Math Problem Statement

apply a horizontal compression by a factor of 2 and a shift 5 units to the right to the equation y=x^4 + x^2

Solution

To apply the given transformations to the equation y=x4+x2y = x^4 + x^2, follow these steps:

1. Horizontal compression by a factor of 2:

When applying a horizontal compression by a factor of 2, you replace xx with x2\frac{x}{2} in the equation. This compresses the graph horizontally by a factor of 2.

So, the new equation becomes: y=(x2)4+(x2)2y = \left(\frac{x}{2}\right)^4 + \left(\frac{x}{2}\right)^2 Simplifying: y=x416+x24y = \frac{x^4}{16} + \frac{x^2}{4}

2. Horizontal shift 5 units to the right:

To shift the graph 5 units to the right, replace xx with x5x - 5 in the equation.

Now the equation becomes: y=(x52)4+(x52)2y = \left(\frac{x - 5}{2}\right)^4 + \left(\frac{x - 5}{2}\right)^2

Simplifying this: y=(x5)416+(x5)24y = \frac{(x - 5)^4}{16} + \frac{(x - 5)^2}{4}

Final equation:

y=(x5)416+(x5)24y = \frac{(x - 5)^4}{16} + \frac{(x - 5)^2}{4}

This is the equation after applying the horizontal compression by a factor of 2 and a horizontal shift 5 units to the right.


Do you want more details or have any questions?

Here are 5 related questions you could explore:

  1. How do vertical compressions and stretches affect an equation?
  2. What is the difference between horizontal shifts and vertical shifts?
  3. How does a vertical shift by 3 units upwards change the equation?
  4. Can you apply both vertical and horizontal compressions at the same time?
  5. What happens to the graph of y=x2y = x^2 if you apply a horizontal stretch by a factor of 3?

Tip: Remember that horizontal transformations (shifts and compressions) act on the xx-variable, often in reverse of what you might expect! For example, shifting to the right involves subtracting from xx.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Transformations of Functions

Formulas

Horizontal compression by a factor of a: replace x with x/a
Horizontal shift by h units to the right: replace x with x - h

Theorems

Transformation Theorem: Horizontal stretches/compressions and shifts affect the input variable (x) in the reverse way expected.

Suitable Grade Level

Grades 10-12