Contoh Soal 5 Transformasi Geometri

$\begin{array}{l}\\ 21.& \text{Jika setiap titik pada grafik dengan}\\ & \text{dengan persamaan}y=\sqrt{x}\text{dicerminkan}\\ & \text{terhadap garis}y=x,\text{maka persamaan}\\ & \text{grafik yang dihasilkan adalah}...\\ & \begin{array}{l}\\ \text{a}.y=x^2,x\ge 0& & \\ \text{b}.y=-\sqrt{x},x\ge 0& \\ \text{c}.y=-x^2,x\le 0& \\ \text{d}.y=\sqrt{-x},x\le 0\\ \text{e}.y=-\sqrt{-x},x\le 0\end{array}\\ \\ & \mathbf{\text{UMB Tahun 2011 Kode 152}}\\ \\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{a}}\\ & \begin{array}{rl}\text{Dike}& \text{tahui bahwa}:\\ y& =\sqrt{x},\text{atau}{y}^2=x\\ \mathbf{\text{Alt}}& \mathbf{\text{ernatif 1}}\\ \text{mak}& \text{a}\text{saat dicerminkan terhadap}\\ \text{gari}& \text{s}y=x,\text{adalah}{x}^2=y\\ \text{atau}& y=x^2.\\ \mathbf{\text{Alt}}& \mathbf{\text{ernatif 2}}\\ \text{Jika}& \text{ingin dikerjakan dengan rumus}\\ \left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)& =M_{x=y}\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =\left(\begin{array}{c}y\\ x\end{array}\right)\\ \text{Sela}& \text{njutnya hasilnya disubstitusikan}\\ \text{ke p}& \text{ersamaan}y=\sqrt{x}\Rightarrow x^{\prime }=\sqrt{y^{\prime }}\\ \sqrt{y^{\prime }}& =x^{\prime }\text{maka}\\ y^{\prime }& ={\left(x^{\prime }\right)}^2\text{selanjutnya}\\ y& =x^2\end{array}\end{array}$.

Sebelum dicerminkan terhadap garis y=x

Gambar kurva/grafik setelah cerminkan terhadap garis y=x

$\begin{array}{l}\\ 22.& \text{Transformasi}T\text{adalah pencerminan}\\ & \text{terhadap garis}y={\displaystyle \frac{x}{3}\text{dilanjutkan oleh}}\\ & \text{pencerminan terhadap garis}y=-3x.\\ & \text{Matriks yang bersesuian dengan}\\ & \text{transformasi}T\text{adalah}...\\ & \begin{array}{l}\\ \text{a}.\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)& & \\ \text{b}.\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)& \\ \text{c}.\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)& \\ \text{d}.\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\\ \text{e}.\left(\begin{array}{cc}0& -1\\ -1& 0\end{array}\right)\end{array}\\ \\ & \mathbf{\text{SBMPTN Tahun 2013 Kode 433}}\\ \\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{b}}\\ & \begin{array}{rl}\text{Dike}& \text{tahui bahwa}:\\ \text{sebu}& \text{ah persamaan garis lurus}\\ \text{dapa}& \text{t dituliskan dengan}:y=mx\\ \text{Dike}& \text{tahui pula bahwa ada 2 garis}:\\ y_1& ={\displaystyle \frac{1}{3}x\text{dan}{y}_2=-3x}\\ \text{seba}& \text{gai representasi transformasi}T.\\ \text{Kare}& \text{na}{m}_1\times m_2=\left({\displaystyle \frac{1}{3}}\right)(-3)=-1\\ \text{bera}& \text{rti 2 garis di atas saling tegak}\\ \text{luru}& \text{s dan hal ini seperti rotasi 2}\\ \text{kali}& {90}^{\circ }\text{atau}{180}^{\circ }\\ \text{Jadi},& T=\left(\begin{array}{cc}\cos {180}^{\circ }& -\sin {180}^{\circ }\\ \sin {180}^{\circ }& \cos {180}^{\circ }\end{array}\right)\\ \iff & T=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)\end{array}\end{array}$.

DAFTAR PUSTAKA

1. Johanes, Kastolan, Sulasim, 2006. Kompetensi Matematika 3A SMA Kelas XII Program IPA Semester Pertama. Jakarta: YUDHISTIRA.
2. Nugroho, P. A. Gunarto, D. 2013. Big Bank Soal-Bahas MAtematika SMA/MA. Jakarta: WAHYUMEDIA.
3. Sharma,S.N., dkk. 2017. Jelajah Matematika SMA Kelas XI Program Wajib. Jakarta: YUDHISTIRA.