# 2.1 Linear functions  (Page 8/17)

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## 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 $b=1000.$

Rather than solving for $m,$ 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.

$P\left(w\right)=40w+1000$

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 $\left(2,1080\right)$ and $\left(6,1240\right)$

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 $f\left(x\right)=mx+b,$ and solve for $b.$

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, $x$ months since the measurements began. Write a linear function, $H\left(x\right),$ where $x$ is the number of months since the start of the experiment.

 $x$ 0 2 4 8 12 $H\left(x\right)$ 12.5 13.5 14.5 16.5 18.5

$H\left(x\right)=0.5x+12.5$

Access this online resource for additional instruction and practice with linear functions.

## Key equations

 slope-intercept form of a line $f\left(x\right)=mx+b$ slope point-slope form of a line $y-{y}_{1}=m\left(x-{x}_{1}\right)$

## Key concepts

• 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 $m$ and initial value $b$ 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] .

how to convert general to standard form with not perfect trinomial
Rectangle coordinate
how to find for x
it depends on the equation
Robert
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama