Problem 38 - Algebra Form and Function

Question

Give the slope and \(y\) -intercept for the graphs of the functions in Problems \(38-43\). $$ f(x)=220-12 x $$

Step-by-Step Solution

Answer

Answer: The slope of the function is -12, and the y-intercept is 220.

Step 1: Identify the linear function

The given function is a linear function of the form f(x) = 220 - 12x.

Step 2: Find the slope of the function

The slope of the function is the coefficient of x in the linear function. In this case, the coefficient of x is -12. So, the slope of the function is -12.

Step 3: Find the y-intercept of the function

The y-intercept of the function is the constant term in the linear function. In this case, the constant term is 220. So, the y-intercept of the function is 220.

Step 4: Write the final answer

The slope of the function f(x) = 220 - 12x is -12, and the y-intercept is 220.

Key Concepts

Understanding Slope

The concept of slope is fundamental to understanding linear equations. The slope is a measure of how steep a line is, and it conveys how the line rises or falls. In a linear function of the form \(f(x) = mx + b\), the slope is represented by the coefficient \(m\). In simple terms, the slope tells us how much the \(y\)-value of a function changes for a one-unit increase in the \(x\)-value.
In our example, the linear function is \(f(x) = 220 - 12x\). Here, our slope \(m\) is \(-12\). This means that for each unit increase in \(x\), the \(y\)-value decreases by 12 units. A negative slope like this indicates the line is slanting downwards, creating a downhill effect.
When dealing with slope, remember:

• A positive slope means the line rises as \(x\) increases.
• A negative slope means the line falls as \(x\) increases.
• A slope of zero means the line is horizontal, with no rise or fall.

Demystifying the Y-Intercept

The y-intercept is an essential component in understanding a linear function. It is the point where the line crosses the y-axis. In the function \(f(x) = mx + b\), \(b\) is the y-intercept. This point is where \(x\) is zero, meaning that it represents the initial value of \(y\) when nothing else has affected it yet.
In our linear function \(f(x) = 220 - 12x\), the y-intercept \(b\) is 220. This value tells us that when \(x = 0\), the \(y\)-value is 220. Effectively, it represents the starting point of the line on the graph, a critical piece of information for graphing.
Understanding where the y-intercept lies helps us:

• Identify the starting position of the line on a graph.
• Interpret the initial value or condition in various real-world scenarios.
• Gain a complete understanding of the linear function's behavior.

Linear Function Essentials

Linear functions are a category of function characterized by their straight-line graphs. Such functions are expressed in the standard form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Linear functions are a cornerstone of algebra and appear frequently in both mathematical problems and real-world applications.
In the problem given, \(f(x) = 220 - 12x\) is a classic linear function. With a negative slope of \(-12\) and a y-intercept of 220, this linear function defines a direct relationship where for every increase of one in \(x\), \(y\) decreases by 12.
Some important points about linear functions:

• They represent constant rates of change.
• They graph to straight lines, making them easy to predict and analyze.
• They are essential for modeling real-world scenarios where relationships are proportional or constant over time.