**Given the function f(x) = x2 and k = 3, which of the following represents the graph becoming more narrow?**

A. f(x)+k B. kf(x) C. f(x+k) D. f(k-x)

**Answer**

Functions A and C and D are translations - this means that they are exactly the same shape as f(x), but are displaced on the axes.

A translation in the y-direction by k units is represented by f(x) + k, and a translation in the x-direction by k units is represented by f(x - k).

This can be summarised in column vector A translation of the graph y = f(x), by the vector [tex] \left[\begin{array}{ccc}a\\b\end{array}

ight] [/tex] results in the graph defined by y - b = f(x -

a).

This is done by replacing x with (x -

a) and y with (y -

b).

Finally it is rearranged to make y the subject, as is often the y = f(x -

a) + b.

Since the graph needs to change shape, this requires a 'stretch'.

A stretch is performed by multiplying x or y by a constant, which here is k.

To stretch in the x-direction by a factor n, replace x with (1/n)x.

Similarly to stretch in the y-direction by a factor n, replace y with (1/n)y.

For example to stretch y = f(x) by factor n in the x-direction, it would become y = f((1/n)x).

A simple quadratic graph like y = x^2 behaves slightly differently to most other graphs, in that it can narrowed by a stretch in the x or y direction.

This means that a more convenient option is to perform a stretch in the y direction, since the multiplier lies outside the function notation.

The stretching factor is 3, so we replace y with (1/3) (1/3)y = f(x) y = 3f(x) Since k = 3, this becomes y = kf(x) ... and hence the correct answer is B

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