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

To apply the given transformations to the equation \( y = x^4 + x^2 \), follow these steps:

### 1. Horizontal compression by a factor of 2:
When applying a horizontal compression by a factor of 2, you replace \( x \) with \( \frac{x}{2} \) in the equation. This compresses the graph horizontally by a factor of 2.

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

### 2. Horizontal shift 5 units to the right:
To shift the graph 5 units to the right, replace \( x \) with \( x - 5 \) in the equation.

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

Simplifying this:
\[
y = \frac{(x - 5)^4}{16} + \frac{(x - 5)^2}{4}
\]

### Final equation:
\[
y = \frac{(x - 5)^4}{16} + \frac{(x - 5)^2}{4}
\]

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

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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 \( y = x^2 \) if you apply a horizontal stretch by a factor of 3?

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

Learn how to apply a horizontal compression by a factor of 2 and shift the equation y = x^4 + x^2 by 5 units to the right. This step-by-step solution covers key transformation rules for polynomial functions, ideal for high school students in Grades 10-12.

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