Calculus 1

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

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  1. 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.

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