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Linear functions are some of the simplest types of functions one can encounter in algebra. They can be represented in the form of \( f(t) = mt + b \), where \( m \) and \( b \) are constants. The variable \( t \) is raised to the power of 1, which is why they are called 'linear'. These functions graph as straight lines in the coordinate plane.
A key characteristic of linear functions is that they provide a constant rate of change. This means that as the variable \( t \) increases by a unit, the function's output increases by a constant amount \( m \), known as the slope.
In the exercise's function \( f(t) = 7t \), the absence of the constant \( b \) suggests the line passes through the origin \((0,0)\), and its slope or rate of change is 7. Because the coefficient 7 is positive, the line slants upwards as it moves from left to right.

The degree of a polynomial function is essentially the largest exponent of the variable in the function. For linear functions, this degree is 1 since the variable is raised only to the first power.
Understanding the degree is important because it tells us about the number of possible roots the polynomial might have, and hints at the function's "end behavior."

• Degree of 0 means a constant function.
• Degree of 1 means a linear function.
• Higher degrees (e.g., 2, 3) imply quadratic, cubic forms, and higher polynomial functions.

Since our function \( f(t) = 7t \) is linear, it is a first-degree polynomial. First-degree polynomials have straightforward, linear end behaviors, characterized by constant increase or decrease without bounds.

The coefficient in a polynomial function is the number that multiplies the variable. Coefficients are vital because they directly influence the steepness and direction of a line or curve graphed from a function.

• If the coefficient is positive, the line slopes upwards.
• If negative, the line slopes downwards.
• The absolute value of the coefficient tells us how steep the line is.

In \( f(t) = 7t \), the coefficient is 7. This positive value indicates that as \( t \) increases, so does \( f(t) \), extending indefinitely when \( t \) approaches infinity. Conversely, it decreases without bound as \( t \) approaches negative infinity. Hence, the number 7 sets both the steepness of the line and dictates the end behavior being both increasing and decreasing.