Math Problem Statement
Solution
To solve this problem, we need to match each equation with the corresponding graph, using the transformations of the base function . Here are the steps to do that:
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Equation (a):
- This represents a horizontal shift to the right by 4 units.
- Look for the graph where the function is shifted to the right.
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Equation (b):
- This represents a vertical shift upwards by 3 units.
- Look for the graph where the function is moved upward by 3 units.
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Equation (c):
- This represents a horizontal shift to the left by 6 units and a vertical stretch by a factor of 2.
- Look for the graph that is stretched and shifted to the left.
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Equation (d):
- This represents a reflection over the x-axis and a horizontal compression by a factor of 2.
- Look for the graph that is flipped upside down and horizontally compressed.
Now, let's analyze the graphs:
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Graph (1):
- This graph appears to be horizontally shifted to the right by 4 units.
- This corresponds to equation (a): .
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Graph (2):
- This graph is stretched vertically and shifted to the left by 6 units.
- This corresponds to equation (c): .
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Graph (3):
- This graph is shifted upwards by 3 units.
- This corresponds to equation (b): .
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Graph (4):
- This graph is reflected over the x-axis and horizontally compressed.
- This corresponds to equation (d): .
Final Matching:
- (a) : Graph 1
- (b) : Graph 3
- (c) : Graph 2
- (d) : Graph 4
Would you like more details or have any questions?
Here are 5 related questions you might explore next:
- How do horizontal shifts affect the graph of a function?
- What is the effect of vertical stretching on a graph?
- How does a reflection over the x-axis change the graph of a function?
- Can you determine the transformation if a function is compressed horizontally?
- What is the combined effect of multiple transformations on a graph?
Tip: Always check the order of transformations, as some transformations (like shifts) are affected by others (like reflections or stretches).
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Math Problem Analysis
Mathematical Concepts
Graph transformations
Shifting and stretching functions
Formulas
Horizontal shift: y = f(x - h)
Vertical shift: y = f(x) + k
Vertical stretch: y = a*f(x)
Reflection: y = -f(x)
Horizontal compression: y = f(bx)
Theorems
Transformation rules for functions
Suitable Grade Level
High School (Grades 9-12)
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