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Magazine Chapter 2: Problem 63
Graph the linear function and label the \(x\) -intercept. $$f(x)=-x-2$$
Short Answer
Expert verified
The x-intercept is \((-2, 0)\).
Step by step solution
01
Identify the Linear Function
The given linear function is \( f(x) = -x - 2 \). This is in the slope-intercept form \( y = mx + b \), where \( m = -1 \) is the slope and \( b = -2 \) is the y-intercept.
02
Find the Y-Intercept
From the equation \( f(x) = -x - 2 \), the y-intercept \( b \) is \(-2\). This means the graph will cross the y-axis at the point \((0, -2)\).
03
Find the X-Intercept
To find the x-intercept, we set \( f(x) = 0 \) and solve for \( x \). Thus, \( 0 = -x - 2 \) which simplifies to \( x = -2 \). This means the x-intercept is \((-2, 0)\).
04
Plot the Intercepts
Plot the y-intercept at \((0, -2)\) and the x-intercept at \((-2, 0)\) on the graph. These points are critical for drawing the line.
05
Draw the Line
Using the intercepts, draw a line through these two points. The slope tells us that the line moves down 1 unit vertically for every 1 unit it moves horizontally to the right.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the X-Intercept
The x-intercept of a function is the point where the graph of the function crosses the x-axis. At this point, the value of the function is zero because it lies on the x-axis. To find the x-intercept, you substitute zero for the function value and solve the equation for the independent variable \( x \).
For example, in the function \( f(x) = -x - 2 \), to find the x-intercept, we set \( f(x) = 0 \):
For example, in the function \( f(x) = -x - 2 \), to find the x-intercept, we set \( f(x) = 0 \):
- Start with the equation \( 0 = -x - 2 \).
- Add 2 to both sides to get \( 2 = -x \).
- Multiply both sides by -1 to solve for \( x \), resulting in \( x = -2 \).
Understanding the Y-Intercept
The y-intercept of the function is the point where the line intersects the y-axis. It signifies the value of the output when the input \( x \) is zero. When a linear function is given in the slope-intercept form \( y = mx + b \), the constant \( b \) represents the y-intercept.
In our example, the function \( f(x) = -x - 2 \), helps to identify the y-intercept by looking at \( b \). In this case, \( b = -2 \).
In our example, the function \( f(x) = -x - 2 \), helps to identify the y-intercept by looking at \( b \). In this case, \( b = -2 \).
- This tells us that the line crosses the y-axis at the point \((0, -2)\).
The Slope-Intercept Form
Understanding the slope-intercept form is central to graphing linear functions. The equation \( y = mx + b \) is the general representation of a linear equation in slope-intercept form where:
The slope-intercept form is advantageous because it instantly reveals both the slope and y-intercept, making it easier to graph a linear function by hand.
- \( m \) represents the slope of the line, which describes the line's steepness and direction.
- \( b \) is the y-intercept, marking the point where the line crosses the y-axis.
The slope-intercept form is advantageous because it instantly reveals both the slope and y-intercept, making it easier to graph a linear function by hand.
Plotting Points and Drawing the Line
To graph a linear function successfully, it's crucial to plot key points carefully and connect them to form a straight line. Start by plotting the intercepts as they are easily obtained from the equation. For our function \( f(x) = -x - 2 \):
Always check: The slope determines the angle and direction of the line. In our roll, the slope \( m = -1 \) ensures that for every step along the x-axis, the line drops one step down vertically, confirming the line's path is correct.
- Plot the y-intercept \((0, -2)\) on the graph.
- Plot the x-intercept \((-2, 0)\).
Always check: The slope determines the angle and direction of the line. In our roll, the slope \( m = -1 \) ensures that for every step along the x-axis, the line drops one step down vertically, confirming the line's path is correct.
