To find the linear function of a graph, you need to determine its slope and y-intercept and then express it in slope-intercept form (y = mx + b).
Steps to Determine the Linear Function
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Identify the Y-Intercept:
- The y-intercept is the point where the line crosses the y-axis. This is the point where x = 0. Note down the y-value at this point. This value is 'b' in the equation y = mx + b.
- Example: If the line crosses the y-axis at (0, 3), then the y-intercept is 3, and b = 3.
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Calculate the Slope (m):
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The slope represents the rate of change of the line (rise over run). You can determine the slope in a couple of ways:
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Counting Increments (Rise over Run): If the graph has clear, easily readable integer points, you can count the "rise" (vertical change) and the "run" (horizontal change) between two points on the line. The slope (m) is then calculated as rise/run.
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Using Two Points: If counting is difficult or the values are not integers, identify two distinct points on the line, (x₁, y₁) and (x₂, y₂). Use the slope formula:
m = (y₂ - y₁) / (x₂ - x₁) -
Example: Let's say we have points (1, 5) and (2, 7).
- m = (7 - 5) / (2 - 1) = 2 / 1 = 2. Therefore, the slope is 2.
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Write the Equation in Slope-Intercept Form:
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Once you have the slope (m) and the y-intercept (b), plug these values into the slope-intercept form of a linear equation:
y = mx + b -
Example: If m = 2 and b = 3, the linear function is y = 2x + 3.
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Example
Let's say we have a graph where the line:
- Crosses the y-axis at (0, -1). Therefore, b = -1.
- Passes through the point (2, 3).
To find the slope, we can use the y-intercept (0, -1) and the given point (2, 3):
m = (3 - (-1)) / (2 - 0) = 4 / 2 = 2
Therefore, the linear function is y = 2x - 1.
In summary, to find the linear function of a graph, first identify the y-intercept (b), then calculate the slope (m) using either counting increments or the slope formula, and finally, plug 'm' and 'b' into the equation y = mx + b.
