Explain how the following graphs are obtained from the graph of y=f(x). (Select all that apply.)
(d) y= -2f(x)
shift 2 units upward
shift 2 units downward
shift 2 units to the right
shift 2 units to the left
stretch the graph vertically by a factor of 2
shrink the graph vertically by a factor of 2
stretch the graph horizontally by a factor of 2
shrink the graph horizontally by a factor of 2
reflect about the x-axis
reflect about the origin
(e) y=f(6x)
shift 6 units upward
shift 6 units downward
shift 6 units to the right
shift 6 units to the left
stretch the graph vertically by a factor of 6
shrink the graph vertically by a factor of 6
stretch the graph horizontally by a factor of 6
shrink the graph horizontally by a factor of 6
reflect about the x-axis
reflect about the origin
(f) y= 9f(x)-7
shift 9 units upward
shift 7 units upward
shift 9 units downward
shift 7 units downward
shift 9 units to the right
shift 7 units to the right
shift 9 units to the left
shift 7 units to the left
stretch the graph vertically by a factor of 9
stretch the graph vertically by a factor of 7
shrink the graph vertically by a factor of 9
shrink the graph vertically by a factor of 7
stretch the graph horizontally by a factor of 9
stretch the graph horizontally by a factor of 7
shrink the graph horizontally by a factor of 9
shrink the graph horizontally by a factor of 7
reflect about the x-axis
reflect about the origin
just try the steps.
(d): -2f(x)
reflect about the x-axis
stretch the graph vertically by a factor of 2
the order does not matter here
(e): f(6x)
stretch the graph horizontally by a factor of 6
6x changes 6 times as fast as x, so x only has to be 1/6 as big to get the same graph.
(f): 9f(x)-7
stretch the graph vertically by a factor of 9
shift 7 units downward
Here the order of steps matters.
(d) The graph y = -2f(x) is obtained by reflecting the graph of y = f(x) about the x-axis and then stretching it vertically by a factor of 2.
(e) The graph y = f(6x) is obtained by shrinking the graph of y = f(x) horizontally by a factor of 6.
(f) The graph y = 9f(x) - 7 is obtained by stretching the graph of y = f(x) vertically by a factor of 9, shifting it 7 units downward, and then reflecting it about the x-axis.
(d) The graph y = -2f(x) is obtained by reflecting the graph of y = f(x) about the x-axis and then stretching it vertically by a factor of 2.
(e) The graph y = f(6x) is obtained by shrinking the graph of y = f(x) horizontally by a factor of 6.
(f) The graph y = 9f(x) - 7 is obtained by stretching the graph of y = f(x) vertically by a factor of 9 and then shifting it downward by 7 units.
To understand how the given graphs are obtained from the original graph of y = f(x), we need to analyze each transformation one by one.
(d) y = -2f(x)
- This transformation involves multiplying the original function f(x) by -2, which results in a vertical reflection and a stretching of the graph vertically by a factor of 2. The negative sign reflects the graph about the x-axis, and the number 2 stretches it vertically.
(e) y = f(6x)
- This transformation involves multiplying the input of the function f(x) by 6, which results in a horizontal compression of the graph by a factor of 6. The graph is squeezed horizontally, making it appear narrower.
(f) y = 9f(x) - 7
- This transformation involves multiplying the original function f(x) by 9, which stretches the graph vertically by a factor of 9. Additionally, subtracting 7 from the function shifts the graph 7 units downward.
To summarize, here are the transformations involved for each function:
(d) y = -2f(x):
- Stretch the graph vertically by a factor of 2
- Reflect the graph about the x-axis
(e) y = f(6x):
- Shrink the graph horizontally by a factor of 6
(f) y = 9f(x) - 7:
- Stretch the graph vertically by a factor of 9
- Shift the graph 7 units downward