What are the key points on the graph of y=x2−16x+48 ? Name the vertex, x-intercepts, and y-intercept.(2 points) vertex: () (smaller) x-intercept: () (larger) x-intercept: () y-intercept: ()
To find the vertex, we can use the formula $x=-\dfrac{b}{2a}$. In our equation, $a=1$ and $b=-16$, so plugging these values in, we get $x=-\dfrac{-16}{2(1)}=8$. To find the y-coordinate of the vertex we substitute this value into the equation: $y=(8)^2-16(8)+48=64-128+48=-16$. Therefore, the vertex is (8, -16).
To find the x-intercepts, we set $y$ equal to zero and solve for $x$. So $0=x^2-16x+48$. Factoring this expression, we get $(x-6)(x-8)$, which means $x=6$ or $x=8$. Therefore, the x-intercepts are (6,0) and (8,0).
Finally, to find the y-intercept, we set $x$ equal to zero and solve for $y$. Plugging in $x=0$ into the equation, we get $y=(0)^2-16(0)+48=0-0+48=48$. Therefore, the y-intercept is (0,48).