Using tabular form to write an equation for a linear function
[link] relates the number of rats in a population to time, in weeks. Use the table to write a linear equation.
w , number of weeks
0
2
4
6
P(w) , number of rats
1000
1080
1160
1240
We can see from the table that the initial value for the number of rats is 1000, so
Rather than solving for
we can tell from looking at the table that the population increases by 80 for every 2 weeks that pass. This means that the rate of change is 80 rats per 2 weeks, which can be simplified to 40 rats per week.
If we did not notice the rate of change from the table we could still solve for the slope using any two points from the table. For example, using
and
Is the initial value always provided in a table of values like
[link] ?
No. Sometimes the initial value is provided in a table of values, but sometimes it is not. If you see an input of 0, then the initial value would be the corresponding output. If the initial value is not provided because there is no value of input on the table equal to 0, find the slope, substitute one coordinate pair and the slope into
and solve for
A new plant food was introduced to a young tree to test its effect on the height of the tree.
[link] shows the height of the tree, in feet,
months since the measurements began. Write a linear function,
where
is the number of months since the start of the experiment.
The ordered pairs given by a linear function represent points on a line.
Linear functions can be represented in words, function notation, tabular form, and graphical form. See
[link] .
The rate of change of a linear function is also known as the slope.
An equation in the slope-intercept form of a line includes the slope and the initial value of the function.
The initial value, or
y -intercept, is the output value when the input of a linear function is zero. It is the
y -value of the point at which the line crosses the
y -axis.
An increasing linear function results in a graph that slants upward from left to right and has a positive slope.
A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.
A constant linear function results in a graph that is a horizontal line.
Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant. See
[link] .
The slope of a linear function can be calculated by dividing the difference between
y -values by the difference in corresponding
x -values of any two points on the line. See
[link] and
[link] .
The slope and initial value can be determined given a graph or any two points on the line.
One type of function notation is the slope-intercept form of an equation.
The point-slope form is useful for finding a linear equation when given the slope of a line and one point. See
[link] .
The point-slope form is also convenient for finding a linear equation when given two points through which a line passes. See
[link] .
The equation for a linear function can be written if the slope
and initial value
are known. See
[link] ,
[link] , and
[link] .
A linear function can be used to solve real-world problems. See
[link] and
[link] .
A linear function can be written from tabular form. See
[link] .
