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Magazine Chapter 1: Problem 8
Graph each linear function. Give the (a) \(x\) -intercept, (b) \(y\) -intercept. (c) domain, (d) range, and (e) slope of the line. $$f(x)=-0.5 x$$
Short Answer
Expert verified
The x-intercept and y-intercept are both (0, 0). The slope is -0.5. Domain and range are both all real numbers.
Step by step solution
01
Identify the Function Format
The given function is \( f(x) = -0.5x \), which is in the format \( y = mx + b \). Here, \( m = -0.5 \) is the slope, and since there is no constant term, \( b = 0 \). This means the line passes through the origin (0,0).
02
Find the x-intercept
The x-intercept occurs where \( f(x) = 0 \). Substitute 0 for \( f(x) \) in the equation: \[ 0 = -0.5x \]. Solving for \( x \) gives \( x = 0 \). So, the x-intercept is at (0, 0).
03
Find the y-intercept
The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) in the equation: \[ f(x) = -0.5(0) = 0 \]. So, the y-intercept is at (0, 0).
04
Determine the Domain
The domain of any linear function is always all real numbers \( (-fty, fty) \), since a linear function is defined throughout the entire set of real numbers.
05
Determine the Range
Similar to the domain, for a linear function the range is all real numbers as well \( (-fty, fty) \), because as x-values move from negative infinity to positive infinity, the y-values do as well.
06
Confirm the Slope
The slope \( m \) of the line is given as -0.5. This indicates that for every unit increase in x, the value of f(x) decreases by 0.5. The slope confirms the line is downward sloping.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope
In linear functions, the slope is a crucial concept that determines the steepness of the line graph. The slope is often denoted by the letter \( m \), and it's calculated in linear equations of the form \( y = mx + b \). In the function \( f(x) = -0.5x \), the slope is \( -0.5 \). This negative value tells us something about the direction in which the line moves: it slopes downwards from left to right.
Understanding the slope:
Understanding the slope:
- A positive slope means the line rises as it moves from left to right.
- A negative slope, like our example, means the line falls as it moves from left to right.
- The larger the absolute value of the slope, the steeper the line.
- If the slope is zero, the line is horizontal.
Intercepts
In the context of a linear function, intercepts are the points where the function crosses the axes. For the function \( f(x) = -0.5x \), the intercepts are remarkably straightforward because they both occur at the origin (0,0).
Types of intercepts:
Types of intercepts:
- \(x\)-intercept: This is the point where the graph crosses the \(x\)-axis. It happens where \(y = 0\). For our linear function, setting \(f(x) = 0\) gives \(0 = -0.5x\), leading to \(x = 0\). Thus, the \(x\)-intercept is at (0, 0).
- \(y\)-intercept: This is the point where the line crosses the \(y\)-axis. It occurs where \(x = 0\). For our function, plugging \(x = 0\) into the equation gives \(f(x) = 0\), so the \(y\)-intercept is also at (0, 0).
Domain and Range
Every function has a domain and range, which represents all possible \(x\)-values and \(y\)-values, respectively. Linear functions, unless otherwise restricted, have a domain and range that extend infinitely.
For the function \( f(x) = -0.5x \):
For the function \( f(x) = -0.5x \):
- Domain: Since there are no restrictions like square roots or denominators, the domain is all real numbers, \((-\infty, \infty)\).
- Range: Similarly, because the function is linear and unobstructed, the range is also all real numbers, \((-\infty, \infty)\).
Graphing
Graphing a linear function is straightforward once you understand its basic components like slope and intercepts. Starting with the function \( f(x) = -0.5x \), we know the line will pass through the origin with a slope of \(-0.5\).
Steps to graph:
Steps to graph:
- First, plot the intercept. Here, both the \(x\)-intercept and \(y\)-intercept are at (0, 0).
- From the intercept, use the slope to determine the next point. The slope of \(-0.5\) means that for each 1 unit increase in \(x\), \(f(x)\) decreases by 0.5. So a move 1 unit to the right on the \(x\)-axis should lead 0.5 units down on the \(y\)-axis.
- Draw a line through the points. Extend the line in both directions to illustrate its endless nature given the domain and range of \((-\infty, \infty)\).
