Finding the slope of a line from its graph is a fundamental concept in algebra. This worksheet will guide you through various methods and examples, helping you master this crucial skill. We'll cover different scenarios, including positive, negative, zero, and undefined slopes. Let's dive in!
Understanding Slope
Before we start analyzing graphs, let's review what slope represents. Slope (often denoted by m) measures the steepness and direction of a line. It's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. The formula is:
m = (y₂ - y₁) / (x₂ - x₁)
Where (x₁, y₁) and (x₂, y₂) are coordinates of two distinct points on the line.
Types of Slopes
Understanding the different types of slopes is crucial for accurate interpretation:
- Positive Slope: The line rises from left to right. The slope is a positive number.
- Negative Slope: The line falls from left to right. The slope is a negative number.
- Zero Slope: The line is horizontal. The slope is zero (0).
- Undefined Slope: The line is vertical. The slope is undefined.
Finding the Slope: Step-by-Step Guide
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Identify Two Points: Choose any two distinct points on the line clearly marked on the graph. The clearer the points, the easier the calculation will be.
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Determine the Coordinates: Write down the x and y coordinates of each point. Remember, the x-coordinate comes first (x, y).
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Apply the Slope Formula: Substitute the coordinates into the slope formula: m = (y₂ - y₁) / (x₂ - x₁).
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Simplify: Perform the subtraction and division to find the numerical value of the slope.
Practice Problems
(Note: Since I can't display graphs directly, I will provide coordinate pairs for you to practice with. Imagine these points plotted on a graph and draw the line connecting them.)
Problem 1: Find the slope of the line passing through points (2, 4) and (6, 8).
Problem 2: Find the slope of the line passing through points (-3, 5) and (1, -1).
Problem 3: Find the slope of the line passing through points (0, 2) and (5, 2).
Problem 4: Find the slope of the line passing through points (4, 1) and (4, -3).
Problem 5: A line passes through the points (1,3) and (4,9). What is its slope?
Solutions and Explanations
Solution 1: m = (8 - 4) / (6 - 2) = 4 / 4 = 1 (Positive slope)
Solution 2: m = (-1 - 5) / (1 - (-3)) = -6 / 4 = -3/2 (Negative slope)
Solution 3: m = (2 - 2) / (5 - 0) = 0 / 5 = 0 (Zero slope – horizontal line)
Solution 4: m = (-3 - 1) / (4 - 4) = -4 / 0 (Undefined slope – vertical line)
Solution 5: m = (9-3) / (4-1) = 6/3 = 2 (Positive slope)
Frequently Asked Questions (FAQ)
What if I choose different points on the line?
You should get the same slope regardless of which two points you choose, as long as they lie on the line. This is a key property of linear functions.
What does a slope of 1 mean?
A slope of 1 means that for every 1 unit increase in the x-value, the y-value increases by 1 unit. The line rises at a 45-degree angle.
How can I check my work?
You can visually inspect your answer. Does the slope match the direction and steepness of the line on the graph? A positive slope should correspond to a line rising from left to right, and a negative slope to a line falling from left to right.
This worksheet provides a solid foundation for understanding and calculating slope from a graph. Remember to practice consistently to solidify your skills!
