Writing rules for linear functions
1. A linear function has the form y = mx + b, where m is the slope and b is the y-intercept.
2. The slope of a linear function determines the steepness of the line. A positive slope goes up from left to right and a negative slope goes down from left to right.
3. The y-intercept is the point where the line crosses the y-axis. It is the value of y when x = 0.
4. To graph a linear function, plot the y-intercept on the y-axis and use the slope to find other points on the line. Connect the points to form a straight line.
5. To find the slope of a linear function, use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
6. To find the x-intercept of a linear function, set y = 0 and solve for x.
7. To find the y-intercept of a linear function, set x = 0 and solve for y.
8. To find the equation of a linear function given two points, use the formula y - y1 = m(x - x1), where m is the slope and (x1, y1) is one of the points on the line.
9. Linear functions are continuous, meaning that there are no breaks or gaps in the line.
10. The domain of a linear function is all real numbers and the range is all real numbers.
To properly write rules for linear functions, there are a few important components to consider. Here are the step-by-step guidelines:
Step 1: Understand the form of a linear function
A linear function can be represented in the form of y = mx + b, where:
- "y" represents the dependent variable or the output.
- "x" represents the independent variable or the input.
- "m" represents the slope of the line, which determines the steepness or direction.
- "b" represents the y-intercept, which indicates the point where the line intersects the y-axis.
Step 2: Identify the slope (m)
The slope represents the rate of change of the function. To find the slope, you need two points on the line or the coordinates of a single point along with the slope. You can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are any two points on the line.
Step 3: Determine the y-intercept (b)
The y-intercept is the point where the line intersects the y-axis, or when x = 0. So, substitute x = 0 into the equation and solve for y. The resulting value will be the y-intercept.
Step 4: Write the final rule
Once you have determined the slope and the y-intercept, you can now write the rule for the linear function in the form of y = mx + b.
Example:
Let's say the slope (m) is 2 and the y-intercept (b) is 3. The rule for this linear function would be:
y = 2x + 3
Remember, this equation represents the relationship between the input (x) and the output (y) in the linear function.