Question
The graph shows g(x), which is a transformation 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)=
the graph is an absolute value function starting at -1
Write your answer in the form a|x–h|+k, where a, h, and k are integers or simplified fractions. g(x)=
the graph is an absolute value function starting at -1
Answers
GPT-4o mini
To write the function rule for \( g(x) \) as a transformation of \( f(x) = |x| \), we need to identify the transformations applied to the basic absolute value function.
The general form of an absolute value function is given by:
\[
g(x) = a|x - h| + k
\]
Where:
- \( a \) is the vertical stretch or compression and reflects the graph if negative,
- \( h \) represents the horizontal shift,
- \( k \) represents the vertical shift.
Since you mentioned that the graph starts at \(-1\), it likely means that the vertex of the absolute value function has been shifted.
1. **Vertical Shift (k):** The vertex is at \((0, k)\). If \( g(x) \) starts at \(-1\), the vertical shift \( k \) would be \(-1\).
2. **Horizontal Shift (h):** Since the vertex is typically at \( (0,0) \) for \( f(x) = |x| \), and if there’s no horizontal shift mentioned (the graph is not specified to move left or right), we can assume \( h = 0\).
3. **Vertical Stretch/Reflection (a):** If the function remains the same shape as the original `|x|`, then \( a = 1\). If it's wider or narrower, that would change \( a \), but with no additional information, we will assume \( a = 1\).
Putting it all together, we have:
- \( a = 1 \)
- \( h = 0 \)
- \( k = -1 \)
Thus, the function \( g(x) \) can be written as:
\[
g(x) = 1|x - 0| - 1
\]
Which simplifies to:
\[
g(x) = |x| - 1
\]
So, the final function rule for \( g(x) \) is:
\[
g(x) = |x| - 1
\]
The general form of an absolute value function is given by:
\[
g(x) = a|x - h| + k
\]
Where:
- \( a \) is the vertical stretch or compression and reflects the graph if negative,
- \( h \) represents the horizontal shift,
- \( k \) represents the vertical shift.
Since you mentioned that the graph starts at \(-1\), it likely means that the vertex of the absolute value function has been shifted.
1. **Vertical Shift (k):** The vertex is at \((0, k)\). If \( g(x) \) starts at \(-1\), the vertical shift \( k \) would be \(-1\).
2. **Horizontal Shift (h):** Since the vertex is typically at \( (0,0) \) for \( f(x) = |x| \), and if there’s no horizontal shift mentioned (the graph is not specified to move left or right), we can assume \( h = 0\).
3. **Vertical Stretch/Reflection (a):** If the function remains the same shape as the original `|x|`, then \( a = 1\). If it's wider or narrower, that would change \( a \), but with no additional information, we will assume \( a = 1\).
Putting it all together, we have:
- \( a = 1 \)
- \( h = 0 \)
- \( k = -1 \)
Thus, the function \( g(x) \) can be written as:
\[
g(x) = 1|x - 0| - 1
\]
Which simplifies to:
\[
g(x) = |x| - 1
\]
So, the final function rule for \( g(x) \) is:
\[
g(x) = |x| - 1
\]