Completing the square followed by translating vertically and reflecting along y-axis

Question Sample Titled 'Completing the square followed by translating vertically and reflecting along y-axis'

Let ${f{{\left({x}\right)}}}=\dfrac{{1}}{{2}}{x}^{{2}}-{8}{x}+{43}$ .

 (a) Using the method of completing the square, find the coordinates of the vertex of the graph of ${y}={f{{\left({x}\right)}}}$ . (2 marks) (b) The graph of ${y}={g{{\left({x}\right)}}}$ is obtained by translating the graph of ${y}={f{{\left({x}\right)}}}$ vertically. If the graph of ${y}={g{{\left({x}\right)}}}$ touches the ${x}$-axis, find ${g{{\left({x}\right)}}}$ . (2 marks) (c) Under a transformation, ${f{{\left({x}\right)}}}$ is changed to $\dfrac{{1}}{{2}}{x}^{{2}}+{8}{x}+{43}$. Describe the geometric meaning of the transformation.  (2 marks)

 (a) ${f{{\left({x}\right)}}}$ $=\dfrac{{1}}{{2}}{x}^{{2}}-{8}{x}+{43}$ $=\dfrac{{1}}{{2}}{\left({x}^{{2}}-{16}{x}\right)}+{43}$ $=\dfrac{{1}}{{2}}{\left({x}^{{2}}-{16}{x}-{8}^{{2}}+{8}^{{2}}\right)}+{43}$ $=\dfrac{{1}}{{2}}{\left({x}-{8}\right)}^{{2}}+{11}$ 1A ∴   The coordinates of the vertex of the graph $={\left({8},{11}\right)}$ 1M  (b) The graph of ${y}$ $={g{{\left({x}\right)}}}$ touching the ${x}$-axis means that the vertex lies on the ${x}$-axis, i.e. its ${y}$-coordinate is ${0}$ . The graph of ${y}={g{{\left({x}\right)}}}$ is obtained by translating ${y}={f{{\left({x}\right)}}}$ downwards by ${11}$ units. 1M can be absorbed ∴  ${g{{\left({x}\right)}}}=\dfrac{{1}}{{2}}{\left({x}-{8}\right)}^{{2}}$ 1A  (c) $\dfrac{{1}}{{2}}{x}^{{2}}+{8}{x}+{43}$ $=\dfrac{{1}}{{2}}{\left(-{x}\right)}^{{2}}+{8}{\left(-{x}\right)}+{43}$ $={f{{\left(-{x}\right)}}}$ 1M can be absorbed ∴   The transformation is reflecting ${f{{\left({x}\right)}}}$ along the ${y}$-axis. 1A

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