Math Problem Statement

Write the equation of the image y = f(x) after each transformation. Include the mapping notation.

Solution

The image contains a table showing the equations of transformations applied to functions, including horizontal and vertical stretches (hsf and vsf) and their corresponding mapping notations. Let's break it down:

1. First row:

  • Function: y=3f(x2)y = 3f\left(\frac{x}{2}\right)
  • Stretch:
    • hsf = 2 (horizontal stretch factor)
    • vsf = 3 (vertical stretch factor)
  • Mapping notation:
    • x2xx \to 2x
    • y3yy \to 3y

2. Second row:

  • Function: y=14f(x3)y = \frac{1}{4}f\left(\frac{x}{3}\right)
  • Stretch:
    • hsf = 3
    • vsf = 14\frac{1}{4}
  • Mapping notation:
    • x3xx \to 3x
    • yy4y \to \frac{y}{4}

3. Third row:

  • Function: y=25f(4x)y = \frac{2}{5}f(4x)
  • Stretch:
    • hsf = 14\frac{1}{4}
    • vsf = 25\frac{2}{5}
  • Mapping notation:
    • xx4x \to \frac{x}{4}
    • y25yy \to \frac{2}{5}y

4. Fourth row:

  • Function: y=59f(73x)y = \frac{5}{9}f\left(\frac{7}{3}x\right)
  • Stretch:
    • hsf = 37\frac{3}{7}
    • vsf = 59\frac{5}{9}
  • Mapping notation:
    • x3x7x \to \frac{3x}{7}
    • y5y9y \to \frac{5y}{9}

Would you like further clarification or details on how to apply these transformations?

Here are five related questions to deepen your understanding:

  1. How do you determine whether a function is being horizontally stretched or compressed?
  2. What effect does a vertical stretch factor greater than 1 have on the graph of a function?
  3. How would you describe the difference between a horizontal stretch and a reflection across the y-axis?
  4. What happens to the graph of a function when both horizontal and vertical transformations are applied?
  5. Can you apply a horizontal stretch and a translation to a function at the same time?

Tip: Horizontal transformations affect the x-coordinate inversely; for example, a factor of 12\frac{1}{2} horizontally stretches by 2!

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Math Problem Analysis

Mathematical Concepts

Function Transformations
Horizontal Stretch
Vertical Stretch
Mapping Notation

Formulas

Horizontal stretch: y = f(c*x) -> x' = x/c
Vertical stretch: y = k*f(x) -> y' = k*y

Theorems

Function Transformations

Suitable Grade Level

Grades 9-12