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 always true about a function of the form ( f(x)=ax^2 ).
c. the statement \the axis of symmetry of the graph is ( x = 0 )\ is (\boxed{quad}) about a function of the form ( f(x)=ax^2 ).

Explanation:

Step1: Recall the axis of symmetry formula for a parabola of the form \( f(x) = ax^2 + bx + c \). The axis of symmetry is given by \( x = -\frac{b}{2a} \).

Step2: For the function \( f(x) = ax^2 \), we have \( b = 0 \) (since there is no \( x \)-term). Substitute \( b = 0 \) into the axis of symmetry formula: \( x = -\frac{0}{2a} = 0 \) (as long as \( a

eq 0 \), which it must be for it to be a quadratic function). So regardless of the value of \( a \) (as long as \( a
eq 0 \)), the axis of symmetry is always \( x = 0 \).

Answer:

always true