On The Same Coordinate Plane, Graph A Line With A Slope Of -3 That Passes Through The Point (-6, 7)
Understanding how to graph lines on the coordinate plane is fundamental in algebra and geometry. It enables students and professionals to visualize relationships between variables, analyze functions, and solve real-world problems involving linear equations. Today, we will focus on a specific yet illustrative problem: graphing a line with a slope of -3 that passes through the point (-6, 7). This task involves applying key concepts such as the slope-intercept form, point-slope form, and the coordinate plane's principles.
In this comprehensive guide, we will explore the process step-by-step, provide explanations of essential concepts, and discuss practical tips for accurate graphing. Whether you're a student preparing for an exam, a teacher designing lesson plans, or someone interested in understanding the fundamentals of graphing lines, this article offers valuable insights.
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Understanding the Basics: The Coordinate Plane and Linear Equations
The Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface formed by two perpendicular axes:- The x-axis (horizontal)
- The y-axis (vertical)
Linear Equations and Their Graphs
A linear equation is an algebraic expression that models a straight line when graphed. The general form of a linear equation in two variables is: \[ y = mx + b \] where:- \( m \) is the slope of the line
- \( b \) is the y-intercept, the point where the line crosses the y-axis
Deciphering the Given Data: Slope and Point
The problem specifies:
- The slope (m) as -3
- The point (-6, 7) through which the line passes
This information uniquely determines the line, and our goal is to graph it accurately.
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Formulating the Equation of the Line
To graph the line, we need its equation in a usable form. The most straightforward is the slope-intercept form \( y = mx + b \). Since we are given the slope and a point, we can use the point-slope form to find the equation:
\[ y - y1 = m(x - x1) \]
where:
- \( (x1, y1) \) is the known point (-6, 7)
- \( m \) is the slope (-3)
Step-by-Step Calculation
- Substitute the known values into the point-slope form:
- Simplify the expression:
- Distribute the slope:
- Solve for \( y \):
\[ y = -3x - 11 \]
Thus, the equation of the line in slope-intercept form is:
\[ y = -3x - 11 \]
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Graphing the Line Step-by-Step
1. Identify the Y-Intercept
The y-intercept \( b \) is -11, meaning the line crosses the y-axis at (0, -11).2. Plot the Y-Intercept
Plot the point (0, -11) on the coordinate plane.3. Use the Slope to Find Another Point
The slope \( m = -3 \) indicates that for every 1 unit increase in \( x \), \( y \) decreases by 3 units. Alternatively, for every 1 unit decrease in \( x \), \( y \) increases by 3.- Starting from (0, -11), move:
- 1 unit to the right (\( x = 1 \))
- \( y = -11 + (-3) \times 1 = -11 - 3 = -14 \)
Similarly, moving:
- 1 unit to the left (\( x = -1 \))
- \( y = -11 + 3 \times 1 = -11 + 3 = -8 \)
Plot the point (-1, -8).
4. Draw the Line
Using the two points (0, -11) and (1, -14), draw a straight line through them, extending across the coordinate plane. Make sure the line is straight and extends beyond the plotted points.5. Verify the Line Passes Through the Given Point
Check that the point (-6, 7) lies on the line:- Plug \( x = -6 \) into the line equation:
Since the calculated \( y \) is 7, which matches the point's y-coordinate, the point (-6, 7) indeed lies on the line, confirming our graph's accuracy.
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Additional Tips for Accurate Graphing
- Always plot the y-intercept first to establish a reference point.
- Use the slope to find additional points, ensuring the line is accurate.
- When plotting points, label them clearly.
- Use graphing tools or graph paper for precision.
- Extend the line beyond the plotted points to clearly show its direction.
Applications of Graphing Lines with Negative Slopes
Understanding lines with negative slopes is essential in various real-world contexts:
- Economics: Representing decreasing demand or supply curves.
- Physics: Describing objects with decreasing velocity over time.
- Statistics: Visualizing negative correlations between variables.
- Engineering: Analyzing decline or depreciation trends.
Mastering the process of graphing such lines enhances problem-solving skills and conceptual understanding.
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Practice Problems to Reinforce Learning
- Graph the line with a slope of -2 passing through the point (3, 4).
- Find the equation of the line passing through (-2, 5) with a slope of -4, and graph it.
- Determine whether the point (0, -11) lies on the line \( y = -3x - 11 \). (Answer: Yes, it does.)
- Plot the line with the equation \( y = -3x - 11 \) on graph paper and verify its accuracy.
Conclusion: Mastering Line Graphs with Slope and Point
Graphing a line on the coordinate plane using a given slope and point is a fundamental skill in algebra and geometry. By understanding how to convert between different forms of linear equations, plotting key points, and applying the slope, students can accurately represent lines and analyze their properties.
In this tutorial, we demonstrated how to derive the equation of a line with a slope of -3 passing through (-6, 7), and how to graph it step-by-step. Remember to verify your points and equations to ensure accuracy. Practice regularly with different slopes and points to strengthen your understanding of linear graphs and their applications across various fields.
Whether you're tackling homework, preparing for exams, or applying these concepts in professional scenarios, mastering line graphing techniques is a valuable skill that lays the foundation for advanced math and analytical thinking.
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Keywords: graphing lines, slope -3, point (-6, 7), linear equations, coordinate plane, slope-intercept form, point-slope form, plotting points, algebra, geometry, graphing tips, linear functions, math tutorial
