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Chapter 2: Problem 12

Determine if each function is increasing or decreasing $$ k(x)=-4 x+1 $$

Short Answer

Expert verified
The function \( k(x) = -4x + 1 \) is decreasing.

Step by step solution

01

Identify the Linear Function Format

The given function is in the format of a linear function: \[ k(x) = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. In this case, \( k(x) = -4x + 1 \).
02

Determine the Sign of the Slope

The slope \( m \) of the function \( k(x) = -4x + 1 \) is \(-4\). This value is negative.
03

Conclude About the Function Behavior

Since the slope \( m = -4 \) is negative, the function \( k(x) \) is decreasing. In linear functions, a negative slope indicates a decreasing function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope
The slope of a linear function is a crucial element that determines the direction and steepness of the line. For any linear equation in the form of \( y = mx + b \), the slope is represented by the coefficient \( m \). This coefficient indicates how much the output value (\( y \)) changes for every unit change in the input (\( x \)).
  • If \( m \) is positive, the line ascends from left to right.
  • If \( m \) is zero, the line is flat, parallel to the x-axis.
  • If \( m \) is negative, the line descends from left to right.
In the example function \( k(x) = -4x + 1 \), the slope is \(-4\). This negative value tells us that for each unit increase in \( x \), \( y \) decreases by 4 units, thus giving the line a downward slope.
Increasing and Decreasing Functions
A function can be described as increasing or decreasing based on the behavior of the slope:- **Increasing Function**: Occurs when the slope \( m \) is positive. As \( x \) gets larger, \( y \) increases, making the line rise.- **Decreasing Function**: Happens when the slope \( m \) is negative. In this situation, \( y \) decreases as \( x \) increases, causing the line to fall.In our exercise, the function \( k(x) = -4x + 1 \) is decreasing because its slope is \(-4\). This means as you move from left to right along the x-axis, the value of the function declines.
Function Behavior
The behavior of a function outlines how it responds as the input changes. In linear functions, this behavior is directly linked to the slope. With only one slope across the entire line, the behavior remains consistent:
  • If the slope is positive, the function is always increasing.
  • If the slope is negative, like in \( k(x) = -4x + 1 \), the function is always decreasing.
This consistent behavior means that once you determine the slope of a linear function, you can predict how it will behave overall. In the case of our example, \(-4\) ensures that \( k(x) \) steadily decreases without any fluctuations as \( x \) increases.

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Most popular questions from this chapter

Solve each inequality. $$ |3 x+9|<4 $$

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