Math Problem Statement

The graph of y = (x + 1)^2 is transformed by the composition of two transformations to the graph of y = 2(x - 4)^2. Find these two transformations.

Solution

To transform the graph of y=(x+1)2y = (x + 1)^2 into y=2(x4)2y = 2(x - 4)^2, we need to identify two transformations applied to the original graph. These transformations include changes in position (horizontal and vertical shifts) and scaling (vertical stretching or compressing).

Step 1: Transformation from y=(x+1)2y = (x + 1)^2 to y=(x4)2y = (x - 4)^2

This involves a horizontal shift:

  • The expression x+1x + 1 becomes x4x - 4.
  • This indicates a shift 7 units to the right because: x+1x4    xx7x + 1 \to x - 4 \quad \implies \quad x \to x - 7

Step 2: Transformation from y=(x4)2y = (x - 4)^2 to y=2(x4)2y = 2(x - 4)^2

This involves a vertical scaling:

  • The coefficient of the quadratic term changes from 11 to 22.
  • This represents a vertical stretch by a factor of 2, meaning the graph becomes twice as steep.

Final Answer:

  1. Horizontal shift by 7 units to the right.
  2. Vertical stretch by a factor of 2.

Verification:

  • Starting from y=(x+1)2y = (x + 1)^2, shifting it 7 units to the right gives y=(x4)2y = (x - 4)^2.
  • Scaling this vertically by a factor of 2 gives y=2(x4)2y = 2(x - 4)^2.

Do you have any further questions or want a detailed explanation?

Related Questions:

  1. What would the transformations be if y=(x+1)2y = (x + 1)^2 were transformed to y=12(x3)2y = \frac{1}{2}(x - 3)^2?
  2. How do you graphically represent a vertical stretch on a parabola?
  3. If the transformations were reversed, how would y=2(x4)2y = 2(x - 4)^2 transform back to y=(x+1)2y = (x + 1)^2?
  4. What happens to the vertex of y=(x+1)2y = (x + 1)^2 after these transformations?
  5. How can horizontal shifts and vertical stretches be combined into a single transformation matrix?

Tip: When dealing with transformations, always track the changes in the vertex and the shape of the graph as a starting point!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Transformations of functions
Quadratic functions

Formulas

y = a(x - h)^2 + k (vertex form of a quadratic function)

Theorems

Transformation properties of quadratic graphs

Suitable Grade Level

Grades 9-12