Question
The graph shows g(x), which is a transformation of f(x) = |x|. write the function ruloe for g(x)
a graph ranges from 10 to -10 on both the x and y axises. the graph is a lineit goes from 9,0 and goes downwards towards 0,-8.
write your answer in the form a|x-h|+k, where a, h, and k are integers or simplified fractions.
a graph ranges from 10 to -10 on both the x and y axises. the graph is a lineit goes from 9,0 and goes downwards towards 0,-8.
write your answer in the form a|x-h|+k, where a, h, and k are integers or simplified fractions.
Answers
GPT-4o mini
To write the function rule for \( g(x) \) based on the transformation of \( f(x) = |x| \), we can start by analyzing the given points of the graph and the transformation properties.
1. The given points are \( (9, 0) \) and \( (0, -8) \). The graph appears to be a downward opening linear graph, which implies a reflection and a vertical shift of the absolute function.
2. The vertex of the absolute function \( f(x) = |x| \) is at the origin, which means we need to determine how these points relate to the transformation of the basic absolute value function.
3. The point \( (9, 0) \) is likely the new vertex of \( g(x) \). This means that \( h = 9 \).
4. The point \( (0, -8) \) indicates the graph is declining towards that point, therefore we should confirm the slope and ensure it matches a vertical reflection.
Since the vertex form of an absolute function can be modeled as:
\[
g(x) = a|x - h| + k
\]
where \( (h, k) \) is the vertex, we already have \( h = 9 \), and since the graph goes downwards from \( (9, 0) \), we can hypothesize that \( k = 0 \).
To find \( a \), we need to calculate the slope from the vertex to the other point:
The slope between \( (9, 0) \) and \( (0, -8) \) is:
\[
\text{slope} = \frac{-8 - 0}{0 - 9} = \frac{-8}{-9} = \frac{8}{9}
\]
However, since the graph is reflected and opens downwards:
\[
a = -\frac{8}{9}
\]
Now we can put everything together to find \( g(x) \):
\[
g(x) = -\frac{8}{9} |x - 9| + 0
\]
Thus, the function rule for \( g(x) \) in the specified form is:
\[
g(x) = -\frac{8}{9} |x - 9|
\]
1. The given points are \( (9, 0) \) and \( (0, -8) \). The graph appears to be a downward opening linear graph, which implies a reflection and a vertical shift of the absolute function.
2. The vertex of the absolute function \( f(x) = |x| \) is at the origin, which means we need to determine how these points relate to the transformation of the basic absolute value function.
3. The point \( (9, 0) \) is likely the new vertex of \( g(x) \). This means that \( h = 9 \).
4. The point \( (0, -8) \) indicates the graph is declining towards that point, therefore we should confirm the slope and ensure it matches a vertical reflection.
Since the vertex form of an absolute function can be modeled as:
\[
g(x) = a|x - h| + k
\]
where \( (h, k) \) is the vertex, we already have \( h = 9 \), and since the graph goes downwards from \( (9, 0) \), we can hypothesize that \( k = 0 \).
To find \( a \), we need to calculate the slope from the vertex to the other point:
The slope between \( (9, 0) \) and \( (0, -8) \) is:
\[
\text{slope} = \frac{-8 - 0}{0 - 9} = \frac{-8}{-9} = \frac{8}{9}
\]
However, since the graph is reflected and opens downwards:
\[
a = -\frac{8}{9}
\]
Now we can put everything together to find \( g(x) \):
\[
g(x) = -\frac{8}{9} |x - 9| + 0
\]
Thus, the function rule for \( g(x) \) in the specified form is:
\[
g(x) = -\frac{8}{9} |x - 9|
\]