Algebra - Transformations

Section 4.6 : Transformations

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

\[f\left( x \right) = {x^3} - 2\]

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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,

\[f\left( x \right) = {x^3} - 2 = g\left( x \right) - 2\]

Therefore, we can see that the graph of $f\left( x \right)$ is simply going to be the graph of $g\left( x \right)$ shifted down by 2.

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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 $f\left( x \right) = {x^3} - 2$, the red solid curve. Both curves have the same shape, flattening out as they pass through their center point and rising steeply on either end, but the solid curve sits 2 units below the dashed one and flattens out at $\left( {0, - 2} \right)$ instead of the origin.$