Contoh Soal 2 Fungsi Kuadrat (Kelas X/Fase E Semester Genap) Tahun 2024

 $\begin{array}{ll}\\ 6.&\textrm{Fungsi kuadrat}\: \: f(x)=(2x+p)^{2}+q\\ &\textrm{dengan titik balik minimum}\: \:  (-1,3).\\ &\textrm{Nilai}\: \: p+q\: \: \textrm{adalah}\: ....\\ &\begin{array}{lllllllll}\textrm{A}.&\displaystyle 2&&&&&\textrm{D}.&\displaystyle 6\\ \textrm{B}.&\displaystyle 4&&\textrm{C}.&\color{red}\displaystyle 5&&\textrm{E}.&7 \end{array}\\\\ &\textbf{Jawab}:\textbf{C}\\ &\begin{aligned}&\textrm{Diketahui FK}:f(x)=(2x+p)^{2}+q,\: \: \textrm{dengan}\\ &\textrm{titik}\: \: \left ( x_{ss},y_{ss} \right )=(-1,3),\: \: \textrm{maka}\\ &f(x)=4x^{2}+4px+p^{2}+q\: \: \textrm{dengan}\\ &x_{ss}=-1=-\displaystyle \frac{b}{2a}\Leftrightarrow 1=\displaystyle \frac{4p}{2.4}\Leftrightarrow p=2\\ &\textrm{Selanjutnya}\\ &f(-1)=(2.(-1)+2)^{2}+q=3\Leftrightarrow q=3\\ &\textrm{maka}\\ &p+q=\color{red}2+3=5 \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 7.&\textrm{Jika grafik fungsi kuadrat}\: \: f(x)=ax^{2}+x+c\\ &\textrm{dengan titik balik minimum}\: \:  (-1,3)\: \: \textrm{dan melalui}\\ &(2,12)\: \: \textrm{maka}\: \: a+b+c\: \: \textrm{sama dengan}\: ....\\ &\begin{array}{lllllllll}\textrm{A}.&\color{red}\displaystyle 7&&&&&\textrm{D}.&\displaystyle 13\\ \textrm{B}.&\displaystyle 9&&\textrm{C}.&\displaystyle 11&&\textrm{E}.&15 \end{array}\\\\ &\textbf{Jawab}:\textbf{A}\\&\begin{aligned}&\textrm{Diketahui FK}:f(x)=ax^{2}+bx+c,\: \: \textrm{dengan}\\ &\textrm{titik}\: \: \left ( x_{ss},y_{ss} \right )=(-1,3)\: \: \textrm{dan melalui titik}\\ &(2,12)\: ,\: \textrm{maka}\\ &\begin{array}{c|c}\hline \begin{aligned} &12=4a+2b+c\\ &3=a-b+c\\&\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ -\\ &9=3a+3b\\ &\Leftrightarrow 3=a+b \end{aligned}&\begin{aligned}&-\displaystyle \frac{b}{2a}=-1,\: \: \textrm{maka}\\&2a-b=0,\: \: \textrm{dan ingat}\\ &a+b=3\\ &\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ +\\ &3a=3\\ &\Leftrightarrow a=1,\: \: \textrm{maka}\: \: b=2 \end{aligned} \end{array}\\ &\textrm{dan}\\ &a-b+c=3\Rightarrow 1-2+c=3\Rightarrow c=4\\ &\textrm{Jadi, nilai}\: \: a+b+c=\color{red}1+2+4=7  \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 8.&\textrm{Nilai minimum grafik fungsi}\: \: f(x)=ax^{2}-2x+8\\ &\textrm{adalah 5. Nilai }\: \:  6a\: \: \: \textrm{sama dengan}\: ....\\ &\begin{array}{lllllllll}\textrm{A}.&\displaystyle 1&&&&&\textrm{D}.&\displaystyle 9\\ \textrm{B}.&\color{red}\displaystyle 2&&\textrm{C}.&\displaystyle 4&&\textrm{E}.&12 \end{array}\\\\ &\textbf{Jawab}:\textbf{B}\\ &\begin{aligned}&\textrm{Diketahui FK}:f(x)=ax^{2}-2x+8,\: \: \textrm{dengan}\\ &\textrm{titik}\: \: \color{red}\left ( x_{ss},y_{ss} \right )=\left (-\displaystyle \frac{b}{2a},5  \right )\: \: \color{black}\textrm{maka}\\ &\bullet \quad x_{ss}=-\displaystyle \frac{b}{2a}=-\frac{-2}{2a}=\color{red}\frac{1}{a}\\ &\bullet \quad y_{ss}=f\left ( x_{ss} \right )=a\left ( \displaystyle \frac{1}{a} \right )^{2}-2\left ( \displaystyle \frac{1}{a} \right )+8=\color{red}5\\ &\qquad\quad \Leftrightarrow \displaystyle \frac{1}{a}-\frac{2}{a}=5-8\Leftrightarrow -\displaystyle \frac{1}{a}=-3\\ &\qquad\quad \Leftrightarrow a=\displaystyle \frac{1}{3}\\ &\textrm{maka nilai}\: \: 6a=6\left ( \displaystyle \frac{1}{3} \right )=\color{red}2  \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 9.&\textrm{Jika kurva fungsi}\: \: f(x)=x^{2}+bx+c\\ &\textrm{memotong sumbu-X di}\: \:  (1,0)\: \: \: \textrm{dan}\: \: (5,0),\\ &\textrm{maka nilai}\: \: b^{2}-c^{2}\: \: \textrm{sama dengan}\: ....\\ &\begin{array}{lllllllll}\textrm{A}.&\displaystyle -11&&&&&\textrm{D}.&\color{red}\displaystyle 11\\ \textrm{B}.&\displaystyle -3&&\textrm{C}.&\displaystyle 6&&\textrm{E}.&13 \end{array}\\\\ &\textbf{Jawab}:\textbf{D}\\ &\begin{aligned}&\textrm{Diketahui FK}:f(x)=x^{2}+bx+c,\: \: \textrm{memotong}\\ &\textrm{sumbu-X di}\: \: (1,0)\:  \& \: (5,0)\: \: \textrm{artinya}\: x_{1}=1\: \&\: x_{2}=5\\ &\textrm{maka}\\ &x_{ss}=\displaystyle \frac{-b}{2.1}=\displaystyle \frac{x_{1}+x_{2}}{2}=\frac{1+5}{2}\Leftrightarrow b=-6\\ &\textrm{dan kita juga memiliki}\\ &f(1)=1+b+c=0\Rightarrow c=-b-1=6-1=5\\ &\textrm{Sehingga}\\ &b^{2}-c^{2}=(-6)^{2}-5^{2}=36-25=\color{red}11 \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 10.&\textrm{Perhatikan gambar berikut} \end{array}$.

$.\quad\begin{array}{ll}\\ &\textrm{Persamaan grafik fungsi kuadrat pada}\\ &\textrm{pada gambar tersebut di atas adalah}\: ....\\ &\begin{array}{lllllllll}\textrm{A}.&\displaystyle y=6x^{2}-12x+18&&&&&\\ \textrm{B}.&\displaystyle y=6x^{2}+12x+16&&\\ \textrm{C}.&y=6x^{2}-24x+17\\ \textrm{D}.&\color{red}y=6x^{2}-24x+19\\ \textrm{E}.&y=6x^{2}-24x+29 \end{array}\\\\ &\textbf{Jawab}:\textbf{D}\\ &\begin{aligned}&\textrm{Diketahui FK}:f(x)=ax^{2}+bx+c,\: \: \textrm{dengan}\\ &\textrm{koordinat}\: \: \left ( x_{ss},y_{ss} \right )=(2,-5)\: \: \textrm{dan melalui}\\ &\textrm{titik}\: \: (3,1)\: ,\: \textrm{maka}\\ &y=a(x-x_{ss})^{2}+y_{ss}\Leftrightarrow 1=a(3-2)^{2}+(-5)\\ &\Leftrightarrow 6=a.1^{2}\Leftrightarrow a=6\\ &\textrm{maka persamaan fungsi kuadratnya adalah}:\\ &y=a(x-x_{ss})^{2}+y_{ss}\\ &\Leftrightarrow y=6(x-2)^{2}-5\Leftrightarrow y=6(x^{2}-4x+4)-5\\ &\Leftrightarrow y=\color{red}6x^{2}-24x+19 \end{aligned} \end{array}$