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

 $\begin{array}{l}\\ 6.& \text{Fungsi kuadrat}f(x)=(2x+p{)}^2+q\\ & \text{dengan titik balik minimum}(-1,3).\\ & \text{Nilai}p+q\text{adalah}....\\ & \begin{array}{llllllll}\text{A}.& {\displaystyle 2}& & & & & \text{D}.& {\displaystyle 6}\\ \text{B}.& {\displaystyle 4}& & \text{C}.& {\displaystyle 5}& & \text{E}.& 7\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{C}}\\ & \begin{array}{rl}& \text{Diketahui FK}:f(x)=(2x+p{)}^2+q,\text{dengan}\\ & \text{titik}(x_{ss},y_{ss})=(-1,3),\text{maka}\\ & f(x)=4x^2+4px+p^2+q\text{dengan}\\ & x_{ss}=-1=-{\displaystyle \frac{b}{2a}\iff 1={\displaystyle \frac{4p}{2.4}\iff p=2}}\\ & \text{Selanjutnya}\\ & f(-1)=(2.(-1)+2{)}^2+q=3\iff q=3\\ & \text{maka}\\ & p+q=2+3=5\end{array}\end{array}$.

$\begin{array}{l}\\ 7.& \text{Jika grafik fungsi kuadrat}f(x)=a{x}^2+x+c\\ & \text{dengan titik balik minimum}(-1,3)\text{dan melalui}\\ & (2,12)\text{maka}a+b+c\text{sama dengan}....\\ & \begin{array}{llllllll}\text{A}.& {\displaystyle 7}& & & & & \text{D}.& {\displaystyle 13}\\ \text{B}.& {\displaystyle 9}& & \text{C}.& {\displaystyle 11}& & \text{E}.& 15\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{A}}\\ & \begin{array}{rl}& \text{Diketahui FK}:f(x)=a{x}^2+bx+c,\text{dengan}\\ & \text{titik}(x_{ss},y_{ss})=(-1,3)\text{dan melalui titik}\\ & (2,12),\text{maka}\\ & \overline{\begin{array}{cc}\begin{array}{rl}& 12=4a+2b+c\\ & 3=a-b+c\\ & \mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}-\\ & 9=3a+3b\\ & \iff 3=a+b\end{array}& \begin{array}{rl}& -{\displaystyle \frac{b}{2a}=-1,\text{maka}}\\ & 2a-b=0,\text{dan ingat}\\ & a+b=3\\ & \mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}+\\ & 3a=3\\ & \iff a=1,\text{maka}b=2\end{array}\end{array}}\\ & \text{dan}\\ & a-b+c=3\Rightarrow 1-2+c=3\Rightarrow c=4\\ & \text{Jadi, nilai}a+b+c=1+2+4=7\end{array}\end{array}$.

$\begin{array}{l}\\ 8.& \text{Nilai minimum grafik fungsi}f(x)=a{x}^2-2x+8\\ & \text{adalah 5. Nilai }6a\text{sama dengan}....\\ & \begin{array}{llllllll}\text{A}.& {\displaystyle 1}& & & & & \text{D}.& {\displaystyle 9}\\ \text{B}.& {\displaystyle 2}& & \text{C}.& {\displaystyle 4}& & \text{E}.& 12\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{B}}\\ & \begin{array}{rl}& \text{Diketahui FK}:f(x)=a{x}^2-2x+8,\text{dengan}\\ & \text{titik}(x_{ss},y_{ss})=(-{\displaystyle \frac{b}{2a},5})\text{maka}\\ & \bullet x_{ss}=-{\displaystyle \frac{b}{2a}=-\frac{-2}{2a}=\frac{1}{a}}\\ & \bullet y_{ss}=f\left(x_{ss}\right)=a{\left({\displaystyle \frac{1}{a}}\right)}^2-2\left({\displaystyle \frac{1}{a}}\right)+8=5\\ & \iff {\displaystyle \frac{1}{a}-\frac{2}{a}=5-8\iff -{\displaystyle \frac{1}{a}=-3}}\\ & \iff a={\displaystyle \frac{1}{3}}\\ & \text{maka nilai}6a=6\left({\displaystyle \frac{1}{3}}\right)=2\end{array}\end{array}$.

$\begin{array}{l}\\ 9.& \text{Jika kurva fungsi}f(x)=x^2+bx+c\\ & \text{memotong sumbu-X di}(1,0)\text{dan}(5,0),\\ & \text{maka nilai}{b}^2-c^2\text{sama dengan}....\\ & \begin{array}{llllllll}\text{A}.& {\displaystyle -11}& & & & & \text{D}.& {\displaystyle 11}\\ \text{B}.& {\displaystyle -3}& & \text{C}.& {\displaystyle 6}& & \text{E}.& 13\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{D}}\\ & \begin{array}{rl}& \text{Diketahui FK}:f(x)=x^2+bx+c,\text{memotong}\\ & \text{sumbu-X di}(1,0)\mathrm{\&}(5,0)\text{artinya}{x}_1=1\mathrm{\&}{x}_2=5\\ & \text{maka}\\ & x_{ss}={\displaystyle \frac{-b}{2.1}={\displaystyle \frac{x_1+x_2}{2}=\frac{1+5}{2}\iff b=-6}}\\ & \text{dan kita juga memiliki}\\ & f(1)=1+b+c=0\Rightarrow c=-b-1=6-1=5\\ & \text{Sehingga}\\ & b^2-c^2=(-6{)}^2-5^2=36-25=11\end{array}\end{array}$.

$\begin{array}{l}\\ 10.& \text{Perhatikan gambar berikut}\end{array}$.

$.\begin{array}{l}\\ & \text{Persamaan grafik fungsi kuadrat pada}\\ & \text{pada gambar tersebut di atas adalah}....\\ & \begin{array}{lllllll}\text{A}.& {\displaystyle y=6x^2-12x+18}& & & & & \\ \text{B}.& {\displaystyle y=6x^2+12x+16}& & \\ \text{C}.& y=6x^2-24x+17\\ \text{D}.& y=6x^2-24x+19\\ \text{E}.& y=6x^2-24x+29\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{D}}\\ & \begin{array}{rl}& \text{Diketahui FK}:f(x)=a{x}^2+bx+c,\text{dengan}\\ & \text{koordinat}(x_{ss},y_{ss})=(2,-5)\text{dan melalui}\\ & \text{titik}(3,1),\text{maka}\\ & y=a(x-x_{ss}{)}^2+y_{ss}\iff 1=a(3-2{)}^2+(-5)\\ & \iff 6=a{.1}^2\iff a=6\\ & \text{maka persamaan fungsi kuadratnya adalah}:\\ & y=a(x-x_{ss}{)}^2+y_{ss}\\ & \iff y=6(x-2{)}^2-5\iff y=6(x^2-4x+4)-5\\ & \iff y=6x^2-24x+19\end{array}\end{array}$