Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether each function is increasing or decreasing. $$ h(x)=-2 x+4 $$

Step-by-Step Solution

Answer

The function is decreasing.

Step 1: Understand the Function

First, identify the type and components of the given function. The function provided is a linear function of the form \( h(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Identify the Slope

In the function \( h(x) = -2x + 4 \), the coefficient of \( x \) is the slope \( m \). Here, \( m = -2 \).

Step 3: Analyze the Slope

Determine the behavior of the function based on the slope. If \( m > 0 \), the function is increasing. If \( m < 0 \), the function is decreasing. Since \( m = -2 < 0 \), the function is decreasing.

Key Concepts

Slope of a Line

The slope of a line is crucial in understanding how linear functions behave. Think of the slope as the steepness or tilt of a line on a graph. It is represented by the letter \( m \) in the equation of a line, which is often written in the form \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis.

• If the slope \( m \) is positive, it means that as you move from left to right on a graph, the line goes upwards. This is like climbing a hill.
• If \( m \) is negative, the line moves downwards, similar to going down a hill.
• A slope of zero means the line is perfectly horizontal, indicating no rise or fall.

Understanding the slope helps you predict the direction and tilt of the line, which is fundamental for analyzing linear functions.

Function Analysis

Function analysis is about understanding what a function represents and how it behaves. In the context of linear functions, which are functions of the form \( f(x) = mx + b \), analysis involves identifying key features like slope and y-intercept.

Here's how to approach this task:

• Identify the type of function: Verify whether it's linear, quadratic, etc. In our case, the function \( h(x) = -2x + 4 \) is linear.
• Determine the slope and y-intercept: In the equation \( h(x) = mx + b \), \( m \) is \(-2\) and \( b \) is \(4\).
• Understand what these values mean: A slope of \(-2\) suggests that for every 1 unit increase in \( x \), \( h(x) \) decreases by 2 units.

Function analysis helps to break down a function into understandable parts, enabling you to determine characteristics like increasing or decreasing behavior.

Increasing and Decreasing Functions

Determining whether a function increases or decreases is key to understanding its graph and how it changes values. Specifically for linear functions represented by \( y = mx + b \):

• When the slope \( m > 0 \), the function is considered increasing. This means as \( x \) increases, the function's output also increases.
• Conversely, when the slope \( m < 0 \), the function is decreasing. This indicates that as \( x \) increases, the output decreases.

For our function \( h(x) = -2x + 4 \), the slope is \(-2\), hence it's a decreasing function. Recognizing whether a function increases or decreases helps in predicting and modeling real-world phenomena, as well as in various applications like trend analysis and forecasting.