Math Problem Statement
Solution
Let's break down and solve the questions based on the graph and the provided instructions.
1. Graph Analysis
The graph of is a piecewise linear function. From the visual representation, we can observe the behavior of the function at key points:
- The function is symmetric and forms a "V" shape.
- appears to have the following critical points:
The graph appears to consist of straight lines connecting these points.
Problem Breakdown:
(a) Find :
By observing the graph, at , the function has a value of .
(b) Given , find :
- We already know that .
- Therefore, .
(c) Sketch the graph of :
- The graph of is simply a vertical translation of the graph of by 2 units upward.
- All points on the graph of are moved 2 units higher. For example:
- , so
- , so
- , so
- , so
- , so
The new graph of will be an identical shape to , just shifted upward by 2 units.
Do you want more details or explanations for any part of this solution?
Here are five related questions to explore this further:
- How would the graph of look like?
- What is the slope of the line segments in the graph of ?
- Can you express as a piecewise function based on the graph?
- How would the domain and range of differ from those of ?
- What happens to the graph if ?
Tip: When dealing with transformations of graphs, always look for shifts, stretches, and reflections to understand how the shape changes.
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Math Problem Analysis
Mathematical Concepts
Graphing
Piecewise Functions
Function Transformation
Formulas
g(x) = f(x) + 2
Theorems
Vertical Translation of Functions
Suitable Grade Level
Grades 10-12
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