Question

a large door is shaped like an arch. the arch is modeled by the equation y = -2x² + 8x where y is the height of the door, in feet, and x is the width of the door, in feet. use the coordinate grid to graph this equation. click on the point that represents the maximum height of the door from the ground. then, click on the point representing an x - intercept. to graph a parabola, first click on the point that represents the maximum height, then click on an x - intercept. you must graph the parabola using only these two points; any other graphed points will cause your graph to score as incorrect.

Explanation:

Step1: Find the x - coordinate of the vertex

For a quadratic function $y = ax^{2}+bx + c$, the x - coordinate of the vertex is given by $x=-\frac{b}{2a}$. In the equation $y=-2x^{2}+8x$, where $a=-2$ and $b = 8$. So, $x=-\frac{8}{2\times(-2)}=\frac{-8}{-4}=2$.

Step2: Find the y - coordinate of the vertex

Substitute $x = 2$ into the equation $y=-2x^{2}+8x$. Then $y=-2\times(2)^{2}+8\times2=-2\times4 + 16=-8 + 16=8$. So the maximum height of the door (the vertex of the parabola) is at the point $(2,8)$.

Step3: Find the x - intercepts

Set $y = 0$ in the equation $y=-2x^{2}+8x$. So, $-2x^{2}+8x=0$. Factor out $-2x$: $-2x(x - 4)=0$. Then $-2x=0$ or $x - 4=0$. Solving these gives $x = 0$ or $x = 4$. The x - intercepts are the points $(0,0)$ and $(4,0)$.

Answer:

The point representing the maximum height is $(2,8)$. An x - intercept point can be $(0,0)$ or $(4,0)$.