-7) + 9
This function also involves two transformations: a horizontal shift and a vertical shift, followed by a vertical compression.
a) Horizontal shift: f(x-7)
The graph of f is shifted 7 units to the right. This means that every point on the graph of f will be shifted 7 units to the right to obtain the corresponding point on the new graph.
b) Vertical shift: 1/4f(x-7)
The graph of f is multiplied by 1/4, which vertically compresses the graph. Every y-coordinate of the graph of f will be multiplied by 1/4 to obtain the corresponding y-coordinate on the new graph.
c) Vertical shift: 1/4f(x-7) + 9
Finally, the entire graph is shifted 9 units upward. This means that every y-coordinate of the graph obtained in the previous step will be increased by 9.
To obtain the graph of 1/4f(x-7) + 9 from the graph of f, you need to shift the original graph 7 units to the right, vertically compress it by a factor of 1/4, and then shift it 9 units upward.
Note: These explanations assume that the vertical scaling factor is applied before the vertical shifting. The order of operations may vary depending on the conventions followed or context of the specific problem.