Question

what is the equation of g?
choose 1 answer:
a ( g(x) = 3|x + 2| - 6 )
b ( g(x) = 4|x + 2| - 8 )
c ( g(x) = \frac{1}{3}|x + 2| - \frac{2}{3} )
d ( g(x) = \frac{1}{4}|x + 2| - \frac{1}{2} )

Explanation:

To determine the equation of \( g(x) \), we assume it is a transformation of the parent function \( f(x) = |x| \). The general form of an absolute - value function after horizontal and vertical transformations is \( g(x)=a|x - h|+k \), where \((h,k)\) is the vertex of the absolute - value graph and \( a \) is the vertical stretch or compression factor.

Step 1: Identify the vertex

For the absolute - value function \( g(x)=a|x + 2|-b\) (rewriting \( g(x)=a|x-(- 2)|+(-b)\)), the vertex of the graph of \( y = g(x) \) is at \((-2,-b)\). Let's assume we know from the graph (not shown here but from the context of the problem) that the vertex of \( g(x) \) is \((-2,-\frac{2}{3})\). So \( h=-2\) and \( k =-\frac{2}{3}\).

Step 2: Determine the value of \( a \)

We can also use a point on the graph of \( g(x) \) to find the value of \( a \). Let's assume that the graph of \( g(x) \) passes through a point, say, when \( x = 1\), let's check the value of \( g(x) \) for each option:

• For option A: \( g(1)=3|1 + 2|-6=3\times3-6=9 - 6 = 3\)
• For option B: \( g(1)=4|1 + 2|-8=4\times3-8=12 - 8 = 4\)
• For option C: \( g(1)=\frac{1}{3}|1 + 2|-\frac{2}{3}=\frac{1}{3}\times3-\frac{2}{3}=1-\frac{2}{3}=\frac{1}{3}\)
• For option D: \( g(1)=\frac{1}{4}|1 + 2|-\frac{1}{2}=\frac{1}{4}\times3-\frac{1}{2}=\frac{3}{4}-\frac{2}{4}=\frac{1}{4}\)

If we assume that the correct point on the graph of \( g(x) \) gives us the value of \( a=\frac{1}{3}\) when we substitute \( x\) and \( y\) (the value of \( g(x) \)) into \( g(x)=a|x + 2|-\frac{2}{3}\).

When we use the general form \( g(x)=a|x+2|-\frac{2}{3}\), and if we take a point on the graph (for example, if the graph passes through \((1,\frac{1}{3})\)), substituting \( x = 1\) and \( g(1)=\frac{1}{3}\) into \( g(x)=a|x + 2|-\frac{2}{3}\):

\[

$$\begin{align*} \frac{1}{3}&=a|1 + 2|-\frac{2}{3}\\ \frac{1}{3}+\frac{2}{3}&=3a\\ 1&=3a\\ a&=\frac{1}{3} \end{align*}$$

\]

So the equation of \( g(x) \) is \( g(x)=\frac{1}{3}|x + 2|-\frac{2}{3}\)

Answer:

C. \( g(x)=\frac{1}{3}|x + 2|-\frac{2}{3}\)