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A **Polynomial Function** is a mathematical expression consisting of variables, coefficients, and exponents. The name "polynomial" comes from "poly," meaning many, and "nomial," meaning terms. Therefore, it is a combination of many terms. A typical polynomial function can be written as:\[a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\]where:

• \(a_n, a_{n-1}, \ldots, a_0\) are constants known as coefficients.
• \(x\) is the variable.
• The highest power of \(x\) (n) is called the degree of the polynomial.

Polynomial functions are smooth and continuous everywhere. For the given function, \( g(t) = t^4 - t^3 + t^2 \), you have a polynomial of degree 4. This implies the graph will have at most 3 turning points, providing a broad idea of its shape.

**Evaluating Functions** refers to finding the value of a function for specific inputs of its variable(s). It involves substituting the given value into the function and simplifying. For example, if you have a function \( g(t) = t^4 - t^3 + t^2 \) and you want to find \( g(-2) \), substitute \( t = -2 \):
\[g(-2) = (-2)^4 - (-2)^3 + (-2)^2\]
Calculate each term:

• \((-2)^4 = 16\)
• \((-2)^3 = -8\)
• \((-2)^2 = 4\)

Add these results together to obtain\[16 - (-8) + 4 = 28\]. Evaluating functions is crucial for solving and graphing polynomials.

The **Difference Quotient** is a formula that provides the average rate of change of the function over a specific interval. It's a fundamental concept in calculus, as it forms the basis for the derivative, which represents instantaneous rate of change.
For a function \( f(x) \), the difference quotient over the interval \([a, b]\) is:
\[\frac{f(b) - f(a)}{b - a}\]This formula calculates how much the function's output changes compared to how much the input changes. It's particularly useful in determining how steep or flat a function's graph is over an interval.
In the context of the given exercise, it helps find the average rate of change of \( g(t) \) from \( t = -2 \) to \( t = 2 \). That's why \[\frac{g(2) - g(-2)}{2 - (-2)}\] obtains the result of \(-4\).

The **Rate of Change** is a fundamental concept in mathematics, describing how one quantity changes concerning another. In simpler terms, it's how fast or slow a variable changes over a given period.

• For linear functions, the rate of change is constant, usually known as the slope.
• For non-linear functions like polynomials, the rate of change varies along the function's graph.

The average rate of change between two points on a curve measures the slope of the secant line connecting those points. It answers the question: if this change were linear, how fast would the change happen?
In the exercise, the average rate of change between \( t = -2 \) and \( t = 2 \) was calculated:\[\frac{g(2) - g(-2)}{2 - (-2)} = -4\]This result indicates that, on average, the function's value decreases by 4 units as \( t \) increases by 1 unit from -2 to 2.