Answer:
Part 1) The axis of symmetry is x=2
Part 2) The vertex is the point (2,8)
Part 3) The domain is all real numbers
Part 4) The range is all real numbers less than or equal to 8
Step-by-step explanation:
we have
[tex]y=-2x^{2}+8x[/tex]
This is a vertical parabola open downward (because the leading coefficient is negative)
The vertex represent a maximum
step 1
Find the vertex
Convert the quadratic equation in vertex form
Factor -2 leading coefficient
[tex]y=-2(x^{2}-4x)[/tex]
Complete the square
[tex]y=-2(x^{2}-4x+4)+8[/tex]
rewrite as perfect squares
[tex]y=-2(x-2)^{2}+8[/tex]
so
The vertex is the point (2,8)
step 2
Find the axis of Symmetry
we know that
In a vertical parabola, the axis of symmetry is equal to the x-coordinate of the vertex
the vertex is the point (2,8)
therefore
The axis of symmetry is x=2
step 3
Find the domain
The domain of the quadratic equation is the interval ------> (-∞,∞)
The domain is all real numbers
step 4
Find the range
The range of the quadratic equation is the interval ------> (-∞,8]
[tex]y\leq 8[/tex]
The range is all real numbers less than or equal to 8
