Question

Please help! The graph of function g(x) is a transformation of the graph of function f(x) = x^2.

g(x) = _________

A.) 2x^2+3
B.) -2x^2-3
C.) -1/2x^2-3
D.) 1/2x^2-3

Answer 1

B is the correct function

Answer 2

B.) -2x^2-3

Explanation:

You only need to look at the values of the offered answers for x=1 in order to determine the correct one.

A.) for x = 1, 2·1+3 = 5

B.) for x = 1, -2·1-3 = -5 . . . . matches the graph

C.) for x = 1, -1/2·1-3 = -3 1/2

D.) for x = 1, 1/2·1-3 = -2 1/2

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Comparing the values at x=1 to the values at x=0, you find that f(1)-f(0) = 1, while g(1)-g(0) = -2. This tells you the multiplier is -2, so matches choice B.

The fact that the vertex of g(x) is 3 units below that of f(x) tells you that -3 has been added to the product -2x^2.

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Comment on vertical scale factor

Since 1^2 = 1, it is often convenient to look at a point that is 1 unit away from the vertex of the function. The function value relative to the vertex value at that point will tell you the vertical scale factor.

For example, we know that the coefficient of x^2 is 1 in f(x), and we see on the graph that f(1) is 1 unit above the vertex at (0, 0). The value of g(1) is -5, which is 2 units below the vertex of g(x) at (0, -3). So, the vertical scale factor is -5-(-3) = -2.