Question

What are the y and x intercepts of y=3(x-6)²-9 and y=-2(x+5)²+3 .. Thanks

Answer 1

1.
$y=3(x-6)^2-9 \\ \\ \hbox{x-intercept - y=0} \\ 0=3(x-6)^2-9 \\ 9=3(x-6)^2 \ \ \ |\div 3 \\ 3=(x-6)^2 \\ \sqrt{3}=\sqrt{(x-6)^2} \\ \pm \sqrt{3}=x-6 \\ \pm \sqrt{3}+6=x \\ x=6+\sqrt{3} \ \lor \ x=6-\sqrt{3}} \\ \\ \hbox{y-intercept - x=0} \\ y=3(0-6)^2-9 \\ y=3 \times (-6)^2-9 \\ y=3 \times 36-9 \\ y=108-9 \\ y=99 \\ \\ \boxed{\hbox{x-intercepts: } 6+\sqrt{3} \hbox{ and } 6-\sqrt{3}} \\ \boxed{\hbox{y-intercept: } 99}$

2.
$y=-2(x+5)^2+3 \\ \\ \hbox{x-intercept - y=0} \\ 0=-2(x+5)^2+3 \\ -3=-2(x+5)^2 \ \ \ |\div (-2) \\ \frac{3}{2}=(x+5)^2 \\ \sqrt{\frac{3}{2}}=\sqrt{(x+5)^2} \\ \pm \sqrt{\frac{3}{2}}=x+5 \\ \pm \sqrt{\frac{3}{2}}-5=x \\ x=\sqrt{\frac{3}{2}}-5 \ \lor \ x=-\sqrt{\frac{3}{2}}-5 \\ \\ \hbox{y-intercept - x=0} \\ y=-2(0+5)^2+3 \\ y=-2 \times 5^2 +3 \\ y=-2 \times 25+3 \\ y=-50+3 \\ y=-47 \\ \\ \boxed{\hbox{x-intercepts: } \sqrt{\frac{3}{2}}-5 \hbox{ and } -\sqrt{\frac{3}{2}}-5} \\ \boxed{\hbox{y-intercept: } -47}$