Question

consider the function $g(x)=|x + 4|+6$. complete the statements. the graph of $g(x)$ is $\boxed{ }$. the vertex of $g(x)$ is $\boxed{ }$. $g(x)$ is increasing on the interval(s) $\boxed{ }$ and $g(x)$ is positive on the interval(s) $\boxed{ }$.

Explanation:

Step1: Analyze the parent function

The parent function of an absolute - value function is \(y = |x|\), which has a V - shaped graph, vertex at \((0,0)\), is decreasing on \((-\infty,0)\) and increasing on \((0,+\infty)\), and is non - negative (\(y\geq0\)) for all real \(x\).

Step2: Analyze the transformation of \(g(x)=|x + 4|+6\)

For the absolute - value function \(y=a|x - h|+k\), the vertex is at \((h,k)\), the graph is a V - shape (if \(a>0\), it opens upwards; if \(a < 0\), it opens downwards). Here, \(a = 1>0\), \(h=-4\), \(k = 6\).

• Graph shape: Since \(a = 1>0\), the graph of \(g(x)\) is a V - shaped graph opening upwards (a transformation of the parent absolute - value function \(y = |x|\)).
• Vertex: Using the formula for the vertex of \(y=a|x - h|+k\), with \(h=-4\) and \(k = 6\), the vertex of \(g(x)\) is \((-4,6)\).
• Increasing interval: For \(y=|x - h|+k\) with \(a>0\), the function is increasing when \(x>h\). Here \(h=-4\), so \(g(x)\) is increasing on the interval \((-4,+\infty)\).
• Positive interval: We want to find where \(g(x)=|x + 4|+6>0\). Since \(|x + 4|\geq0\) for all real \(x\), then \(|x + 4|+6\geq6>0\) for all real \(x\). So \(g(x)\) is positive on the interval \((-\infty,+\infty)\).

Now, looking at the graphs:
The first graph (blue) has a vertex at \(x=- 4\) (since the vertex is at \(x=-5\) is not correct, wait, let's re - check the vertex calculation. Wait, \(g(x)=|x + 4|+6=|x-(-4)|+6\), so the vertex is at \((-4,6)\). The first graph (left - top) has a vertex around \(x=-4\) (since when \(x=-4\), \(y = |-4 + 4|+6=6\)). The \(y\) - intercept of \(g(x)\) is when \(x = 0\), \(g(0)=|0 + 4|+6=4 + 6 = 10\), which matches the first graph (the blue graph has a \(y\) - intercept of 10). So the graph of \(g(x)\) is the first (blue) V - shaped graph opening upwards.

Answer:

• The graph of \(g(x)\) is the V - shaped graph (the first/blue one) opening upwards.
• The vertex of \(g(x)\) is \((-4,6)\).
• \(g(x)\) is increasing on the interval \((-4,+\infty)\).
• \(g(x)\) is positive on the interval \((-\infty,+\infty)\).