for all functions of the form $f(x)=ax^{2}+bx + c$, which is true when $b = 0$?\nthe graph will always have…

for all functions of the form $f(x)=ax^{2}+bx + c$, which is true when $b = 0$?\nthe graph will always have zero x - intercepts.\nthe function will always have a minimum.\nthe y - intercept will always be the vertex.\nthe axis of symmetry will always be positive.

for all functions of the form $f(x)=ax^{2}+bx + c$, which is true when $b = 0$?\nthe graph will always have zero x - intercepts.\nthe function will always have a minimum.\nthe y - intercept will always be the vertex.\nthe axis of symmetry will always be positive.

Answer

Answer:

C. The y - intercept will always be the vertex.

Explanation:

Step1: Recall the vertex - form and axis of symmetry formula

The vertex of a quadratic function (y = ax^{2}+bx + c) has (x) - coordinate (x=-\frac{b}{2a}). The (y) - intercept is found by setting (x = 0), so (y(0)=c).

Step2: Analyze when (b = 0)

When (b = 0), the (x) - coordinate of the vertex (x=-\frac{0}{2a}=0). The (y) - value of the vertex is (y(0)=a(0)^{2}+b(0)+c=c). The (y) - intercept is also (y(0)=c). So when (b = 0), the (y) - intercept is the vertex.

Step3: Analyze option A

The number of (x) - intercepts is determined by the discriminant (\Delta=b^{2}-4ac). When (b = 0), (\Delta=-4ac). If (ac>0), there are no (x) - intercepts; if (ac = 0), there is one (x) - intercept; if (ac<0), there are two (x) - intercepts. So option A is false.

Step4: Analyze option B

The function (y = ax^{2}+bx + c) has a minimum when (a>0) and a maximum when (a<0). It does not depend on (b) alone. So option B is false.

Step5: Analyze option D

The axis of symmetry is (x =-\frac{b}{2a}). When (b = 0), the axis of symmetry is (x = 0) (not positive). So option D is false.