Question

The function $f$ is defined by $f\left(x\right)=\left(x+3\right)\left(x+1\right)$.
The graph of $f$ in the $xy-plane$ is a parabola. Which of the following intervals contains the $x-coordinate$ of the vertex of the graph of $f$??

• A. $-4<x<-3$
• B. $-3<x<1$
• C. $1<x<3$
• D. $3<x<4$

Answer 1

The correct option is B $-3<x<1$

Since the graph of function f is a parabola,

Thus, equation of parabola is $y=(x+3)(x+1)$

$\Rightarrow y=x^2+4x+3\Rightarrow y-3+2^2=x^2+2\times x\times 2+2^2$

$\Rightarrow y+1=(x+2{)}^2$

This can also be written as $(x+2{)}^2=4\times \frac{1}{4}\times (y+1)$

Comparing the above equation to the equation of a parabola, $(x-h{)}^2=4a(y-k)$, where (h,k) is the coordinates of vertex of parabola, we have,

$(h,k)\equiv (-2,-1)$

So, the x coordinate of the vertex is $-2$ which lies in the interval $-3<x<1$