Question

What word best describes the function $$ f(x)=-\frac{4}{13} x-\frac{7}{5} ? $$ A. negative B. increasing C. linear D. symmetric

Short answer

Linear

Step-by-step solution

- Understand the Problem

The function given is a mathematical expression of the form \( f(x) = ax + b \). Identify the type and characteristics of this function.

- Identify the Form of the Function

The function \( f(x) = -\frac{4}{13} x - \frac{7}{5} \) is in the form of \( f(x) = ax + b \), which is a linear function. In a linear function, the highest power of \( x \) is 1.

- Determine the Characteristics of Linear Functions

Linear functions have a constant rate of change. This means that for any value of \( x \), the function's rate of change is characterized by the coefficient of \( x \). Thus, the function is linear and not quadratic, cubic, or another higher-order polynomial.

- Compare with Choices

The choices are: A. Negative, B. Increasing, C. Linear, D. Symmetric. The term that best fits a function of the form \( f(x) = ax + b \) is 'linear'. It is not inherently negative, increasing, or symmetric based solely on its general form.

- Conclusion

After understanding that the given function is a straight line (has a linear form\( f(x) = -\frac{4}{13} x - \frac{7}{5} \)), we can determine that the best word to describe it is 'linear'.

Key concepts

Rate of change

In linear functions, the 'rate of change' is a critical concept. It tells us how much one quantity changes for a unit change in another quantity. For the linear function \( f(x) = -\frac{4}{13} x - \frac{7}{5} \), the rate of change is represented by the coefficient of \( x \), which is \( -\frac{4}{13} \).
This coefficient indicates how rapidly the value of \( f(x) \) changes as \( x \) increases. We interpret this as:

• A negative coefficient \( (-\frac{4}{13}) \) means that \( f(x) \) decreases as \( x \) increases.
• The magnitude, \( \frac{4}{13} \), shows how steep the decline is.

The concept of the rate of change is fundamental in understanding the behavior of linear functions and an essential aspect of calculus and real-world applications.

Constant rate

Linear functions are defined by their 'constant rate' of change. This means that for every unit increase in \( x \), there is a consistent change in \( f(x) \). For example, in our function \( f(x) = -\frac{4}{13} x - \frac{7}{5} \), the rate of change does not vary; it is always \( -\frac{4}{13} \). This constancy has several implications:

• Graphically, it results in a straight line.
• In practical problems, it represents predictable and uniform changes.

Understanding that linear functions maintain a constant rate of change allows us to make accurate predictions and simple calculations in various fields, from physics to economics.

Mathematical expressions

Linear functions are represented by simple 'mathematical expressions' of the form \( ax + b \). In our case, the expression is: \( f(x) = -\frac{4}{13} x - \frac{7}{5} \). Here’s a breakdown of the components:

• \( a \) (\( -\frac{4}{13} \)): The slope. It determines the direction (positive/negative) and steepness.
• \( b \) (\( -\frac{7}{5} \)): The y-intercept. It’s the value of \( f(x) \) when \( x \) is zero.

Such expressions are straightforward and make it easy to understand how the function behaves. Additionally, they are essential in both algebra and calculus for solving equations and modeling real-world scenarios. By mastering these expressions, we can easily interpret and manipulate mathematical relations.