which equation describes the line graphed above

Guri Bee

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22-10-2024

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Unlocking the Equation of a Line: A Step-by-Step Guide

Have you ever been presented with a graph and wondered: "What's the equation of this line?" It's a common question in mathematics, especially when dealing with linear relationships. Let's dive into a method for finding the equation of a line using information directly from its graph.

The Essential Components

To write the equation of a line, we need two key pieces of information:

1. Slope: The slope tells us how steep the line is and its direction (upward or downward).
2. Y-intercept: This is the point where the line crosses the y-axis.

Finding the Slope

Question: How can I calculate the slope from a graph? Answer: (From GitHub user "MathIsFun") The slope of a line is the ratio of the "rise" to the "run" between any two points on the line.

Let's break this down.

1. Choose two points: Select any two points on the line.
2. Calculate the rise: This is the vertical change between the two points. If the line goes upward, the rise is positive. If it goes downward, the rise is negative.
3. Calculate the run: This is the horizontal change between the two points. The run is positive if the line moves to the right and negative if it moves to the left.
4. Slope = Rise / Run: Divide the rise by the run to get the slope.

Example:

Imagine a line passes through points (1, 2) and (3, 6).

• Rise: 6 - 2 = 4 (positive because the line goes upward)
• Run: 3 - 1 = 2 (positive because the line moves to the right)
• Slope: 4 / 2 = 2

Finding the Y-intercept

Question: How do I find the y-intercept from a graph? Answer: (From GitHub user "MathIsFun") The y-intercept is the point where the line crosses the y-axis.

This means we need to identify the point on the graph where the line intersects the vertical y-axis. The y-coordinate of that point is the y-intercept.

Putting it Together: The Equation

The equation of a line can be written in the "slope-intercept form":

• y = mx + b

Where:

• y: Represents the dependent variable (usually on the vertical axis)
• x: Represents the independent variable (usually on the horizontal axis)
• m: Represents the slope
• b: Represents the y-intercept

Example:

Using our previous example where the slope is 2 and the y-intercept is (0, 1), the equation of the line is:

• y = 2x + 1

Practical Applications

Understanding how to find the equation of a line has many practical applications in real life:

• Predicting Trends: In finance, analyzing stock market trends often involves identifying patterns in linear graphs.
• Modeling Growth: In biology, the growth of a population can be modeled using linear equations.
• Calculating Costs: In business, linear equations can help calculate total costs based on a fixed cost and a variable cost per unit.

Key Takeaways

• Finding the equation of a line requires determining its slope and y-intercept.
• The slope is calculated as the rise over the run between any two points on the line.
• The y-intercept is the point where the line crosses the y-axis.
• The slope-intercept form of a linear equation is y = mx + b.

By following these steps, you'll be able to confidently find the equation of any line graphed before you.