Math Problem Statement
What are the transformations from the parent function?
Solution
When referring to "transformations from the parent function," this usually involves changes to a basic or "parent" function that result in a new graph. The parent function represents the simplest form of a particular type of function, and transformations change its appearance, position, or size.
Here are the common types of transformations:
-
Translation (Shift):
- Vertical Shift: Moves the graph up or down.
- : Shifts the graph upward by units.
- : Shifts the graph downward by units.
- Horizontal Shift: Moves the graph left or right.
- : Shifts the graph right by units.
- : Shifts the graph left by units.
- Vertical Shift: Moves the graph up or down.
-
Reflection:
- Reflection over the x-axis:
- : Flips the graph upside down (over the x-axis).
- Reflection over the y-axis:
- : Flips the graph horizontally (over the y-axis).
- Reflection over the x-axis:
-
Stretching and Compressing:
- Vertical Stretch/Compression:
- : Stretches the graph vertically by a factor of if , or compresses it if .
- Horizontal Stretch/Compression:
- : Compresses the graph horizontally by a factor of if , or stretches it if .
- Vertical Stretch/Compression:
-
Rotation:
- Although less common in basic function transformations, certain combinations of reflections and stretches/compressions can appear to rotate graphs about a point.
Example:
For a parent function (a parabola), applying the transformation involves:
- A reflection over the x-axis (the negative sign).
- A vertical stretch by a factor of 2.
- A horizontal shift right by 3 units.
- A vertical shift upward by 4 units.
Relative questions:
- What are the transformations for the parent function with ?
- How does a horizontal stretch differ from a horizontal shift?
- What effect does a reflection over the y-axis have on the graph of ?
- Can multiple transformations occur simultaneously, and how would you represent them?
- How would the graph of change if it were reflected and stretched vertically?
Tip: Always pay attention to the order of transformations. Horizontal changes (shifts and stretches/compressions) are applied before vertical ones when interpreting function transformations.
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Math Problem Analysis
Mathematical Concepts
Function Transformations
Parent Functions
Graphing
Formulas
Vertical Shift: f(x) + c or f(x) - c
Horizontal Shift: f(x - c) or f(x + c)
Reflection: -f(x) or f(-x)
Vertical Stretch/Compression: a * f(x)
Horizontal Stretch/Compression: f(bx)
Theorems
Transformation rules for functions
Reflection theorem
Suitable Grade Level
Grades 9-12
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