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 $\mathrm{a}, \mathrm{h}$, and k are integers or simplified fractions.

Solution

Solution Steps

Step 1: Determine the slope

The graph of g(x) passes through points (9,0) and (0,9). The slope is calculated as: $m = \frac{9-0}{0-9} = -1$

Step 2: Determine the vertex

The vertex of the absolute value function g(x) is at (9, 0), which represents the horizontal and vertical shifts, h = 9 and k = 0, respectively.

Step 3: Write the equation

The general form of an absolute value function is $g(x) = a|x-h|+k$ . Substituting the slope $a=-1$ , h=9 and k=0 into the general form, we get: $g(x) = -|x-9|+0$ , which simplifies to $g(x) = -|x-9|$ .

Final Answer

$g(x) = -|x-9|$

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