Algebra - Transformations

Section 4.6 : Transformations

5. Use transformations to sketch the graph of the following function.

\[f\left( x \right) = - {x^3}\]

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Let’s first identify the “base” function (i.e. the function we are transforming). In this case it looks like we are transforming $g\left( x \right) = {x^3}$ .

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The function that have here looks like it can be written as,

\[f\left( x \right) = - {x^3} = - g\left( x \right)\]

Therefore, we can see that the graph of $f\left( x \right)$ is simply going to be the graph of $g\left( x \right)$ reflected about the $x$-axis.

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Here is a sketch of both the base function (blue dashed curve) and the function we were asked to graph (red solid curve).

$The graphs of the base function $g\left( x \right) = {x^3}$, the blue dashed curve, and $f\left( x \right) = - {x^3}$, the red solid curve. The solid curve is the dashed curve reflected about the x-axis, so the dashed curve rises from the lower left to the upper right while the solid curve falls from the upper left to the lower right. The two cross at the origin.$