Question

the graph shows \\(g(x)\\), which is a translation of \\(f(x) = |x|\\). write the function rule for \\(g(x)\\).

write your answer in the form \\(a|x - h| + k\\), where \\(a\\), \\(h\\), and \\(k\\) are integers or simplified fractions.

\\(g(x) = \\)

Explanation:

Identify the vertex of the graph

The vertex of the absolute value function \(g(x)\) is the lowest point on the V-shaped graph. Looking at the grid, this turning point is located at \((1, -2)\).

Determine the general vertex form

The general equation for a transformed absolute value function is:

$$g(x) = a|x - h| + k$$

where \((h, k)\) is the vertex of the graph. Substituting \(h = 1\) and \(k = -2\) gives:

$$g(x) = a|x - 1| - 2$$

Find the value of the vertical stretch factor

To find the value of \(a\), we can choose another clear point on the graph. The graph passes through the point \((3, 0)\). Substituting \(x = 3\) and \(g(x) = 0\) into our equation:

$$0 = a|3 - 1| - 2$$

$$0 = 2a - 2$$

$$2a = 2$$

$$a = 1$$

Write the final function rule

Substituting \(a = 1\) back into the equation yields:

$$g(x) = |x - 1| - 2$$

Answer:

The graph shows \(g(x)\), which is a translation of \(f(x) = |x|\). Write the function rule for \(g(x)\).

Write your answer in the form \(a|x - h| + k\), where \(a\), \(h\), and \(k\) are integers or simplified fractions.

\(g(x) =\) <blank>\(|x - 1| - 2\)</blank>