A linear function is a mathematical expression that describes a straight line on a graph. It can be expressed in the standard form ( y = mx + b ), where ( m ) is the slope of the line, ( b ) is the y-intercept, and ( x ) and ( y ) are the variables.
In this article, we will explore how to write a linear function given certain values.
Step 1: Understand the Given Values
Before we can write a linear function, we need to identify the values provided. These typically include:
- Two points on the line, which can be represented as ( (x_1, y_1) ) and ( (x_2, y_2) ).
- Alternatively, one point and the slope can be given.
Example Values
Let's say we have the following two points:
- Point A: ( (1, 2) )
- Point B: ( (3, 4) )
Step 2: Calculate the Slope
The slope ( m ) of a linear function can be calculated using the formula:
[ m = \frac{y_2 - y_1}{x_2 - x_1} ]
Substituting the values from our example points:
[ m = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1 ]
Step 3: Use the Point-Slope Form
Now that we have the slope, we can use the point-slope form of a linear equation, which is:
[ y - y_1 = m(x - x_1) ]
Using point A ( (1, 2) ) and the slope ( m = 1 ):
[ y - 2 = 1(x - 1) ]
Step 4: Simplify the Equation
Now we can simplify the equation to get it into the slope-intercept form ( y = mx + b ):
[ y - 2 = x - 1 ]
Adding 2 to both sides gives us:
[ y = x + 1 ]
Conclusion
The linear function that passes through the points ( (1, 2) ) and ( (3, 4) ) is:
[ y = x + 1 ]
Recap
To summarize, the steps to write a linear function with given values are:
- Identify the given points or slope.
- Calculate the slope using the provided points.
- Use the point-slope form to derive the equation.
- Simplify to the slope-intercept form.
Now you can confidently write a linear function using any two points or a point and a slope!
