How to write a rule for a linear function?
To write a rule for a linear function, you will need two pieces of information: the slope of the line and the y-intercept.
1. Determine the slope: The slope (m) represents the rate at which the line increases or decreases. It can be found by calculating the change in y-coordinates (vertical change) divided by the change in x-coordinates (horizontal change), often expressed as rise over run.
For example, if you have two points on the line, (x1, y1) and (x2, y2), the slope (m) can be calculated as:
m = (y2 - y1) / (x2 - x1)
If you are given the slope directly, you can skip this step.
2. Identify the y-intercept: The y-intercept (b) is where the line intersects the y-axis. To find this value, you will need to know at least one point on the line or another equation such as the standard form of a linear equation (Ax + By = C).
If you have a point (x, y) on the line, substituting the values into the equation y = mx + b will allow you to solve for b. Alternatively, if you have the equation in standard form, you can rearrange it to solve for y in terms of x and then identify the constant term.
For example, if you are given the point (3, 5), substituting x = 3 and y = 5 into the equation y = mx + b, you can solve for b:
5 = m(3) + b
b = 5 - 3m
If you are given another form of the equation, you will need to rearrange it to solve for y in terms of x and identify the constant term.
3. Write the rule: Once you have determined the slope (m) and the y-intercept (b), you can write the equation of the linear function in slope-intercept form (y = mx + b).
For example, if the slope is 2 and the y-intercept is -3, the rule for the linear function would be:
y = 2x - 3
This equation can be used to determine the y-coordinate for any given x-coordinate on the line.
Remember, it's important to have either the slope and the y-intercept or a point on the line to write a rule for a linear function.