Answer:
A rule for a linear function can be expressed in the form:
f(x) = mx + b
where m is the slope of the line and b is the y-intercept. The slope is the rate at which the line changes vertically for every unit change in x, and the y-intercept is the point where the line crosses the y-axis.
Explanation:
Certainly, I can help you with that. Here's a step-by-step guide to writing a rule for a linear function:
1. Identify the variables: In a linear function, there are two variables: the independent variable (usually denoted as x) and the dependent variable (usually denoted as y).
2. Identify the slope: The slope is the rate at which the dependent variable changes with respect to the independent variable. To find the slope, you need to identify two points on the line. You can then use the slope formula, which is:
slope = (change in y) / (change in x)
3. Plug in the coordinates of one of the points: Choose one of the points you identified in step 2 and plug in its x and y coordinates into the point-slope form of the equation:
y - y1 = m(x - x1)
Here, m is the slope and (x1, y1) is the coordinate of the point you chose. Plug in the values and simplify.
4. Convert to slope-intercept form: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept (the point at which the line intersects the y-axis). To convert the equation from point-slope form to slope-intercept form, simply solve for y by isolating it on one side of the equation.
y - y1 = m(x - x1)
y - y1 = mx - mx1
y = mx + (y1 - mx1)
Here, (y1 - mx1) represents the y-intercept.
That's it! By following these steps, you can write a rule for any linear function.