constant rate
Each linear function has its own constant rate of change.
Suppose is a linear function. Let and be numbers in the domain of . Then and are the corresponding range values.
Since is a linear function, we know that . And, this works for ANY two domain numbers.
Otherwise, it is not a linear function.
Somewhere in history, became a popular choice to represent the constant rate of change of a linear function.
No matter which two numbers you select from the domain of , the rate of change always turns out to be . Each linear function has its own - its own constant rate of change.
A Formula
Let be a linear function. That means it has is own constant rate of change. Let’s call
it .
Let be one specific pair in .
Let represent any other pair in the function . Then the rate of change from to must equal .
Since is a pair in , we know that . And, since represents any other pair in , we know that . Replacing these in the equation for constant slope gives
Solving this for gives
Note: In advanced mathematics, a similar idea called linear maps are required to include the pair . This would result in We will not require this for Calculus.
Suppose is a linear function with constant rate of change equal to and is one pair in
.
Create a formula for .
The template tell us that a formula for looks like
We could multiply this out and collect like terms and obtain the equivalent formula
Perhaps, we do not like as the variable for our formula. Perhaps suits our situation better.
Or, .
Or, .
It is always advantageous to select a variable that is a nice reminder of the domain measurement.
Suppose is a linear function containing and .
Then the template: will need a rate of change
A formula for is
A Graph
A linear function has a line as its graph. The line includes a point for each pair in the function. And, since it is a line, only two points are needed to draw the graph. Any two distinct points will do.
We used the point to create the equation. We could also have used .
Same equation. You can use any point on the line.
Linear Equations
A linear equation in and is an equation that is equivalent to
and are the variables.
and are constants called coefficients.
is called the constant term.
We often write solution pairs as an ordered pair: .
One of the solution numbers is designated for and one is designated for . (That is
what ordered means.) Upon substituting these numbers into the equation for and ,
the resulting statement is a true statement, i.e. the equation is satisfied.
If the order is not understood, then we might write .
With the order understood, each solution pair can be interpreted as coordinates for a point on the Cartesian plane and plotted as a dot.
Consider the linear equation .
Let’s select to be the first variable and the second variable.
With this agreement, is a solution to the equation. can be viewed as a point and plotted as a dot on the Cartesian plane.
If we plot a dot for each and every solution pair for the equation, then we obtain the graph of the equation.
The graph of a linear equation is a line, which can be drawn using only two points from two solution pairs.
Graphs of lines don’t care which axis is “vertical” and which axis is “horizontal”. The
graph is just a collection of points whose coordinates satisfy the linear equation. The
situation changes once we wish to interpret one of the variables as a function of the
other variable.
Once we designate one of the variables as the dependent variable or the function,
then its axis becomes the “vertical” axis. The other axis measures the independent
variable and represents the domain of the function.
Except for vertical lines, all lines are graphs of linear functions.
In the example above, can be rewritten as .
To emphasize that we are now thinking in terms of functions, we might write .
We could just as easily have chosen as the domain variable and as the function variable. In this case, can be rewritten as . We might write this as
In either case, the function formula matches the “” template. is the slope of the line as well as the rate of change of the function.
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more examples can be found by following this link
More Examples of Linear Functions