Question

unit algebra: concepts and connections/ quadratic functions: standard form for all functions of the form $f(x)=ax^2 + bx + c$, which is true when $b = 0$? the graph will always have zero $x$-intercepts. the function will always have a minimum. the $y$-intercept will always be the vertex. the axis of symmetry will always be positive.

Explanation:

Step1: Substitute $b=0$ into function

The function becomes $f(x)=ax^2 + c$.

Step2: Analyze vertex and y-intercept

The x-coordinate of the vertex is $x=-\frac{b}{2a}=0$. Substitute $x=0$ into $f(x)$: $f(0)=c$, so the vertex is $(0,c)$. The y-intercept is found by setting $x=0$, which is also $y=c$.

Step3: Eliminate other options

• A function has a minimum only if $a>0$; if $a<0$, it has a maximum.
• The graph has x-intercepts only if $ax^2 + c=0$ has real solutions, i.e., $ac<0$; if $ac>0$, there are no real x-intercepts.
• The axis of symmetry is $x=-\frac{b}{2a}=0$, which is not positive.

Answer:

The y-intercept will always be the vertex.