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Magazine Chapter 3: Problem 54
Write an equation of the line with the given slope and \(y\) -intercept and graph it. See Example 2. Slope \(-3, y\) -intercept \(\left(0,-\frac{1}{2}\right)\)
Short Answer
Expert verified
The equation of the line is \(y = -3x - \frac{1}{2}\).
Step by step solution
01
Understand the equation of a line
The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this problem, we need to write the equation of the line using the given slope and y-intercept.
02
Substitute the slope and y-intercept into the equation
Here, the slope \( m \) is given as \(-3\) and the y-intercept \( b \) is given as \(-\frac{1}{2}\). Substitute these values into the equation: \( y = -3x - \frac{1}{2} \).
03
Plot the y-intercept on the graph
To graph this equation, first plot the y-intercept on the y-axis. The y-intercept \( b = -\frac{1}{2} \) corresponds to the point \((0, -\frac{1}{2})\).
04
Use the slope to find another point on the line
The slope \(-3\) indicates a rise over run of \(-3/1\). From the y-intercept point \((0, -\frac{1}{2})\), move down 3 units and 1 unit to the right to find the next point \((1, -\frac{7}{2})\).
05
Draw the line
Plot the second point \((1, -\frac{7}{2})\) on the graph. Draw a straight line through the points \((0, -\frac{1}{2})\) and \((1, -\frac{7}{2})\) to represent the equation of the line.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope
The concept of "slope" is a fundamental idea in understanding lines on a graph. Slope essentially measures the steepness or inclination of a line. It tells us how much the line goes up or down as you move from left to right. The slope is represented by the letter \( m \) in the equation of a line written in the slope-intercept form: \( y = mx + b \). In this formula, \( m \) is the slope, and it plays a critical role in determining the direction and steepness of the line.
- If the slope \( m \) is positive, the line rises as it moves from left to right.
- If the slope is negative, like in our exercise where it is \(-3\), the line falls as it moves from left to right.
- If the slope is zero, the line is horizontal, indicating no vertical change as you move across the line.
- A larger absolute value of the slope signifies a steeper line, while a smaller absolute value represents a gentler incline.
Y-Intercept
The "y-intercept" is another essential component when discussing the equation of a line. The y-intercept is where the line crosses the y-axis. This point is crucial because it indicates the starting point of the line on the graph when the value of \( x \) is zero. In the equation \( y = mx + b \), the y-intercept is represented by the letter \( b \). For our specific problem, the y-intercept is given as \(-\frac{1}{2}\). This means that when \( x = 0 \), \( y \) will be \(-\frac{1}{2}\).
In a graphical representation:
In a graphical representation:
- The y-intercept is the point \((0, b)\). For example, in our exercise, it is \((0, -\frac{1}{2})\).
- It provides a crucial reference point that helps in plotting the line on a graph, serving as a starting point for drawing.
- The y-intercept can be positive, negative, or zero, each affecting the line's position on the graph.
Graphing Lines
Graphing lines is a visual way to understand linear equations. It involves translating the mathematical description (using slope and y-intercept) into a picture on a coordinate plane.To graph a line like the one you've calculated \( y = -3x - \frac{1}{2} \), follow these steps:
- Start with the y-intercept: Begin by plotting the y-intercept, which for our example is \((0, -\frac{1}{2})\). This point is placed directly on the y-axis.
- Use the slope: From the y-intercept, apply the slope to determine the next point. A slope of \(-3\) means moving 3 units down and 1 unit to the right from the y-intercept to find the next point \((1, -\frac{7}{2})\).
- Draw the line: With at least two points plotted, draw a smooth line through these points to extend across the graph. This visual representation confirms the equation \( y = -3x - \frac{1}{2} \).
