Question

algebra 2

1. selected response: 1 point for correct answer

g is a quadratic function defined by the equation below:
g(x)=-(x + 3)^2+1
which of the statements below are true?
i. the vertex (-3,1) is a minimum.
ii. g(x) has two non - real roots.
a. i only
b. ii only
c. both i and ii
d. neither i nor ii

Explanation:

Step1: Analyze the vertex - type

The quadratic function is in vertex - form $y = a(x - h)^2+k$, where $(h,k)$ is the vertex. Here $g(x)=-(x + 3)^2+1$, so $a=-1$, $h=-3$, $k = 1$ and the vertex is $(-3,1)$. Since $a=-1<0$, the parabola opens downwards, and the vertex is a maximum, not a minimum. So statement I is false.

Step2: Find the roots of the function

Set $g(x)=0$, then $-(x + 3)^2+1 = 0$. Rearrange it to $(x + 3)^2=1$. Take the square - root of both sides: $x+3=\pm1$. Solve for $x$: $x=-3\pm1$. So $x=-2$ or $x=-4$, which are real roots. So statement II is false.

Answer:

D. Neither I nor II