Question

select the correct answer. which equation represents a parabola that opens upward, has a minimum value of 3, and has an axis of symmetry at ( x = 3 )? a. ( f(x) = (x + 3)^2 - 6 ) b. ( f(x) = (x + 3)^2 + 3 ) c. ( f(x) = (x - 3)^2 + 3 ) d. ( f(x) = (x - 3)^2 - 6 )

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is $f(x)=a(x-h)^2+k$, where:

• $a$ determines direction: $a>0$ opens upward
• $(h,k)$ is the vertex; axis of symmetry is $x=h$, $k$ is the min/max value

Step2: Match given conditions to parameters

Given: opens upward ($a>0$, all options satisfy this), minimum value $k=3$, axis of symmetry $x=3$ so $h=3$.
Substitute $h=3$, $k=3$ into vertex form:
$f(x)=(x-3)^2+3$

Step3: Match to options

This matches option C.

Answer:

C. $f(x) = (x - 3)^2 + 3$