Question

for problems 15 & 16, circle the graph that best represents the given function. 15. f(x)=x² - 2? a. b. c. d. 16. g(x)=|x + 3|?

Explanation:

Step1: Analyze \(f(x)=x^{2}-2\)

The general form of a quadratic function is \(y = ax^{2}+bx + c\), here \(a = 1\), \(b=0\), \(c=-2\). The vertex - form of a quadratic function is \(y=a(x - h)^{2}+k\), and for \(y=x^{2}-2\), the vertex is \((0, - 2)\) (since \(h = 0,k=-2\)) and the parabola opens upwards because \(a = 1>0\).

Step2: Analyze \(g(x)=|x + 3|\)

The general form of an absolute - value function is \(y=a|x - h|+k\). For \(y = |x + 3|\), we can rewrite it as \(y=|x-(-3)|\), so the vertex is \((-3,0)\). When \(x>-3\), \(y=x + 3\) (a line with slope \(m = 1\)), and when \(x<-3\), \(y=-(x + 3)=-x - 3\) (a line with slope \(m=-1\)).

Answer:

1. The graph of \(f(x)=x^{2}-2\) is a parabola opening upwards with vertex at \((0,-2)\), so it should match the graph that has a U - shape and crosses the y - axis at \(-2\).
2. The graph of \(g(x)=|x + 3|\) has a vertex at \((-3,0)\) and is in a V - shape. You need to find the graph with a V - shape and vertex at \(x=-3\). Without seeing the specific details of each option precisely, based on the above - described characteristics, you can identify the correct graphs among the given choices.