6 5 4 3 2 1 -6 -5 -4 -3 -2 -1 -2 -3 1 -5 2 3 4 5 6 1₂ The graph above is a transformation of the function f(x) = |x| Write an equation for the function graphed above g(x) =
6 5 4 3 2 1 -6 -5 -4 -3 -2 -1 -2 -3 1 -5 2 3 4 5 6 1₂ The graph above is a transformation of the function f(x) = |x| Write an equation for the function graphed above g(x) =
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.4: Quadratic Functions; Translation And Reflection
Problem 51E: LIFE SCIENCE APPLICATIONS APPLY IT Splenic Artery Resistance Blood flow to the fetal spleen is of...
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![**Understanding Transformations of Absolute Value Functions**
The graph above demonstrates a transformation of the absolute value function \( f(x) = |x| \).
The graph shows how the function \( f(x) = |x| \) has been altered. The transformation includes a translation and a reflection. Here are the details of the graph:
### Graph Description:
- The original absolute value function \( f(x) = |x| \) forms a 'V' shape where the point of the V is located at the origin (0, 0).
- In the transformed graph, the point where the 'V' shape changes direction is at (2, -4).
### Observations:
1. **Shift to the Right:** The vertex of the transformed graph is moved 2 units to the right.
2. **Vertical Shift Downward:** The vertex of the transformed graph is moved 4 units downward.
3. **Reflection:** The ‘V’ shape is inversely reflected along the y-axis because the arms of the V open downwards instead of upwards.
### Equation Derivation:
Based on these observations, the transformation can be described by the equation \( g(x) = -|x - 2| - 4 \):
1. **Right Shift:** To shift the graph 2 units to the right, replace \( x \) with \( (x - 2) \).
2. **Reflection and Vertical Shift:** The negative sign outside the absolute value represents a reflection, and subtracting 4 indicates that the entire graph shifts downward by 4 units.
### Formula for the Transformed Function:
\[ g(x) = -|x - 2| - 4 \]
### Prompt:
Write an equation for the function graphed above:
#### Answer:
\[ g(x) = -|x - 2| - 4 \]
By understanding these transformations, students can learn how basic function graphs are altered through translations and reflections.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6691cd2-55cd-462f-8290-5296a9107f11%2F28ebd845-9c61-4d3b-91c0-e93fc7070d06%2Fv9zeh2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Transformations of Absolute Value Functions**
The graph above demonstrates a transformation of the absolute value function \( f(x) = |x| \).
The graph shows how the function \( f(x) = |x| \) has been altered. The transformation includes a translation and a reflection. Here are the details of the graph:
### Graph Description:
- The original absolute value function \( f(x) = |x| \) forms a 'V' shape where the point of the V is located at the origin (0, 0).
- In the transformed graph, the point where the 'V' shape changes direction is at (2, -4).
### Observations:
1. **Shift to the Right:** The vertex of the transformed graph is moved 2 units to the right.
2. **Vertical Shift Downward:** The vertex of the transformed graph is moved 4 units downward.
3. **Reflection:** The ‘V’ shape is inversely reflected along the y-axis because the arms of the V open downwards instead of upwards.
### Equation Derivation:
Based on these observations, the transformation can be described by the equation \( g(x) = -|x - 2| - 4 \):
1. **Right Shift:** To shift the graph 2 units to the right, replace \( x \) with \( (x - 2) \).
2. **Reflection and Vertical Shift:** The negative sign outside the absolute value represents a reflection, and subtracting 4 indicates that the entire graph shifts downward by 4 units.
### Formula for the Transformed Function:
\[ g(x) = -|x - 2| - 4 \]
### Prompt:
Write an equation for the function graphed above:
#### Answer:
\[ g(x) = -|x - 2| - 4 \]
By understanding these transformations, students can learn how basic function graphs are altered through translations and reflections.
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