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 sometimes true about a function of the form $f(x)=ax^2$
b. the statement \the vertex of the graph is $(0,0)$\ is about a function of the form $f(x)=ax^2$

always true
sometimes true
never true

Explanation:

For the function \( f(x) = ax^2 \), the vertex form of a parabola is \( f(x)=a(x - h)^2 + k \), where \((h,k)\) is the vertex. In \( f(x)=ax^2 \), we can rewrite it as \( f(x)=a(x - 0)^2+0 \), so \( h = 0 \) and \( k = 0 \). This means the vertex is always \((0,0)\) regardless of the value of \( a \) (as long as \( a
eq0 \); if \( a = 0 \), it's not a quadratic function, but the form \( f(x)=ax^2 \) implies \( a\) is a non - zero coefficient for the quadratic term in the context of parabolas). So the statement "the vertex of the graph is \((0,0)\)" is always true for the function \( f(x)=ax^2 \).

Answer:

always true