Questions: 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 ax-h+k, where a, h, and k are integers or simplified fractions.

$The graph shows g(x), which is a translation of f(x)=x. Write the function rule for g(x). Write your answer in the form ax-h+k, where a, h, and k are integers or simplified fractions.$

Transcript text: 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 $\mathrm{a}|\mathrm{x}-\mathrm{h}|+\mathrm{k}$, where a , h , and k are integers or simplified fractions.

Solution

Solution Steps

Step 1: Find the vertex

The vertex of the absolute value function g(x) is at (-6, 0). This means h = -6 and k = 0.

Step 2: Determine the slope

The graph of g(x) passes through the point (0,6), which is to the right of the vertex. We can calculate the slope using the vertex (-6,0) and the point (0,6). The slope is (6-0)/(0-(-6)) = 6/6 = 1. This means a = 1.

Step 3: Write the function

Substituting the values of a, h, and k into the general form _a_|x - _h_| + _k_, we get 1|x - (-6)| + 0. This simplifies to |x + 6|.