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:

  1. Translation (Shift):

    • Vertical Shift: Moves the graph up or down.
      • f(x)+cf(x) + c: Shifts the graph upward by cc units.
      • f(x)cf(x) - c: Shifts the graph downward by cc units.
    • Horizontal Shift: Moves the graph left or right.
      • f(xc)f(x - c): Shifts the graph right by cc units.
      • f(x+c)f(x + c): Shifts the graph left by cc units.
  2. Reflection:

    • Reflection over the x-axis:
      • f(x)-f(x): Flips the graph upside down (over the x-axis).
    • Reflection over the y-axis:
      • f(x)f(-x): Flips the graph horizontally (over the y-axis).
  3. Stretching and Compressing:

    • Vertical Stretch/Compression:
      • af(x)a \cdot f(x): Stretches the graph vertically by a factor of aa if a>1a > 1, or compresses it if 0<a<10 < a < 1.
    • Horizontal Stretch/Compression:
      • f(bx)f(bx): Compresses the graph horizontally by a factor of bb if b>1b > 1, or stretches it if 0<b<10 < b < 1.
  4. 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 f(x)=x2f(x) = x^2 (a parabola), applying the transformation g(x)=2(x3)2+4g(x) = -2(x - 3)^2 + 4 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:

  1. What are the transformations for the parent function f(x)=sin(x)f(x) = \sin(x) with g(x)=2sin(xπ)+1g(x) = 2\sin(x - \pi) + 1?
  2. How does a horizontal stretch differ from a horizontal shift?
  3. What effect does a reflection over the y-axis have on the graph of f(x)=x3f(x) = x^3?
  4. Can multiple transformations occur simultaneously, and how would you represent them?
  5. How would the graph of f(x)=xf(x) = \sqrt{x} 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