Questions and Answers
What is the first transformation for the function given the expression -3 f(x) + 1?
What transformation occurs with an 'a' value greater than 1?
When the 'b' value is negative, the function is reflected over the ______.
y-axis
What does the 'd' value represent in transformations?
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Identify the transformations for the function f(x) = (x-3)².
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What transformation is described when the 'a' value is negative?
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What does a 'b' value of 2 indicate regarding compression in transformations?
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Identify the transformations for the function g(x) = (x-2)²+3.
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What is the effect of a horizontal stretch factor of 4?
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What is the transformation of the function m(x) = (¼x)³?
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Study Notes
Function Transformations Overview
- Transformations manipulate the position and shape of a graph.
- Common types of transformations: vertical shifts, horizontal shifts, reflections, stretches, and shrinks.
Types of Transformations
- Vertical Stretch: Occurs when the "a" value > 1, making the graph taller.
- Vertical Shrink: Happens when the "a" value is between 0 and 1, resulting in a shorter graph.
- Horizontal Stretch: Takes place when the "b" value is between 0 and 1, causing the graph to widen.
- Horizontal Shrink: Results when the "b" value > 1, leading to a compressed graph horizontally.
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Reflections:
- Across the x-axis indicated by a negative "a" value.
- Across the y-axis shown by a negative "b" value.
Movement Instructions
- Vertical Translation: Controlled by the "d" value; positive moves the graph up, while negative moves it down.
- Horizontal Translation: Adjusted by the "c" value; (x - c) shifts right, while (x + c) shifts left.
Specific Transformations in Functions
- Functions can combine transformations, affecting their resultant graphs.
- Sequence of transformations matters; transformations are typically applied in the order of reflections, stretches/shrinks, then translations.
Function Examples
- f(x) = (x - 3)² involves a right translation by 3 units.
- h(x) = -2x² combines a vertical stretch (factor of 2) and reflection over the x-axis.
- g(x) = (-x + 1)² performs a left translation by 1 unit and a horizontal reflection.
Key Component Values
- "a" value reflects vertical transformations (stretch/shrink, reflection).
- "b" value governs horizontal transformations (stretch/shrink, reflection).
- "c" value indicates left/right shifts.
- "d" value controls up/down shifts.
Identifying Transformations
- Formulations help recognize transformations by their structure:
- Positive "d" value = Upward shift
- Negative "d" value = Downward shift
- Positive "c" value = Right shift
- Negative "c" value = Left shift
- A sequence is critical for understanding cumulative effects on the graph shape and position.
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Description
Test your understanding of function transformations with this quiz. Identify and describe the various transformations such as vertical stretches, translations, and reflections. Perfect for students looking to strengthen their grasp on mathematical functions.
