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The graph of F(x), shown below, resembles the graph of G(x) = x², but it has been stretched and shifted. Which of the following could be the equation of F(x)?

A. F(x) = 2/5 x² +6
B. F(x) = (1/3x)^2 +2
C. F(x) = 3x² -2
D. F(x) = (9x)² +2 ​

The graph of F(x), shown below, resembles the graph of G(x) = x², but it has been-example-1

1 Answer

7 votes

Answer:

D. f(x) = (9x)² +2

Explanation:

The function transformations we usually study include horizontal and vertical expansion (or compression), and horizontal and vertical translation. These can be summarized in the equation ...

g(x) = v·f((x -a)/h) +b

where v and h are vertical and horizontal expansion factors, respectively, and (a, b) is the (right, up) translation.

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analysis

The graph of F(x) has had its vertex moved 2 units upward (b=2), It is distinctly narrower than G(x), which could be the result of vertical expansion (v>1), or horizontal compression (h<1). Since the graph of F(x) has been compressed horizontally, the value of h must be less than 1.

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answer selections

There are two answer choices with b=2.

Choice B has h=3, a widening of the graph. Choice D has h=1/9, a narrowing of the graph.

The equation of the graph could be F(x) = (9x)² +2. The attachment shows this is an appropriate choice.

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Additional comment

You will notice that in this function, the vertical expansion factor is the reciprocal of the square of the horizontal expansion factor. In this case, h=1/9 is equivalent to v=81, a vertical expansion by a factor of 81.

F(x) = (x/(1/9))² +2 = 81x² +2

The given graphs are not detailed enough to be able to judge either of these factors directly. We can estimate that h=1/9 is approximately correct by looking at the values of x where the graph has its upper boundary. At those points, x≈3 on the G(x) graph, and x<1/2 on the F(x) graph, consistent with a value of about x=1/3. The "expansion" factor is then (1/3)/3 = 1/9.

The graph of F(x), shown below, resembles the graph of G(x) = x², but it has been-example-1
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