To solve the expression given by [tex]\( y = -25x^{-3} + 4x^2 \)[/tex], let's break it down step-by-step.
1. Understanding the terms:
- The expression consists of two different terms: [tex]\( -25x^{-3} \)[/tex] and [tex]\( 4x^2 \)[/tex].
2. Analyze each term separately:
- The term [tex]\( -25x^{-3} \)[/tex]:
- This is an inverse power term, which can also be written as [tex]\( -\frac{25}{x^3} \)[/tex].
- The term [tex]\( 4x^2 \)[/tex]:
- This is a quadratic term, which represents a typical parabola in the [tex]\( y = ax^2 \)[/tex] form.
3. Combining the terms:
- When you combine these terms, you have the expression [tex]\( y = -25x^{-3} + 4x^2 \)[/tex].
- In another form, it could be written as [tex]\( y = 4x^2 - \frac{25}{x^3} \)[/tex], which highlights the balance of a quadratic element and an inverse cubic element.
4. Graphical interpretation:
- The function [tex]\( 4x^2 \)[/tex] tends to dominate for large values of [tex]\( x \)[/tex], driving [tex]\( y \)[/tex] upwards.
- The term [tex]\( -25x^{-3} \)[/tex] becomes significant for small values of [tex]\( x \)[/tex], driving [tex]\( y \)[/tex] downwards.
- Together, these two terms form a unique curve that reflects both characteristics.
5. Final expression:
- Thus, combining these steps, we end up with the expression which is [tex]\( y = 4x^2 - \frac{25}{x^3} \)[/tex].
So, the detailed solution to the given function [tex]\( y = -25x^{-3} + 4x^2 \)[/tex] results in the combined expression [tex]\( y = 4x^2 - \frac{25}{x^3} \)[/tex].
