Question

Determine if a line with the following description has a positive slope, a negative slope, or an undefined slope. Line rises from left to right

Short answer

The line has a positive slope.

Step-by-step solution

Analyze the given description of the line

The line rises from left to right. This description refers to its graphical representation in the coordinate system.

Understand slope directions

In a Cartesian coordinate system, a line that rises from left to right demonstrates a positive slope. If a line falls from left to right, it indicates a negative slope. Vertical lines have an undefined slope, and horizontal lines have a slope of zero.

Determine the slope of the line

As the given line rises from left to right, it corresponds to a positive slope scenario in the Cartesian coordinate system.

Key concepts

Positive Slope

When discussing the slope of a line in the context of algebra and geometry, a positive slope is a fundamental concept. Imagine walking upwards on a hill; the same concept applies to a line rising from left to right on a graph. In mathematical terms, the slope, commonly denoted as 'm', is calculated with the formula \( m = \frac{rise}{run} \).

If the 'rise', which is the vertical change, and the 'run', the horizontal change between two points on the line, are both positive or both negative, their ratio will be positive. This means that as you move along the line from left to right, the line ascends in the Cartesian coordinate system, much like the incline of an up-sloping hill.

Cartesian Coordinate System

The Cartesian coordinate system is an essential framework for understanding the graphical representation of lines. This system is composed of two perpendicular number lines: the horizontal x-axis and the vertical y-axis, which intersect at the origin (0,0).

Every point in this plane can be described by an ordered pair of numbers (x, y), where 'x' represents the position along the x-axis and 'y' represents the position along the y-axis. It's important for students to familiarize themselves with this coordinate plane as it allows for precise placement of points and aids in visualizing relationships between geometrical figures, including the slope of lines.

Graphical Representation of Lines

The graphical representation of lines in the Cartesian coordinate system makes the abstract concept of slope visually intuitive. To plot a line, one must identify at least two points that satisfy the line's equation and mark them on the graph.

By connecting these points, the inclination or decline of the line reveals its slope. The graphical method is not just for finding the slope; it's also used to interpret the slope's impact on the line's direction. For instance, lines with similar slopes appear parallel on the graph, and the greater the slope, the steeper the line.

Undefined Slope

An undefined slope arises in a unique situation on the Cartesian coordinate system. Specifically, when we have a vertical line, the 'run' or horizontal difference between any two points on the line is zero, as it does not traverse left or right but only moves up or down.

In the slope formula \( m = \frac{rise}{run} \), dividing by zero is not possible in mathematics, leading to an undefined slope. In other words, since the slope can't be calculated for vertical lines, they do not have a slope. Recognizing vertical lines are necessary for understanding that not all lines can be categorized by a numeric slope value.