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Chapter 2: Problem 56

Graph each linear or constant function. Give the domain and range. $$ F(x)=-\frac{1}{4} x+1 $$

Short Answer

Expert verified
Plot the y-intercept (0, 1) and another point (4, 0), then draw a line through them. Domain: \( (-\infty, +\infty) \), Range: \( (-\infty, +\infty) \).

Step by step solution

01

Identify the slope and y-intercept

The given function is in the slope-intercept form, which is written as \( F(x) = -\frac{1}{4} x + 1 \). Here, the slope \( m \) is \( -\frac{1}{4} \), and the y-intercept \( b \) is 1.
02

Plot the y-intercept

Start by plotting the y-intercept (0, 1) on the Cartesian plane. This is the point where the function crosses the y-axis.
03

Use the slope to find another point

The slope \( -\frac{1}{4} \) means that for every 4 units you move to the right (positive x-direction), you move 1 unit down (negative y-direction). Starting from (0, 1), move 4 units to the right to (4, 1), then 1 unit down to (4, 0). Plot this point.
04

Draw the line

Draw a straight line through the two points (0, 1) and (4, 0). This line represents the function \( F(x) = -\frac{1}{4} x + 1 \).
05

Determine the domain

The domain of a linear function is all real numbers. Therefore, the domain is \( (-\infty, +\infty) \).
06

Determine the range

The range of a linear function is also all real numbers. Therefore, the range is \( (-\infty, +\infty) \).

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Key Concepts

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

Understanding the Slope-Intercept Form
To graph a linear function, we first need to understand its equation format, known as the slope-intercept form. This form is written as \( y = mx + b \), where:
  • \( m \) is the slope of the line
  • \( b \) is the y-intercept, the point where the line crosses the y-axis
In the given equation \( F(x) = -\frac{1}{4} x + 1 \), we identify the slope \( m \) as \( -\frac{1}{4} \) and the y-intercept \( b \) as 1. The negative slope indicates the line falls as we move from left to right on the graph. Simply put, for every 4 units you move to the right on the x-axis, you will move 1 unit down on the y-axis.
Plotting Points on the Graph
After identifying the slope and y-intercept, the next step is plotting points to draw the line. Start with the y-intercept:
  • Locate (0, 1) on the graph, as this is where the line crosses the y-axis.
Next, use the slope to find another point:
  • The slope \( -\frac{1}{4} \) means for every 4 units you move to the right, you move 1 unit down. From (0, 1), move to (4, 1) horizontally, then drop down to (4, 0) vertically.
By plotting these points, you give yourself a clear pathway to draw the entire line representing the linear function \( F(x) = -\frac{1}{4} x + 1 \). Finally, draw a straight line through the plotted points, thereby graphing the function correctly and effectively.
Domain and Range of Linear Functions
Understanding the domain and range helps us know how the function behaves over different inputs. For linear functions like \( F(x) = -\frac{1}{4} x + 1 \):
  • The domain is all possible x-values. For linear functions, the graph extends infinitely in both left and right directions, hence the domain is all real numbers, \( (-\infty, +\infty) \).
  • The range is all possible y-values. Similarly, the line stretches up and down infinitely, making the range all real numbers, \( (-\infty, +\infty) \) as well.
By understanding and visualizing the domain and range, we grasp how the linear function continues endlessly in both directions, which is a key characteristic of linear equations.

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Most popular questions from this chapter

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