Chapter6: Exponential And Logarithmic Functions
Section: Chapter Questions
Problem 12RE: The graph below shows transformations of the graph of f(x)=2x. What is the equation for the...
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My answer was wrong, im not sure how to do this. thank you!
Transcribed Image Text:6
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-6 -5 -4 -3 -2 -1
1 2 3 4
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--1
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The graph above is a transformation of the function f(x) = |x|
Write an equation for the function graphed above
g(x) =
|x − 1|
X
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![### Transformation of the Absolute Value Function
**Graph Explanation:**
The graph provided is a modification of the function \( f(x) = |x| \).
- The original function \( f(x) = |x| \) is an absolute value function, which forms a symmetric V-shape with its vertex at the origin (0,0).
- In the transformed graph, the vertex of the V-shape has moved to the point (-1, 1).
**Finding the Equation:**
The transformed function can be found by identifying the changes made to the original absolute value function.
**Vertical and Horizontal Shifts:**
- The horizontal shift: The transformation shifts the vertex of the graph one unit to the left, which suggests \( x \) is replaced by \( x + 1 \) inside the absolute value.
- The vertical shift: The transformation shifts the vertex of the graph one unit upwards, which suggests adding 1 to the entire function.
Given these transformations, the transformed function is:
\[ g(x) = |x + 1| + 1 \]
**Validation of Equation:**
The provided options include two functions. The function that matches the transformations is:
\[ \boxed{|x + 1| + 1} \]
The incorrect option is:
\[ \frac{1}{2}|x + 1| + 1 \]
Hence, the correct equation for the transformed function is \( g(x) = |x + 1| + 1 \).](https://content.bartleby.com/qna-images/question/90d1e96e-989a-4849-aeb9-585c1e797a4d/92f3093b-4a0b-4fcd-bb86-e26d68bb877d/8uudp8j_processed.png)
Transcribed Image Text:### Transformation of the Absolute Value Function
**Graph Explanation:**
The graph provided is a modification of the function \( f(x) = |x| \).
- The original function \( f(x) = |x| \) is an absolute value function, which forms a symmetric V-shape with its vertex at the origin (0,0).
- In the transformed graph, the vertex of the V-shape has moved to the point (-1, 1).
**Finding the Equation:**
The transformed function can be found by identifying the changes made to the original absolute value function.
**Vertical and Horizontal Shifts:**
- The horizontal shift: The transformation shifts the vertex of the graph one unit to the left, which suggests \( x \) is replaced by \( x + 1 \) inside the absolute value.
- The vertical shift: The transformation shifts the vertex of the graph one unit upwards, which suggests adding 1 to the entire function.
Given these transformations, the transformed function is:
\[ g(x) = |x + 1| + 1 \]
**Validation of Equation:**
The provided options include two functions. The function that matches the transformations is:
\[ \boxed{|x + 1| + 1} \]
The incorrect option is:
\[ \frac{1}{2}|x + 1| + 1 \]
Hence, the correct equation for the transformed function is \( g(x) = |x + 1| + 1 \).
Solution
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