Podcast
Questions and Answers
What is the first transformation for the function given the expression -3 f(x) + 1?
What is the first transformation for the function given the expression -3 f(x) + 1?
- Vertically reflect (correct)
- Translate 1 unit up (correct)
- Translate down 1 unit
- Vertically stretch factor 3 (correct)
What transformation occurs with an 'a' value greater than 1?
What transformation occurs with an 'a' value greater than 1?
- Vertical Stretch (correct)
- Vertical Shrink
- Horizontal Stretch
- Horizontal Shrink
When the 'b' value is negative, the function is reflected over the ______.
When the 'b' value is negative, the function is reflected over the ______.
y-axis
What does the 'd' value represent in transformations?
What does the 'd' value represent in transformations?
Identify the transformations for the function f(x) = (x-3)².
Identify the transformations for the function f(x) = (x-3)².
What transformation is described when the 'a' value is negative?
What transformation is described when the 'a' value is negative?
What does a 'b' value of 2 indicate regarding compression in transformations?
What does a 'b' value of 2 indicate regarding compression in transformations?
Identify the transformations for the function g(x) = (x-2)²+3.
Identify the transformations for the function g(x) = (x-2)²+3.
What is the effect of a horizontal stretch factor of 4?
What is the effect of a horizontal stretch factor of 4?
What is the transformation of the function m(x) = (¼x)³?
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.
- 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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