Question

The functions f(x) and g(x) are shown on the graph.

The graph shows a v-shaped graph, labeled f of x, with a vertex at the origin, a point at negative 1 comma 1, and a point at 1 comma 1. The graph shows another v-shaped graph, labeled g of x, which has a narrower opening than f of x, with a vertex at the origin, a point at negative 1 comma 4, and a point at 1 comma 4.

What transformation of f(x) will produce g(x)?

g of x equals f of one fourth times x
g of x equals negative one fourth times f of x
g(x) = f(4x)
g(x) = −4f(x)

Answer 1

(c) g(x) = f(4x)

Explanation:

You want the transformation of f(x) = |x| that produces the narrower graph g(x) with the same vertex and through point (1, 4).

Scaling

The narrower graph can be obtained by compressing the graph horizontally by a factor of 4, or by expanding it vertically by a factor of 4.

Horizontal compression by a factor of k gives ...

g(x) = f(kx)

Vertical expansion by a factor of k gives ...

g(x) = k·f(x)

When k > 0 and f(x) = |x|, these are equivalent:

g(x) = |k·x| = k·|x|

Application

Here, the transformation multiplies the y-value by 4: point (1, 1) is on f(x), and point (1, 4) is on g(x). So, k = 4.

g(x) = 4·f(x) . . . . not a choice

g(x) = f(4x) . . . . . the third choice shown

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Additional comment

Effectively the vertical scaling of this graph is expressed in the equation as a horizontal scaling. If the scaling were actually horizontal, we would expect the point given for the graph of f(x) to be (4, 4). Horizontal compression by a factor of 4 would move it to (1, 4).

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