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How to Find the Slope of a Graphed Line on a Coordinate Grid
Mathematics
Grade 8 (Junior High School)
Question Content
Find the slope of the line graphed below. (The graph shows a line on a coordinate grid with two marked points: (-4, -2) and (2, 2))
Correct Answer
2/3
Detailed Solution Steps
1
Step 1: Identify two clear points on the line from the grid. The marked points are (-4, -2) and (2, 2).
2
Step 2: Use the slope formula: m = (y₂ - y₁)/(x₂ - x₁), where (x₁, y₁) = (-4, -2) and (x₂, y₂) = (2, 2).
3
Step 3: Substitute the values into the formula: m = (2 - (-2))/(2 - (-4)) = (2 + 2)/(2 + 4) = 4/6.
4
Step 4: Simplify the fraction 4/6 to its lowest terms by dividing numerator and denominator by their greatest common divisor (2), resulting in 2/3.
Knowledge Points Involved
1
Slope of a Line
The slope of a line is a measure of its steepness, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. It is represented by the formula m = (y₂ - y₁)/(x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two distinct points on the line. Positive slopes mean the line rises from left to right, negative slopes mean it falls, zero slope means it is horizontal, and an undefined slope means it is vertical.
2
Coordinate Grid Points
A coordinate grid uses ordered pairs (x, y) to locate points, where x is the horizontal value (x-coordinate) and y is the vertical value (y-coordinate). When finding slope from a graph, you must identify two distinct points with clear integer coordinates to ensure an accurate calculation.
3
Simplifying Fractions
Simplifying a fraction involves dividing both the numerator and denominator by their greatest common divisor (GCD) to get an equivalent fraction in the lowest possible terms. This is important for presenting the slope in its standard, simplified form.
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