Question

tell whether each statement about a function of the form $f(x)=ax^2$ is always true, sometimes true, or never true.
a. the graph is a parabola that opens upward.
b. the vertex of the graph is $(0,0)$.
c. the axis of symmetry of the graph is $x=0$.
a. the statement \the graph is a parabola that opens upward\ is about a function of the form $f(x)=ax^2$.

Explanation:

Step1: Analyze statement a

For $f(x)=ax^2$, if $a>0$, the parabola opens upward; if $a<0$, it opens downward. So this is only true under some conditions.

Step2: Analyze statement b

The vertex of $f(x)=ax^2$ is found by setting the derivative (or using vertex formula for $ax^2+bx+c$, here $b=0,c=0$) to 0. The vertex occurs at $x=0$, $f(0)=0$, so vertex is always $(0,0)$.

Step3: Analyze statement c

The axis of symmetry for $f(x)=ax^2$ (a quadratic with $b=0$) is given by $x=-\frac{b}{2a}=0$, so this is always true.

Answer:

a. sometimes true
b. always true
c. always true