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A function table is a simple way to display the inputs and outputs of a function. It organizes our work so we can clearly see how different input values translate into output values. By setting up a function table, you can quickly evaluate a function for several different values of the variable.
When creating a function table, start by identifying the function you are working with. In this case, we use the polynomial function: \[ f(x) = x^3 + 2x^2 - 4x - 13 \] This table specifically facilitates calculations for a series of values systematically. Here, the table begins at \( x = -3 \) and increases by a step size of 2 each time.

• Choose a starting point for your input values.
• Define a consistent step size for each subsequent input.
• Calculate the corresponding output for each input using the given function.

This approach simplifies complex calculations, making your work more manageable and organized.

A polynomial function is an expression consisting of variables and coefficients, structured together using only addition, subtraction, multiplication, and non-negative integer exponents of variables. Polynomial functions can have one or more terms, with each term having a variable raised to an integer power.
In our example, \[ f(x) = x^3 + 2x^2 - 4x - 13 \]This is a polynomial function of degree three because the highest exponent of our variable \( x \) is three.

• The coefficient for \( x^3 \) is 1, for \( x^2 \) it's 2, and for \( x \) it's -4.
• The constant term is -13, which is the part of the function that doesn't change with \( x \).
• The dominant term here is \( x^3 \), as it influences the function's behavior most dramatically as \( x \) increases or decreases.

Understanding polynomial functions helps predict the curve's shape as it is graphed, providing insight into how the function values will change in response to shifts in \( x \).

In the context of function tables and sequences, a step size refers to the amount by which the input variable \( x \) is increased as you move from one value to the next in your calculations.
In this exercise, the step size is given as \( \Delta \mathrm{Tbl} = 2 \).This means each new value of \( x \) in the table is found by adding 2 to the previous value.

• Starting at the initial value (here \( x = -3 \)), a step increment defines a predictable pattern.
• By consistently adding the step size, we ensure the table covers an interval of outputs that thoroughly explores the function’s behavior across different input values.
• Ensures a structured approach that helps in visualizing and understanding the function significantly.

Using a step size allows us to comprehend how small changes in \( x \) affect \( f(x) \), which can aid in identifying patterns, trends, or key behaviors of the function.