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Chapter 2: Problem 51

Show that a linear function is decreasing if and only if the slope of its graph is negative.

Short Answer

Expert verified
A linear function \(f(x) = mx + b\) is decreasing if and only if the slope of its graph is negative. This is proven by showing that if the function is decreasing, the slope must be negative (Part 1), and if the slope is negative, the function must be decreasing (Part 2). The key lies in analyzing the relationship between the coordinates of two points on the graph, (x1, y1) and (x2, y2), and their corresponding slopes. When the slope is negative, \(y2 < y1\) whenever \(x2 > x1\), confirming that the function is decreasing.

Step by step solution

01

Part 1: If a function is decreasing, then the slope is negative

Let's consider a linear function of the form \(f(x) = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. For a function to be decreasing, the output must decrease as the input increases. In other words, for any given pair of points (x1, y1) and (x2, y2) on the graph, if x2 > x1, then y2 < y1. Following the linear function form, we have \(y1 = mx1 + b\) and \(y2 = mx2 + b\). Now, let's subtract the first equation from the second equation: \(y2 - y1 = m(x2 - x1)\) We know that for a decreasing function, if x2 > x1, then y2 < y1. Therefore, \(x2 - x1 > 0\) and \(y2 - y1 < 0\). To satisfy this inequality, the slope \(m\) must be negative, and so it proves that if a linear function is decreasing, then the slope of its graph is negative.
02

Part 2: If the slope is negative, then the function is decreasing

Now, let's assume that the slope of the linear function's graph is negative, meaning \(m < 0\). We will show that if \(m < 0\), then the function is decreasing. Consider two points, (x1, y1) and (x2, y2) on the graph. Using the linear function form, we have \(y1 = mx1 + b\) and \(y2 = mx2 + b\). Subtract the first equation from the second equation: \(y2 - y1 = m(x2 - x1)\) For a function to be decreasing, we want to show that if \(x2 > x1\), then \(y2 < y1\). Since the slope \(m\) is negative, \(m(x2 - x1) < 0\) when \(x2 > x1\). Thus, \(y2 - y1 < 0\) when \(x2 > x1\), which implies that \(y2 < y1\) when \(x2 > x1\). This proves that if the slope of a linear function's graph is negative, then the function is decreasing. To conclude, we have shown that a linear function is decreasing if and only if the slope of its graph is negative.

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Key Concepts

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

Slope
In the context of a linear function, the slope is a measure of how steep the line is. This is represented by the letter \(m\) in the linear equation \(f(x) = mx + b\), where \(b\) is the y-intercept.
The slope describes the rate at which the function changes as the input, or \(x\), changes.
If the slope is positive, the function increases as \(x\) increases. Conversely, if the slope is negative, the function decreases as \(x\) increases.
  • Positive slope: Function goes upwards.
  • Negative slope: Function goes downwards.
To calculate the slope between two points on a line, use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This helps determine whether the line is increasing, decreasing, or constant.
Decreasing Function
A decreasing function is one where the output value diminishes as the input value increases. In simple terms, as you move from left to right along the x-axis, the y-value of the function decreases.
This situation occurs when the slope of the function is negative, meaning the line descends.
Characteristics of a decreasing function include:
  • For any \(x_1 < x_2\), \(f(x_1) > f(x_2)\).
  • The line tilts downwards on a graph.
  • No maximum points, only decreasing intervals.
Decreasing functions are frequently used in real-world scenarios, such as in finance to illustrate a decrease in profit over time.
Graph of a Function
The graph of a function is a visual representation of all the points that satisfy the function's equation. For linear functions, this graph is a straight line.
This line has a constant slope, meaning it does not curve or bend.
In order to draw the graph of a linear function, you need two key components:
  • Y-intercept: The point where the line crosses the y-axis, represented by \(b\) in \(f(x) = mx + b\).
  • Slope: The measure of the line's steepness, shown by \(m\).
Understanding and interpreting the graph helps in predicting the behavior of the function, such as identifying whether it is increasing or decreasing.
Negative Slope
A negative slope in a linear function indicates that as the input variable \(x\) increases, the output variable \(y\) decreases. This results in a straight line slanting downwards from left to right.
Mathematically, a negative slope \(m\) is derived when the difference between the y-values is negative, given \(x_2 > x_1\).
Characteristics of a line with a negative slope include:
  • The angle formed with the x-axis is greater than 90掳 but less than 180掳.
  • The function values drop as you move along the x-axis.
  • In a practical scenario, it might represent a decrease, such as a depreciating product price over time.
Recognizing a negative slope is crucial in determining the nature of the function as a decreasing one.

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