Problem 23 - Precalculus : Building Concepts and Connections

Question

Determine the end behavior of the function. $$f(t)=7 t$$

Step-by-Step Solution

Answer

As \(t\) approaches infinity, \(f(t)\) approaches infinity. As \(t\) approaches negative infinity, \(f(t)\) approaches negative infinity.

Step 1: Identify the Type of Function

First, identify the function type. In this case, \(f(t)=7t\), is a linear function, which is a polynomial of degree 1.

Step 2: Determine the Coefficient of the Function

The coefficient of t in the function is 7. Coefficients can help to predict the end behavior of a function. If the coefficient is positive, as t approaches infinity, the function will also approach infinity. If the coefficient is negative, the function will approach negative infinity as t approaches infinity.

Step 3: Determine End Behavior

Given that the coefficient is positive, the function will rise without bound as t approaches positive infinity, and will decrease without bound as t approaches negative infinity. Therefore, the end behavior can be expressed as: As \(t\) approaches infinity, \(f(t)\) approaches infinity. As \(t\) approaches negative infinity, \(f(t)\) approaches negative infinity.

Key Concepts

Linear Functions

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.

Polynomial Degree

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.

Function Coefficients

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.