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Chapter 2: Problem 26

Graph the function. Find the slope, \(y\) -intercept and \(x\) -intercept, if any exist. . \(f(x)=\frac{1-x}{2}\)

Short Answer

Expert verified
The slope is \(-\frac{1}{2}\), the \(y\)-intercept is \((0, \frac{1}{2})\), and the \(x\)-intercept is \((1, 0)\).

Step by step solution

01

Identify the function type

The given function is linear, as it can be rewritten in the standard linear form, \(f(x) = -\frac{1}{2}x + \frac{1}{2}\). This indicates it is a straight line.
02

Find the slope

The slope of the linear function \(f(x) = mx + b\) is given by \(m\). In our equation, \(m = -\frac{1}{2}\), which means the slope of the line is \(-\frac{1}{2}\).
03

Determine the y-intercept

In the equation \(f(x) = mx + b\), \(b\) represents the \(y\)-intercept. Here, \(b = \frac{1}{2}\). Therefore, the \(y\)-intercept is the point \((0, \frac{1}{2}\)).
04

Calculate the x-intercept

To find the \(x\)-intercept, set \(f(x) = 0\) and solve for \(x\).\[0 = -\frac{1}{2}x + \frac{1}{2}\]To isolate \(x\), add \(\frac{1}{2}x\) to both sides:\[\frac{1}{2}x = \frac{1}{2}\]Multiply both sides by 2:\[x = 1\]Thus, the \(x\)-intercept is \((1, 0)\).
05

Graph the function

To graph the linear function, plot the \(y\)-intercept \((0, \frac{1}{2})\) and the \(x\)-intercept \((1, 0)\) on the coordinate plane. Draw a straight line through these two points, extending it across the graph. The line will represent the function \(f(x) = -\frac{1}{2}x + \frac{1}{2}\).

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Key Concepts

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

Slope of a Line
The slope of a line is a fundamental concept when dealing with linear functions. It essentially tells you how steep the line is, as well as its direction. For any linear equation of the form \( y = mx + b \), the slope \( m \) indicates the change in the \( y \)-values for a one-unit increase in the \( x \)-values. Here, the slope is \(-\frac{1}{2}\).
This means that as you move from one point to the next on the line, the \( y \)-value decreases by half a unit for every full unit that the \( x \)-value increases. Thus, the line slants downwards from left to right.
  • A positive slope indicates that the line rises as you move from left to right.
  • A negative slope means the line falls as you move from left to right.
  • A zero slope results in a horizontal line, while an undefined slope creates a vertical line.
Y-Intercept
The \( y \)-intercept is the point where the line crosses the \( y \)-axis. For the linear equation \( y = mx + b \), the \( y \)-intercept is represented by \( b \). It's the value of \( y \) when \( x = 0 \).
In our example, the \( y \)-intercept is \( \frac{1}{2} \), meaning the line crosses the \( y \)-axis at the point \((0, \frac{1}{2})\).
The \( y \)-intercept makes it easier to start graphing a line. By plotting this point, you have a starting reference to draw the line, using the slope to determine its path.
X-Intercept
Finding the \( x \)-intercept is equally important for graphing linear functions. It provides the point where the line crosses the \( x \)-axis. You find it by setting \( y = 0 \) and solving for \( x \).
For the function \( f(x) = -\frac{1}{2}x + \frac{1}{2} \), the \( x \)-intercept calculation yields \((1, 0)\). This means that when the line crosses the \( x \)-axis, \( x \) is 1.
This point is crucial in accurately drawing the line on a graph. Plot this point along with the \( y \)-intercept to accurately determine the line's position and trajectory.
Linear Equations
Linear equations are equations that form a straight line when graphed. They typically appear in the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the \( y \)-intercept. They describe a constant rate of change between two variables.
The given equation \( f(x)=\frac{1-x}{2} \) can easily be simplified to \( f(x) = -\frac{1}{2}x + \frac{1}{2} \). This form is perfect for identifying core features like slope and intercepts.
  • They can be rewritten or simplified for easier interpretation.
  • Used extensively in various fields, including economics, science, and statistics.
  • Essential in modeling relationships between variables.
Understanding linear equations forms the foundation for exploring more complex mathematical relationships.

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Most popular questions from this chapter

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