write an equation in slope-intercept form for the graph shown

Guri Bee

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21-10-2024

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Unlocking the Secrets of Slope-Intercept Form: A Step-by-Step Guide

Have you ever stared at a graph and wondered, "What's the equation of this line?" Fear not, because the magical world of slope-intercept form can help you unravel the mystery!

This article will explore the power of slope-intercept form (y = mx + b) and how to write the equation for any line, just by looking at its graph. We'll be using examples from real-world applications to solidify your understanding.

What is Slope-Intercept Form?

The slope-intercept form of a linear equation is a powerful tool that allows us to represent the relationship between two variables in a clear and concise way. It provides us with all the information needed to understand the line's behavior.

• y: Represents the dependent variable – its value depends on the independent variable 'x'.
• m: Represents the slope – it tells us how steep the line is. A positive slope indicates an upward trend, while a negative slope signifies a downward trend.
• x: Represents the independent variable – its value can be chosen freely.
• b: Represents the y-intercept – it tells us where the line crosses the y-axis.

Step-by-Step Guide to Finding the Equation

Let's break down the process of writing the equation of a line in slope-intercept form using a real-world example:

Scenario: Imagine you're tracking the growth of a plant over time. You have a graph where the x-axis represents the number of days and the y-axis represents the plant's height in centimeters.

Step 1: Identify the y-intercept (b)

• Look for where the line crosses the y-axis. This point represents the initial height of the plant when the experiment started.
• Let's say the line crosses the y-axis at (0, 2). This means the y-intercept (b) is 2.

Step 2: Determine the slope (m)

• Choose two distinct points on the line. These points could be (0, 2) and (5, 8).
• Calculate the slope using the formula: m = (y2 - y1) / (x2 - x1) m = (8 - 2) / (5 - 0) = 6/5
• The slope (m) is 6/5. This means the plant grows 6 centimeters for every 5 days.

Step 3: Write the Equation

• Plug the values for 'm' and 'b' into the slope-intercept form: y = mx + b y = (6/5)x + 2

Therefore, the equation of the line representing the plant's growth is y = (6/5)x + 2. This equation allows us to predict the plant's height at any given day.

Example from GitHub

This example from GitHub shows a line passing through points (1, 2) and (3, 4): Link to GitHub issue

Solution:

1. Slope: m = (4 - 2) / (3 - 1) = 2/2 = 1
2. Y-intercept: Using the point (1, 2) and the slope of 1, we can find the y-intercept by subtracting the slope multiplied by the x-coordinate from the y-coordinate: 2 - (1 * 1) = 1. Therefore, b = 1.
3. Equation: y = mx + b becomes y = 1x + 1, or simply y = x + 1.

Key Points to Remember

• Understanding the meaning of slope and y-intercept is crucial for interpreting the equation.
• Slope-intercept form offers a powerful way to represent linear relationships and predict future values.
• Practice makes perfect! The more you work with slope-intercept form, the easier it will become.

Further exploration:

• Graphing linear equations: Use online tools like Desmos or GeoGebra to visualize the lines you create.
• Real-world applications: Explore how slope-intercept form is used in fields like finance, physics, and engineering.

By mastering the art of slope-intercept form, you'll unlock a world of possibilities for understanding and expressing linear relationships.