Practice with Graphing Quadratic Equations
Practice
Answers
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.
\(y=-5 x^{2}-60 x-175\)
\(y-\text { inter: }(0,-175)\)
\(\begin{gathered}
x-\text { inter: } 0=-5 x^{2}-60 x-175 \\
0=-5\left(x^{2}+12 x+35\right) \\
0=-5(x+5)(x+7) \\
x+5=0 \quad x+7=0 \\
-5-5 \quad-7-7 \\
\hline x=-5 \quad x=-7
\end{gathered}\)
\((-5,0) \quad(-7,0)\)
\(\text { vertex: } x=\frac{60}{2(-5)}=\frac{60}{-10}=-6\)
\(\begin{aligned}
&y=-5(-6)^{2}-60(-6)-175 \\
&y=-5(36)+360-175 \\
&y=-180+360-175 \\
&y=5 \\
&(-6,5)
\end{aligned}\)
