Question

I need help please, I am so confused.​

Answer 1

Answer:

$\begin{tabular}{| c | c | c |}\cline{1-3} Equation & x-intercepts & x-coordinate of vertex\\\cline{1-3} y=x(x-2) & x=0, x=2 & x=1 \\\cline{1-3} y=(x-4)(x+5) & x=-5, x=4 & x=-0.5 \\\cline{1-3} y=-5x(x-3) & x=0, x=3 & x=1.5 \\\cline{1-3} \end{tabular}$

Step-by-step explanation:

x-intercepts are when the curve intercepts the x-axis, so when y =0.

Therefore, to find the x-intercepts, substitute y = 0 and solve for x.

The vertex is the turning point:  the minimum point of a parabola that opens upward, and the maximum point of the parabola that opens downward.  As a parabola is symmetrical, the x-coordinate of the vertex is the midpoint of the x-intercepts.

Equation:  $y=x(x-2)$

$\implies x(x-2)=0$

$\implies x=0$

$\implies (x-2)=0 \implies x=2$

Therefore, the x-intercepts are x = 0 and x = 2

The midpoint of the x-intercepts is x = 1, so the x-coordinate of the vertex is x = 1

Equation: $y=(x-4)(x+5)$

$\implies (x-4)(x+5)=0$

$\implies (x-4)=0 \implies x=4$

$\implies (x+5)=0 \implies x=-5$

Therefore, the x-intercepts are x = -5 and x = 4

The midpoint of the x-intercepts is x = -0.5, so the x-coordinate of the vertex is x = -0.5

Equation: $y=-5x(3-x)$

$\implies -5x(3-x)=0$

$\implies -5x=0 \implies x=0$

$\implies (3-x)=0 \implies x=3$

Therefore, the x-intercepts are x = 0 and x = 3

The midpoint of the x-intercepts is x = 1.5, so the x-coordinate of the vertex is x = 1.5

$\begin{tabular}{| c | c | c |}\cline{1-3} Equation & x-intercepts & x-coordinate of vertex\\\cline{1-3} y=x(x-2) & x=0, x=2 & x=1 \\\cline{1-3} y=(x-4)(x+5) & x=-5, x=4 & x=-0.5 \\\cline{1-3} y=-5x(x-3) & x=0, x=3 & x=1.5 \\\cline{1-3} \end{tabular}$