91Ó°ÊÓ

In mathematics, particularly in functions, the term 'independent variable' describes a quantity that is controlled or manipulated to observe its effect on another quantity, known as the dependent variable. In our exercise, the independent variable is represented by the symbol \( p \). This variable is independent because its value is given to us, typically as part of the problem's conditions.

You can think of an independent variable as the input of a function. It is the variable you "plug" into the function to see how it affects the outcome or behavior of the function. For example, when we know \( p = 10 \), we use this value to further investigate the resulting behavior of our function \( m(p) \). By first identifying the independent variable, we set the stage for analyzing the function systematically.

The dependent variable in a function is the output that results from substituting a particular value for the independent variable. It "depends" on the value assigned to the independent variable. In our example, \( m(p) \) is the dependent variable because it changes in response to different values of \( p \).

In simpler terms, think of the dependent variable as the "effect" in a cause and effect relationship. When different values are plugged into \( p \), the function \( m(p) = -2p^2 + 20.1 \) evaluates to different output values \( m \). Hence, the function's response will dictate the output or dependent variable for the condition provided within the function's context.

The substitution method is a technique used in evaluating functions by directly substituting the independent variable's value into the function's expression. This procedure involves replacing the variable with its given numerical value and performing arithmetic operations as prescribed by the function.

The primary steps involved in this method include:

• Identify the value of the independent variable, in this case, \( p = 10 \).
• Substitute this value into the function's equation, \( m(p) = -2p^2 + 20.1 \), giving us \( m(10) = -2(10)^2 + 20.1 \).
• Carry out the calculations systematically: compute powers, perform multiplication, and add constants to determine the final output.

This technique is broadly applicable to a wide variety of functions and is essential for determining specific function values under given conditions.

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. These functions are characterized by the polynomial degree, which is determined by the highest exponent of the variable in the expression.

Our example, \( m(p) = -2p^2 + 20.1 \), is a simple polynomial function of degree 2, as indicated by the tallest power, \( p^2 \). The format consists of:

• A squared term, \( -2p^2 \), indicating it is a quadratic polynomial.
• An optional linear (first-degree) or constant term, \( 20.1 \), which does not involve the variable \( p \).

Polynomial functions are versatile and widely used in various applications, from modeling real-world scenarios to solving complex equations. Their ease of plugging in values makes them an excellent choice for learning how to appraise outcomes with functional expressions.