Certainly! Let's work through the expression step by step to understand it clearly.
We are given the expression for [tex]\( y \)[/tex] as follows:
[tex]\[ y = \frac{1}{3} x^3 - x^2 + x - \frac{1}{3} \][/tex]
This expression is a polynomial in [tex]\( x \)[/tex]. Let's break it down and explain each term individually:
1. First Term:
[tex]\[ \frac{1}{3} x^3 \][/tex]
This is a cubic term, and it means that [tex]\( x \)[/tex] is raised to the power of 3 and then multiplied by [tex]\(\frac{1}{3} \)[/tex].
2. Second Term:
[tex]\[ - x^2 \][/tex]
This is a quadratic term, meaning [tex]\( x \)[/tex] is squared and then multiplied by -1. It is subtracted from the expression.
3. Third Term:
[tex]\[ + x \][/tex]
This is a linear term, meaning [tex]\( x \)[/tex] is included in its simplest form without any exponentiation, and it has a positive coefficient of 1.
4. Fourth Term:
[tex]\[ - \frac{1}{3} \][/tex]
This is a constant term, which is a simple number subtracted from the expression.
Now, combining all these terms together, we get:
[tex]\[ y = \frac{1}{3} x^3 - x^2 + x - \frac{1}{3} \][/tex]
To summarize the polynomial's structure:
- The leading term [tex]\(\frac{1}{3} x^3\)[/tex] determines the end behavior of the polynomial graph.
- The [tex]\(- x^2\)[/tex] term affects the curvature of the graph.
- The linear term [tex]\( x \)[/tex] influences the slope.
- The constant term [tex]\(-\frac{1}{3}\)[/tex] shifts the graph vertically.
So, the final expression, written in standard polynomial form, is:
[tex]\[ y = 0.333333333333333 x^3 - x^2 + x - 0.333333333333333 \][/tex]
Here, we use numerical approximations for the fractional coefficients where [tex]\(\frac{1}{3} \approx 0.333333333333333\)[/tex].
This polynomial expression accurately represents the given function for [tex]\( y \)[/tex].
