Answer:
- the stretch factor is "a" based on the equation
- g(x) turning point: (2, -1)
- translation of 4 units to the right
- the magnitude of the stretch factor is 3
Step-by-step explanation:
The equation ...
g(x) = a·f(x -h) +k
indicates a vertical stretch by a factor of "a", a horizontal translation to the right by "h" units, and a translation up by "k" units.
Matching the shapes of the curves, we see that the point of inflection of f(x) is (-2, -1). The corresponding point on g(x) is (2, -1). This is called the "turning point" in your question. It is where the graph turns from being concave downward to being concave upward.
The difference in x-values between g(x) and f(x) for the turning point is ...
2-(-2) = 4
This is the amount by which the graph of f(x) is translated to the right: 4 units.
The vertical difference between the marked points on f(x) and the turning point is 1 unit. On g(x), those same marked points are 3 units away from the turning point vertically. Hence the vertical stretch factor is 3.
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Comment on the transformation of f(x)
Please note that the graph of g(x) is actually related to the graph of f(x) as ...
g(x) = 3·f(x -4) +2
That is, for x=1 on g(x), the y-coordinate is ...
g(1) = 3·f(1 -4) +2 = 3·(-2) +2 = -4 . . . . . . . point (1, -4) on g(x)
For x=3 on g(x), the y-coordinate is ...
g(3) = 3·f(3 -4) +2 = 0 +2 = 2 . . . . . . . . . . point (3, 2) on g(x)
It may seem a little strange that there is a vertical translation of 2 units upward, when the point of inflection has the same vertical location. Actually, that is the clue that there is an upward translation.
The stretch factor operates about the origin, so stretching f(x) by a factor of 3 will make the turning point move from y=-1 to y=3·(-1) = -3. Since it shows on the graph of g(x) at location y=-1, it must have been translated 2 units upward from its stretched location.
