There are three different forms that can be used to write a linear function. The three forms are: function form (slope-intercept), point-slope form, and standard form. There are certain pieces of information that are needed to write a function in each of the three forms.
Slope-Intercept Form (Function Form)
Slope-Intercept Form (Function Form)
When writing a function in slope-intercept form, we use the equation \( y = mx + b \), where \( m \) represents the slope, and \( b \) represents the \(y\)-intercept.
The two pieces of information needed to write a linear equation in slope-intercept form are the slope and the \(y\)-intercept. Sometimes these pieces of information are easily identified, other times these pieces might require a little more work to find. For example, if you were given the graph below, you can use the graph itself to identify the slope (rise over run) and y-intercept (where the line crosses the \(y\)-axis) in order to write the equation of the line that is represented by the graph. Since the slope is \( 2 \) and the y-intercept is located at \( (0, -4) \), the equation of the line in the graph would be written as \( y = 2x - 4 \) or \( f(x) = 2x - 4 \). We simply substituted the \( 2 \) for \( m \) and the \( -4 \) for the \( b \) value.
But what if you are not given a graph? How would you write a linear equation in slope-intercept form? Take a look at the video below!
You can also write a linear function in slope-intercept (function) form by first writing an equation in point-slope form and then CONVERTING the equation into slope-intercept form. Watch the video below for an example using the same points as the video above.
Point-Slope Form
In order to write a linear equation in point-slope form, you need to just that....a point and the slope! Once you have these two pieces of information, you want to write the equation using the point-slope equation.
In order to write a linear equation in point-slope form, you need to just that....a point and the slope! Once you have these two pieces of information, you want to write the equation using the point-slope equation.
The point-slope form of an equation is \( y- y_1 = m(x-x_1 ) \), where \( m \) represents the slope and \( (x_1, y_1) \) represents a point on the line. The appropriate mathematical way of saying \( x_1 \) in the equation above is "x sub 1", but sometimes we just say "x one".
To see how to write an equation in point-slope form, watch this video.
Standard Form
The general equation for standard form is: \( Ax + By = C \), where \(A, B, \) and \( C \) represent constants.
When writing an equation in standard form, the \(x\)-term and \(y\)-term are on the same side of the equal sign, and the constant term is on the other. Learn how to write a linear equation in standard form by watching the video below.
Why write an equation in standard form? Take a look at Writing Linear Functions Target B Guided Learning!
Quick Check
1) Write an equation of a line in function form that passes through the points \( (6, -2) \) and \( (4, -8) \).
2) Write an equation using each of the three forms of the line that represents the graph.
1) Write an equation of a line in function form that passes through the points \( (6, -2) \) and \( (4, -8) \).
2) Write an equation using each of the three forms of the line that represents the graph.
3) Write an equation of a line in standard form that has a slope of \( \large\frac{1}{3} \) and passes through the point \( (-9, 5) \).