Answer:
(a) (9, -54)
(b) D: [-4, 9]; R: [-108, 135]
Explanation:
Given point (54, -18) lies on the graph of P(x), which has domain [-24, 54] and range [-36,45], you want the domain and range of Q(x) = 3P(6x) and the point we know must lie on its graph.
Scaling
A function is expanded vertically by a factor of k by multiplying its values by k:
Q(x) = kP(x) . . . . . . expands P(x) vertically by factor k
A function is compressed horizontally by a factor of m by multiplying its input values by m:
Q(x) = P(mx) . . . . . . compresses P(x) horizontally by factor m
Application
In this problem, we have ...
Q(x) = 3P(6x)
This relation has k=3 and m=6, meaning the graph of Q(x) is expanded vertically by a factor of 3 and compressed horizontally by a factor of 6 from the graph of P(x).
(a) Point
To find the point (x, P(x)) on the graph of Q(x), we must compress the x-value by a factor of 6, and expand the P(x) value by a factor of 3:
(54, -18) ⇒ (54/6, 3(-18)) = (9, -54)
The point (9, -54) must lie on the graph of Q(x).
(b) Domain and Range
Similarly, the domain is compressed by a factor of 6:
D: [-24, 54]/6 = [-4, 9] . . . . the domain of Q(x)
And the range is expanded by a factor of 3:
R: 3[-36, 45] = [-108, 135] . . . . the range of Q(x)
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Additional comment
In the attachment, we have made up the function p(x) so that it has the required characteristics (domain, range, point). The corresponding q(x) is shown, along with the corresponding domain, range, and point.
The given point is at the right end of the function's graph, at the upper end of the domain. Since this point does not correspond to an extreme of range, we know the function has at least one turning point.
Note that identifying the scale factors is basically a problem in pattern matching.