Math Problem Statement
Use transformations of the absolute value function,
f left parenthesis x right parenthesis equals StartAbsoluteValue x EndAbsoluteValuef(x)=x,
to graph the function
h left parenthesis x right parenthesis equals StartAbsoluteValue x plus 1 EndAbsoluteValue plus 3h(x)=x+1+3.
Solution
To graph the function using transformations of the basic absolute value function , we need to understand how each transformation modifies the original graph. Here's how to proceed:
Step-by-Step Explanation:
-
Start with the basic absolute value function:
- The basic graph of is a V-shaped graph that has a vertex at the origin (0, 0) and opens upwards.
-
Apply the transformation inside the absolute value**:
- The transformation represents a horizontal shift. Since the "+1" is inside the absolute value function, it shifts the graph 1 unit to the left. This moves the vertex from (0, 0) to (-1, 0).
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Apply the vertical shift of +3:
- The "+3" outside of the absolute value function shifts the graph upwards by 3 units. This moves the vertex from (-1, 0) to (-1, 3).
Final Graph:
- The resulting graph of is a V-shaped graph with a vertex at the point (-1, 3).
- The slope of the arms of the V remains the same as that of the basic absolute value function, with the left arm having a slope of -1 and the right arm having a slope of +1.
Summary of Transformations:
- Horizontal shift: 1 unit to the left.
- Vertical shift: 3 units upwards.
Would you like a detailed graph or further explanation of how to visualize the graph?
Here are 5 relative questions to expand on this information:
- How does changing the coefficient inside the absolute value function affect the graph?
- What happens to the graph if you subtract a value outside the absolute value function?
- Can you describe the graph of ?
- How does combining horizontal and vertical transformations change the graph of any function?
- What are other functions that have a V-shaped graph, and how do they compare to absolute value functions?
Tip: Always handle horizontal transformations (inside the function) first, as they impact the x-values directly, followed by vertical transformations (outside the function) that affect the y-values.
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Math Problem Analysis
Mathematical Concepts
Algebra
Functions
Transformations
Absolute Value Functions
Formulas
h(x) = |x + 1| + 3
f(x) = |x|
Theorems
-
Suitable Grade Level
Grades 8-10
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