Question

Given the functions f(x) = 4x2 − 1, g(x) = x2 − 8x + 5, and h(x) = –3x2 − 12x + 1, rank them from least to greatest based on their axis of symmetry. g(x), h(x), f(x) f(x), h(x), g(x) g(x), f(x), h(x) h(x), f(x), g(x)

Answer 1

The axis of symmetry of a quadratic function $f(x)=ax^2+bx+c$ is given by the equation $x=h$, where h is the x-coordinate of the vertex and is equal to $\frac{-b}{2a}$.

1.
$f(x)=4x^2-1 \\ a=4 \\ b=0 \\ \Downarrow \\ h=\frac{-0}{2 \times 4}=0 \\ \\ \hbox{the axis of symmetry:} \\ x=0$

2.
$g(x)=x^2-8x+5 \\ a=1 \\ b=-8 \\ \Downarrow \\ h=\frac{-(-8)}{2 \times 1}=\frac{8}{2}=4 \\ \\ \hbox{the axis of symmetry:} \\ x=4$

3.
$h(x)=-3x^2-12x+1 \\ a=-3 \\ b=-12 \\ \Downarrow \\ h=\frac{-(-12)}{2 \times (-3)}=\frac{12}{-6}=-2 \\ \\ \hbox{the axis of symmetry: \\ x=-2$

The functions ranked from least to greatest based on their axis of symmetry: h(x), f(x), g(x).