### Transformation of function graph

Question Sample Titled 'Transformation of function graph'

Which of the following may represent the graph of ${y}={f{{\left({x}\right)}}}$ and the graph of ${y}=-{f{{\left(-{x}\right)}}}$ on the same rectangular coordinate system?

A

${x}$${y}$${y}={f{{\left({x}\right)}}}$${y}=-{f{{\left(-{x}\right)}}}$
B

${x}$${y}$${y}={f{{\left({x}\right)}}}$${y}=-{f{{\left(-{x}\right)}}}$
C

${x}$${y}$${y}={f{{\left({x}\right)}}}$${y}=-{f{{\left(-{x}\right)}}}$
D

${x}$${y}$${y}={f{{\left({x}\right)}}}$${y}=-{f{{\left(-{x}\right)}}}$

 The graph of ${y}=-{f{{\left(-{x}\right)}}}$ can be obtained by reflecting the graph of ${y}={f{{\left({x}\right)}}}$ along the x-axis and y-axis. For example, if the vertex of the graph of the original function ${y}={f{{\left({x}\right)}}}$ is ${\left({2},{2}\right)}$ and the graph opens upwards. Then the vertex of the graph of new function ${y}=-{f{{\left(-{x}\right)}}}$ is ${\left(-{2},-{2}\right)}$ and the graph opens downwards. Note: The order of the two performed transformations is not important.

 Let the graph of the original function ${y}={f{{\left({x}\right)}}}$ be ${\left({x}-{2}\right)}^{{2}}+{2}$ ${f{{\left({x}\right)}}}={\left({x}-{2}\right)}^{{2}}+{2}$ ${f{{\left(-{x}\right)}}}$ $={\left[{\left(-{x}\right)}-{2}\right]}^{{2}}+{2}$ Replaced ${x}$ by ${\left(-{x}\right)}$  $={\left[-{\left({x}+{2}\right)}^{{2}}\right]}+{2}$ Factorize out the negative sign  $={\left({x}+{2}\right)}^{{2}}+{2}$ $-{f{{\left(-{x}\right)}}}$ $=-{\left[{\left({x}+{2}\right)}^{{2}}+{2}\right]}$  $=-{\left({x}+{2}\right)}^{{2}}-{2}$ Therefore, new vertex $={\left(-{2},-{2}\right)}$ and the graph open downwards.

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