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Graph Transformations 2

It is often useful to be able to apply more elaborate graph transformations. The methodology is unchanged: draw the function, see how it changes a few points and extrapolate to get the full graph.

  • $$f(\vert x\vert)$$: Horizontal reflection of the graph of $$f$$ for $$x\ge0$$.
  • $$\vert f(x)\vert$$: Vertical reflection of the negative part of the graph of $$f$$.
  • $$1/f(x)$$: roots of $$f$$ (solutions of $$f(x)=0$$) are transformed into vertical asymptotes; monotonicity is transformed into inverse monotonicity (if $$f$$ is increasing, $$1/f$$ is decreasing); when $$\vert f(x)\vert = 1$$, graphs $$f$$ and $$1/f$$ intersect.
  • $$y^2=f(x)$$: when $$f(x)\lt 0$$, $$y$$ is undefined; if $$f(x)\ge 0$$, $$y$$ is the graph superposition of the graphs of the functions $$\sqrt{f(x)}$$ and $$-\sqrt{f(x)}$$; monotonicity of $$\sqrt{f(x)}$$ and $$f$$ is same; when $$f(x) = 1$$, graphs $$f$$ and $$\sqrt{f}$$ intersect.
Transformation of graph of <span style=$$\sin x$$. A: $$f(\vert x\vert)$$; B: $$\vert f(x)\vert$$; C: $$1/f(x)$$; D: $$y^2=f(x)$$" src="" onload="conditionalLoadImage(this, '', 'dfunction_18.png', true, { width: 191, height: 160 }, '')" name="dfunction_18.png" sub="Transformation of graph of $$\sin x$$. A: $$f(\vert x\vert)$$; B: $$\vert f(x)\vert$$; C: $$1/f(x)$$; D: $$y^2=f(x)$$" />
Transformation of graph of $$\sin x$$. A: $$f(\vert x\vert)$$; B: $$\vert f(x)\vert$$; C: $$1/f(x)$$; D: $$y^2=f(x)$$