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Magazine Chapter 3: Problem 84
Graph the linear function and state the domain and range. $$ g(x)=-4 x+6 $$
Short Answer
Expert verified
Domain: \((-\infty, \infty)\)
Range: \((-\infty, \infty)\)
Step by step solution
01
Identify Slope and Y-Intercept
The linear function is given in the slope-intercept form, which is \( y = mx + b \). Here, \( m = -4 \) is the slope, and \( b = 6 \) is the y-intercept. This means the line will cross the y-axis at the point (0, 6) and the slope \( -4 \) indicates that for every unit increase in \( x \), \( y \) decreases by 4 units.
02
Calculate Additional Points
Choose another value for \( x \) to find a second point on the line. If \( x = 1 \), then \( g(1) = -4(1) + 6 = 2 \). Therefore, another point on the line is (1, 2).
03
Plot Points on Graph
Plot the y-intercept point (0, 6) and the point (1, 2) on a Cartesian plane. These points provide a visual guide for the line.
04
Draw the Line
Use a ruler to draw a straight line through the plotted points (0, 6) and (1, 2). Extend the line in both directions, ensuring it passes through all possible values of \( x \). This line represents the graph of the function \( g(x) = -4x + 6 \).
05
Determine the Domain and Range
For a linear function, the domain is all real numbers, expressed as \( (-\infty, \infty) \), because there are no restrictions on the values \( x \) can take. The range is also all real numbers \( (-\infty, \infty) \) because as \( x \) approaches infinity or negative infinity, \( g(x) \) also approaches infinity or negative infinity, respectively.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
Linear functions are often expressed in the slope-intercept form, which is written as \( y = mx + b \). This formula is quite helpful, as it breaks down the line into two key components: the slope and the y-intercept.
- The slope \( m \) indicates how steep the line is. It tells us how much \( y \) changes for each unit increase in \( x \). For example, if \( m = 2 \), this means for every 1 unit increase in \( x \) values, \( y \) will increase by 2 units.
- The y-intercept \( b \) represents the point where the line crosses the y-axis (i.e., when \( x = 0 \)). This point gives us a fixed starting point on the graph, making it easy to draw the line.
Domain and Range
The domain and range provide a comprehensive view of the input and output values of a function.
- Domain: In the case of linear functions, the domain is always all real numbers, represented as \((-\infty, \infty)\). This means there are no restrictions on the values \( x \) can take, as the line stretches infinitely in both horizontal directions.
- Range: Similarly, the range of a linear function is all real numbers, \((-\infty, \infty)\), indicating that the function can produce any \( y \)-value as \( x \) varies over its domain.
Graphing Linear Equations
Graphing linear equations involves plotting the line on a Cartesian plane based on the function's equation. Here's a simplified breakdown:
- Start with the y-intercept: Plot the y-intercept on the graph, as it is the starting point where your line will pass through the y-axis.
- Use the slope: From the y-intercept, use the slope to find other points on the line. For example, if the slope is \(-4\), from the y-intercept draw a point 4 units down for each 1 unit moved right on the x-axis.
- Draw the line: Connect these dots to form a straight line, extending it in both directions across the graph. Ensure it's straight and accurately reflects the steepness of the slope.
