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Chapter 2: Problem 36

Graph the line that passes through the given point and has the given slope. (2,3) and \(m=0\)

Short Answer

Expert verified
The equation of the line with a slope of 0 passing through the point (2,3) is \(y = 3\). The line graphed is a horizontal line passing through all points where the y-coordinate equals 3.

Step by step solution

01

Understand the Slope and Intercept

Since the slope 'm' is zero, it means the line is horizontal and passing through y-axis (y-intercept) at the point given. So, the equation of the line can be written as \(y = mx + c\). Because slope \(m = 0\), the equation is now \(y = c\). Where is \(c\) the y-intercept.
02

Find the Line Equation

The line passes through point (2,3), substitute \(y = 3\) in the equation \(y = c\) and it obtains \(c = 3\). Therefore, the equation of the line is \(y = 3\).
03

Graph the Line

Plot the line on the graph using the equation \(y = 3\). This line will be a horizontal straight line that passes through the point (0, 3), and it crosses the y-axis at y = 3. All points on this line satisfy \(y = 3\). So, the line also passes through the point (2,3) and any other point where y equals 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Slope
The term "slope" in mathematics refers to the steepness or incline of a line on a graph. It is often represented by the letter \(m\) in equations. Slope is calculated as the change in the vertical direction (rise) over the change in the horizontal direction (run). This relationship is often expressed with the formula:\[m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\]
  • A positive slope means the line ascends as it moves left to right.
  • A negative slope indicates the line descends as it moves left to right.
  • A zero slope results in a horizontal line which doesn鈥檛 rise or fall.
For a line with a zero slope, such as in this exercise, the line remains perfectly horizontal, illustrating that for any two points on the line, the \(y\)-values remain constant. This means there are no vertical changes as we move along the line.
Equation of a Line
The equation of a line provides a precise description of any straight line on a graph. The most common form you will encounter is the slope-intercept form written as:\[y = mx + c\]Here, \(m\) is the slope of the line, and \(c\) is the y-intercept, which is the point at which the line crosses the y-axis.
In cases where the slope \(m\) is zero, the equation simplifies to:\[y = c\]This simplification occurs because there is no change in the \(y\)-direction as \(x\) moves. In our exercise, the equation \(y = 3\) tells us that the line crosses the y-axis at the point \((0, 3)\). Further, it reveals that every point along the line maintains this \(y\)-value.
Exploring Horizontal Lines
Horizontal lines are unique in that their slope is always zero. This means no matter how far they stretch across the graph, they will never rise or descend. Consequently, a horizontal line presents an equation in the form \(y = c\), where \(c\) is a constant value.
In our specific case, the equation \(y = 3\) indicates a horizontal line where every point on the line will have a \(y\)-coordinate of 3.
  • This line will extend infinitely left and right, passing through all points where \(y = 3\), such as \((1, 3)\), \((2, 3)\), and so on.
  • Graphically, you will see this line as a straight, unmoving bar across your graph at \(y = 3\).
  • No matter which point you select on this line, the \(x\) value can change, but the \(y\) value always stays at 3, emphasizing the key feature of a horizontal line.

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Most popular questions from this chapter

Graph the line using the parametric equations $$x=1+2 t, y=3+t$$

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