GKT101: General Knowledge for Teachers – Math

1.

$y=x^{2}-2 x-8$

$y-\operatorname{inter}:(0,-8)$

$\begin{array}{r} x-\text { inter: } 0=x^{2}-2 x-8 \\ 0=(x-4)(x+2) \\ x-4=0 \quad x+2=0 \\ +4+4 \quad-2-2 \\ \hline x=4 \quad x=-2 \end{array}$

$(4,0),(-2,0)$

$\begin{aligned} &\text { vertex: } x=\frac{2}{2(1)}=\frac{2}{2}=1\\ &y=(1)^{2}-2(1)-8\\ &y=1-2-8\\ &y=-9\\ &(1,-9) \end{aligned}$

3.

$y=2 x^{2}-12 x+10$

$y \text {-inter: }(0,10)$

$\begin{array}{r} x-\text { inter: } 0=2 x^{2}-12 x+10 \\ 0=2\left(x^{2}-6 x+5\right) \\ 0=2(x-5)(x-1) \\ x-5=0 \quad x-1=0 \\ +5+5+1+1 \\ \hline x=5 \quad x=1 \end{array}$

$(5,0)(1,0)$

$\begin{aligned} &\text { vertex: } x=\frac{12}{2(2)}=\frac{12}{4}=3 \\ &y=2(3)^{2}-12(3)+10 \\ &y=2(9)-36+10 \\ &y=18-36+10 \\ &y=-8 \\ &(3,-8) \end{aligned}$

5.

$y=-2 x^{2}+12 x-18$

$y-\operatorname{inter}:(0,10)$

$\begin{aligned} x-\text { inter: } 0 &=-2 x^{2}+12 x-18 \\ 0 &=-2\left(x^{2}-6 x+9\right) \\ 0 &=-2(x-3)^{2} \\ x-3 &=0 \\ \hline+3 &+3 \\ x &=3 \end{aligned}$

$(3,0)$

$\text { vertex: } x=\frac{-12}{2(-2)}=\frac{-12}{-4}=3$

$\begin{aligned} &y=-2(3)^{2}+12(3)-18 \\ &y=-2(9)+36-18 \\ &y=-18+36-18 \\ &y=0 \\ &(3,0) \end{aligned}$

7.

$y=-3 x^{2}+24 x-45$

$y-\operatorname{inter}:(0,-45)$

$\begin{array}{r} x-\text { inter: } 0=-3 x^{2}+24 x-45 \\ 0=-3\left(x^{2}-8 x+15\right) \\ 0=-3(x-5)(x-3) \\ x-5=0 \quad x-3=0 \\ +5+5 \quad+3 \quad+3 \\ \hline x=5 \quad x=3 \end{array}$

$(5,0)(3,0)$

$\text { vertex: } x=\frac{-24}{2(-3)}=\frac{-24}{-6}=4$

$\begin{aligned} &y=-3(4)^{2}+24(4)-45 \\ &y=-3(16)+96-45 \\ &y=-48+96-45 \\ &y=3 \\ &(4,3) \end{aligned}$

9.

$y=-x^{2}+4 x+5$

$y \text {-inter: }(0,9)$

$\begin{aligned} x-\text { inter: } 0 &=-x^{2}+4 x+5 \\ 0 &=-1\left(x^{2}-4 x-5\right) \\ 0 &=-1(x-5)(x+1) \\ x-5 &=0 \quad x+1=0 \\ +5 &+5-1-1 \\ \hline x &=5 \quad x=-1 \end{aligned}$

$(5,0) \quad(-1,0)$

$\text { vertex: } x=\frac{-4}{2(-1)}=\frac{-4}{-2}=2$

$\begin{aligned} &y=-(2)^{2}+4(2)+5 \\ &y=-4+8+5 \\ &y=9 \\ &(2,9) \end{aligned}$

11.

$y=-x^{2}+6 x-5$

$y \text {-inter: }(0,-5)$

$\begin{array}{r} x-\text { inter: } 0=-x^{2}+6 x-5 \\ 0=-1\left(x^{2}-6 x+5\right) \\ 0=-1(x-1)(x-5) \\ x-1=0 \quad x-5=0 \\ +1+1 \quad+5+5 \\ \hline x=1 \quad x=5 \end{array}$

$(1,0) \quad(5,0)$

$\text { vertex: } x=\frac{-6}{2(-1)}=\frac{-6}{-2}=3$

$\begin{aligned} &y=-(3)^{2}+6(3)-5 \\ &y=-9+18-5 \\ &y=4 \\ &(3,4) \end{aligned}$

13.

$y=-2 x^{2}+16 x-24$

$y-\text { inter: }(0,-24)$

$\begin{array}{r} x-\text { inter: } 0=-2 x^{2}+16 x-24 \\ 0=-2\left(x^{2}-8 x+12\right) \\ 0=-2(x-2)(x-6) \\ x-2=0 \quad x-6=0 \\ +2+2 \quad+6+6 \\ \hline x=2 \quad x=6 \end{array}$

$(2,0) \quad(6,0)$

$\text { vertex: } x=\frac{-16}{2(-2)}=\frac{-16}{-4}=4$

$\begin{aligned} &y=-2(4)^{2}+16(4)-24 \\ &y=-2(16)+64-24 \\ &y=-32+64-24 \\ &y=8 \\ &(4,8) \end{aligned}$

15.

$y=3 x^{2}+12 x+9$

$y-\text { inter: }(0,9)$

$\begin{aligned} x-\text { inter: } 0 &=3 x^{2}+12 x+9 \\ 0 &=3\left(x^{2}+4 x+3\right) \\ 0 &=3(x+1)(x+3) \\ x+1 &=0 \quad x+3=0 \\ -1 &-1 \quad-3-3 \\ \hline x &=-1 \quad x=-3 \end{aligned}$

$(-1,0) \quad(-3,0)$

$\text { vertex: } x=\frac{-12}{2(3)}=\frac{-12}{6}=-2$

$\begin{aligned} &y=3(-2)^{2}+12(-2)+9 \\ &y=3(4)-24+9 \\ &y=12-24+9 \\ &y=-3 \\ &(-2,-3) \end{aligned}$

17.

$y=5 x^{2}-40 x+75$

$y \text {-inter: }(0,75)$

$\begin{array}{r} x-\text { inter: } 0=5 x^{2}-40 x+75 \\ 0=5\left(x^{2}-8 x+15\right. \\ 0=5(x-3)(x-5) \\ x-3=0 x-5=0 \\ +3+3+5+5 \\ \hline x=3 \quad x=5 \end{array}$

$\text { vertex: } \frac{40}{2(5)}=\frac{40}{10}=4$

$\begin{aligned} &y=5(4)^{2}-40(4)+75 \\ &y=5(16)-160+75 \\ &y=80-160+75 \\ &y=-5 \\ &(4,-5) \end{aligned}$

19.

$y=-5 x^{2}-60 x-175$

$y-\text { inter: }(0,-175)$

$\begin{gathered} x-\text { inter: } 0=-5 x^{2}-60 x-175 \\ 0=-5\left(x^{2}+12 x+35\right) \\ 0=-5(x+5)(x+7) \\ x+5=0 \quad x+7=0 \\ -5-5 \quad-7-7 \\ \hline x=-5 \quad x=-7 \end{gathered}$

$(-5,0) \quad(-7,0)$

$\text { vertex: } x=\frac{60}{2(-5)}=\frac{60}{-10}=-6$

$\begin{aligned} &y=-5(-6)^{2}-60(-6)-175 \\ &y=-5(36)+360-175 \\ &y=-180+360-175 \\ &y=5 \\ &(-6,5) \end{aligned}$