Glossary

11.4 Understand Slope of a Line

2 min read•june 25, 2024

is the backbone of linear equations, showing how steep and which way a line goes. It's calculated using the ###-Over-Run_0### formula, comparing changes in y and x coordinates between two points on the line.

Horizontal lines have , while vertical lines have . Knowing how to graph using a point and slope is key. share the same slope, while have slopes that are negative reciprocals.

Understanding Slope

Calculation of slope

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  • Slope measures the and direction of a line
    • Represented as mm in the equation y=mx+by = mx + b (a )
  • Rise-over- formula calculates slope: m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
    • "Rise" is the vertical change found by subtracting the yy-coordinates of two points on the line
    • "Run" is the horizontal change found by subtracting the xx-coordinates of the same two points
  • Counting grid units on a graph to find slope
    • Count the number of units the line rises vertically between two points (rise)
    • Count the number of units the line runs horizontally between the same two points (run)
    • Slope is the ratio of rise to run expressed as a fraction: riserun\frac{\text{rise}}{\text{run}}
  • indicates the line slants upward from left to right (increasing function)
  • indicates the line slants downward from left to right (decreasing function)

Slope of horizontal and vertical lines

  • Horizontal lines have a slope of zero
    • yy-coordinate remains constant for all points on the line (no vertical change)
    • Equation in the form y=by = b, where bb is the yy-intercept (y-value where line crosses y-axis)
  • Vertical lines have an undefined slope
    • xx-coordinate remains constant for all points on the line (no horizontal change)
    • Equation in the form x=ax = a, where aa is the xx-intercept (x-value where line crosses x-axis)

Graphing with point and slope

  • Start at the given point on the
  • Use the slope to find additional points on the line
    • If slope is a fraction (ab\frac{a}{b}), rise by the numerator (aa) and run by the denominator (bb)
    • If slope is an integer (nn), rise by the integer (nn) and run by 1
  • Plot the additional points and connect them with a straight line using a ruler
  • Extend the line in both directions to cover the entire graph (line continues infinitely)

Relationships between lines

  • Parallel lines have the same slope
  • Perpendicular lines have slopes that are negative reciprocals of each other
  • is a special case where the line passes through the origin, and y is directly proportional to x

Key Terms to Review (22)

Coordinate Plane: The coordinate plane is a two-dimensional grid used to represent and analyze the position and relationships of points, lines, and other geometric shapes. It consists of a horizontal x-axis and a vertical y-axis that intersect at a central point known as the origin.
Direct Variation: Direct variation is a mathematical relationship between two variables where one variable is directly proportional to the other. This means that as one variable increases, the other variable increases proportionally, and vice versa.
Horizontal Line: A horizontal line is a straight line that runs parallel to the x-axis of a coordinate plane, maintaining a constant y-coordinate throughout its length. It is a fundamental concept in the study of linear equations and their graphical representations.
Linear Equation: A linear equation is a mathematical equation in which the variables are raised only to the first power and are connected by addition, subtraction, or equality. These equations represent a straight line when graphed on a coordinate plane.
Negative Slope: Negative slope refers to the downward inclination of a line on a coordinate plane, indicating an inverse relationship between the variables represented on the x and y-axes. It describes a line that decreases from left to right.
Ordered Pair: An ordered pair is a set of two numbers that represent a specific point on a coordinate plane. The two numbers in an ordered pair are called the x-coordinate and the y-coordinate, which together define the unique location of that point.
Parallel Lines: Parallel lines are two or more lines that are equidistant from each other and never intersect, maintaining the same orientation and direction throughout their length. This concept is fundamental in understanding the properties of angles, triangles, and the Pythagorean Theorem, as well as the slope of a line.
Perpendicular Lines: Perpendicular lines are a pair of lines that intersect at right angles, forming a 90-degree angle. This geometric relationship is an important concept in understanding the slope of a line.
Positive Slope: Positive slope refers to the incline or upward direction of a line on a coordinate plane. It indicates that as the independent variable increases, the dependent variable also increases, creating a line that rises from left to right.
Rate of Change: The rate of change is a measure of how a quantity changes over time or with respect to another variable. It describes the speed at which a value is changing and is a fundamental concept in the study of functions and their properties.
Rise: The term 'rise' refers to the vertical change or increase in the y-coordinate of a point on a line. It is a key component in understanding the slope of a line, which describes the steepness and direction of the line's incline or decline.
Rise-Over-Run: Rise-over-run is a way to express the slope of a line, which is a measure of the steepness or incline of the line. It represents the change in the vertical (y) direction compared to the change in the horizontal (x) direction as you move along the line.
Run: In the context of understanding the slope of a line, the term 'run' refers to the horizontal distance or change in the x-coordinate between two points on a line. It is one of the key components used to calculate the slope of a line, along with the 'rise' or change in the y-coordinate between the same two points.
Slope: Slope is a measure of the steepness or incline of a line. It represents the rate of change between two points on a line, indicating how much the y-value changes for a given change in the x-value.
Slope of a Line: The slope of a line is a measure of the steepness or incline of a line. It represents the rate of change in the vertical direction (y-coordinate) compared to the change in the horizontal direction (x-coordinate) as you move along the line.
Slope-Intercept Form: Slope-intercept form is a way to represent a linear equation in the format $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept. This form allows for easy graphing and interpretation of the relationship between the variables $x$ and $y$.
Steepness: Steepness is a measure of the incline or gradient of a line or surface. It describes how rapidly a line or curve changes direction, indicating the rate of change or slope between two points.
Undefined Slope: Undefined slope refers to a line that has no finite slope value, meaning the line is vertical and does not have a constant rate of change between any two points on the line. This concept is crucial in understanding the slope of a line, a fundamental topic in the study of linear functions.
Vertical Line: A vertical line is a straight line that runs perpendicular to the x-axis, passing through the same value on the x-axis at all points along the line. It has a constant x-coordinate and is used to represent equations where the independent variable, x, does not change.
X-Intercept: The x-intercept of a linear equation is the point where the graph of the equation crosses the x-axis. It represents the value of x when the value of y is zero, indicating where the line intersects the horizontal axis.
Y-intercept: The y-intercept is the point where a line or graph intersects the y-axis, representing the value of the function when the independent variable (x) is equal to zero. It is a crucial concept in understanding the behavior and properties of linear equations and their graphical representations.
Zero Slope: The zero slope of a line refers to a line that is perfectly horizontal, meaning it has no incline or decline. A line with zero slope is parallel to the x-axis, indicating that the change in the y-coordinate is constant regardless of the change in the x-coordinate.
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