Answer:
(a) The graph is compressed horizontally by a factor of 2, shifted left 3 units, reflected over the x-axis, and translated up 3 units
Explanation:
The transformations under discussion here are ...
- y = k·f(x) . . . . . . reflection over x-axis when k=-1
- y = f(kx) . . . . . . horizontal compression by a factor of k
- y = f(x -k) . . . . . right shift by k units
- y = f(x) +k . . . . translation upward by k units.
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application
The application of these transformations is a bit easier to see if we rewrite the transformed function to ...
y = -(2(x -(-3)))² +3
The leading minus sign shows a reflection over the x-axis.
The factor of 2 shows a horizontal compression by a factor of 2.
The subtraction of -3 from x shows a translation left of 3 units. (Right by -3 units.)
The addition of 3 to the function value shows a translation upward of 3 units.
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summary
The combined transformation can be described as ...
The graph is compressed horizontally by a factor of 2, shifted left 3 units, reflected over the x-axis, and translated up 3 units.
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Additional comment
The base of the square can be obtained by either of these methods:
x² ⇒ (x +6)² ⇒ (2x +6)² . . . translated left 6 units, then the translated function compressed by a factor of 2
x² ⇒ (2x)² ⇒ (2(x +3))² . . . compressed by a factor of 2, then translated left 3 units.
We choose to present the translation last, because it seems more obvious when looking at the graph. The parent vertex is at x=0; the transformed vertex is at x=-3, an apparent translation of -3 units.
It takes more mental effort to understand this is the same as translation -6 units, then compression by a factor of 2, wherein the compression also affects the translation.