Question

which statement is true concerning the vertex and the axis of symmetry of $g(x) = 5x^2 - 10x$? the vertex is at $(1, -5)$ and the axis of symmetry is $y = 1$. the vertex is at $(0, 0)$ and the axis of symmetry is $x = 1$. the function written in vertex form is $g(x) = 5(x - 1)^2 - 5$. the vertex is at $(1, -5)$ and the axis of symmetry is $x = 1$.

Explanation:

Step1: Find x-coordinate of vertex

For quadratic $ax^2+bx+c$, $x=-\frac{b}{2a}$. Here $a=5,b=-10$:
$x=-\frac{-10}{2\times5}=1$

Step2: Find y-coordinate of vertex

Substitute $x=1$ into $g(x)$:
$g(1)=5(1)^2-10(1)=5-10=-5$

Step3: Convert to vertex form

Complete the square:
$g(x)=5(x^2-2x)=5[(x-1)^2-1]=5(x-1)^2-5$

Step4: Identify axis of symmetry

Axis of symmetry is $x =$ x-coordinate of vertex: $x=1$

Answer:

The function written in vertex form is $g(x) = 5(x - 1)^2 - 5$. The vertex is at $(1, -5)$ and the axis of symmetry is $x = 1$.