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Chapter 5: Problem 8

Give the values for \(b\) and \(m\) for the linear functions. $$ w(c)=0.5 c $$

Short Answer

Expert verified
Answer: For the given linear function, the values of \(b\) and \(m\) are: \(b = 0\) and \(m = 0.5\).

Step by step solution

01

Rewrite the function

Rewrite the given function to match the form (\(y = mx + b\)): $$w(c) = 0.5c$$ can be rewritten as $$y = 0.5x$$
02

Identify the values for \(b\) and \(m\)

Now we can see that the function is written in the form $$y = 0.5x$$ and can be compared to the general linear function form $$y = mx + b$$ Comparing these two, we can identify that: $$m = 0.5$$ and $$b = 0$$
03

Present the values of \(b\) and \(m\)

The values for \(b\) and \(m\) for the given linear function are as follows: $$b = 0$$$$m = 0.5$$

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most common ways to express a linear function. You'll often see it written as \(y = mx + b\). This structure is not only simple but also incredibly intuitive for understanding and graphing linear relationships.

Here's a quick breakdown of what each part means:
  • \(m\): This is the slope of the line. It tells you how steep the line is and in which direction it is heading. A positive \(m\) indicates the line rises as you move from left to right, while a negative \(m\) indicates it falls.
  • \(b\): This is the y-intercept. It shows the point where the line crosses the y-axis. When \(b\) is zero, the line passes through the origin (point \((0,0)\)).
By getting familiar with the slope-intercept form, you'll find it easier to write and interpret linear equations. This understanding will help you in identifying those magic numbers \(m\) and \(b\) that characterize any line.
Coefficient Identification
Identifying the coefficients \(m\) and \(b\) in the slope-intercept equation is a core skill in dealing with linear functions. This task is about recognizing the parts of the equation that define its behavior and position on the coordinate plane.

In the equation \(y = mx + b\), identifying these coefficients is straightforward:
  • The Slope (\(m\)): This comes right before the variable \(x\). In our example with the line \(y = 0.5x\), the slope is \(0.5\). This means for every unit increase in \(x\), \(y\) will increase by half a unit.
  • The Y-Intercept (\(b\)): Typically, it's the constant term, floating on its own without an \(x\). In \(y = 0.5x\), because there's no such term, we know \(b = 0\).
Recognizing these parts helps you quickly understand the behavior of the function鈥攈ow it slopes and where it intercepts the y-axis.
Function Rewriting
Function rewriting is the process of transforming a function into a different, but equivalent, form. This can make interpreting or working with the function much easier.

In order to analyze the function \(w(c) = 0.5c\), it's beneficial to rewrite it to fit the familiar slope-intercept form \(y = mx + b\):
  • You'll substitute \(y\) for the function notation \(w(c)\), reflecting that they both stand for the dependent variable.
  • Change \(c\) to \(x\) to fit the typical format of slope-intercept form, making it \(y = 0.5x + 0\).
By rewriting, we aligned the original function with the standard linear equation, making it easier to identify the slope \(m = 0.5\) and y-intercept \(b = 0\). Rewriting functions in different forms is a vital skill, allowing flexibility in solving problems and interpreting information.

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