Problem 32
Question
Find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible. $$(2.3,-8.5), \quad m=0$$
Step-by-Step Solution
Verified Answer
The equation of the line that passes through the point (2.3, -8.5) and has a slope of 0 is \(y = -8.5\).
1Step 1: Substitute the slope into the slope-intercept form
In the slope-intercept form (y = mx + b), m represents the slope. As we know that the given slope \(m\) is 0, we can substitute this value into the equation to get \(y = 0x + b\). This simplifies to \(y = b\).
2Step 2: Substitute the given point to find the y-intercept
The coordinate (2.3, -8.5) tells us that when x = 2.3, y = -8.5. To find the y-intercept (b), we substitute y and x into the equation from step 1: -8.5 = b. Therefore, b = -8.5.
3Step 3: Write the final equation of the line
The final equation of the line, in slope-intercept form, can now be written by substituting the value for b into the equation from step 1: y = -8.5.
4Step 4: Sketching the line
As the line has a slope of 0, it's a horizontal line and cuts the y-axis at point -8.5. This is used to sketch the line.
5Step 5: Verifying with a graphing utility (optional)
If possible, plotting y = -8.5 on a graphing utility should result in a horizontal line passing through y = -8.5, verifying the hand-drawn sketch.
Key Concepts
Slope-Intercept FormHorizontal LineGraphing Utility
Slope-Intercept Form
Understanding the slope-intercept form can vastly simplify the process of graphing linear equations. It is expressed as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis.
In practical terms, \( m \) tells us how much the line rises or falls for each step you move to the right along the x-axis. A positive slope means the line ascends, and a negative slope means the line descends. If the slope is zero, the line is horizontal, indicating no change in the y-value regardless of x.
For example, if we know that a line passes through the point (2.3, -8.5) with a slope \( m = 0 \), we can begin to construct its equation. First, we acknowledge that the slope multiplier for x is zero, so any x-value, including 2.3, will not affect the outcome. Thus, the line represented will be horizontal.
In practical terms, \( m \) tells us how much the line rises or falls for each step you move to the right along the x-axis. A positive slope means the line ascends, and a negative slope means the line descends. If the slope is zero, the line is horizontal, indicating no change in the y-value regardless of x.
For example, if we know that a line passes through the point (2.3, -8.5) with a slope \( m = 0 \), we can begin to construct its equation. First, we acknowledge that the slope multiplier for x is zero, so any x-value, including 2.3, will not affect the outcome. Thus, the line represented will be horizontal.
Horizontal Line
A horizontal line in coordinate geometry is a straight line that moves left to right, or right to left, and has an equal y-coordinate at all points. The defining characteristic of a horizontal line is that its slope \( m \) is 0, meaning it has no inclination.
Imagine walking on a perfectly flat ground; no matter how far you walk, your altitude doesn't change. This is essentially what a horizontal line represents on a graph. For any given x-value, the y-value will remain constant. In the equation of a horizontal line, it's always in the form of \( y = k \), where \( k \) is the constant y-value across the entire line.
For the given problem, as soon as we discern that the slope is 0, it becomes apparent that the line will be horizontal. Substituting the given point into the slope-intercept equation, we determine that the constant y-value is -8.5. Hence, \( y = -8.5 \) is the equation of a horizontal line that goes through the point (2.3, -8.5).
Imagine walking on a perfectly flat ground; no matter how far you walk, your altitude doesn't change. This is essentially what a horizontal line represents on a graph. For any given x-value, the y-value will remain constant. In the equation of a horizontal line, it's always in the form of \( y = k \), where \( k \) is the constant y-value across the entire line.
For the given problem, as soon as we discern that the slope is 0, it becomes apparent that the line will be horizontal. Substituting the given point into the slope-intercept equation, we determine that the constant y-value is -8.5. Hence, \( y = -8.5 \) is the equation of a horizontal line that goes through the point (2.3, -8.5).
Graphing Utility
Leveraging a graphing utility can be a tremendous aid for students to understand and verify the equations they encounter. Graphing utilities are tools — either online platforms or graphing calculators — that allow us to plot equations and visualize their corresponding lines or curves on the Cartesian plane.
To use a graphing utility, you would typically input your linear equation, such as \( y = -8.5 \), and the tool graphically renders the line for you. In this case, it should show a perfectly horizontal line at y-coordinate -8.5, extending infinitely in both the positive and negative x-directions.
To use a graphing utility, you would typically input your linear equation, such as \( y = -8.5 \), and the tool graphically renders the line for you. In this case, it should show a perfectly horizontal line at y-coordinate -8.5, extending infinitely in both the positive and negative x-directions.
Benefits of Using a Graphing Utility:
- Immediate visualization of the graph which helps in better understanding the concepts.
- Verification of manually drawn sketches to ensure accuracy.
- Ability to explore the effects of changing parameters, like the slope or y-intercept, on the shape and position of the line.
Other exercises in this chapter
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