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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 ( w ) = 40 w + 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 ( 2 , 1080 ) and ( 6 , 1240 )

m = 1240 − 1080 6 − 2    = 160 4    = 40
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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 ( x ) = m x + 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 ( x ) , where x is the number of months since the start of the experiment.

x 0 2 4 8 12
H ( x ) 12.5 13.5 14.5 16.5 18.5

H ( x ) = 0.5 x + 12.5

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

Key equations

slope-intercept form of a line f ( x ) = m x + b
slope m = change in output (rise) change in input (run) = Δ y Δ x = y 2 − y 1 x 2 − x 1
point-slope form of a line y − y 1 = m ( x − x 1 )

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] .

Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
Widad Reply
can you send the book attached ?
Ariel
?
Ariel
What is economics
Widad Reply
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn Reply
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn
what is ecnomics
Jan Reply
this is the study of how the society manages it's scarce resources
Belonwu
what is macroeconomic
John Reply
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
etc
husaini
difference between firm and industry
husaini Reply
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?
Toofiq Reply
explain standard reason why economic is a science
innocent Reply
factors influencing supply
Petrus Reply
what is economic.
Milan Reply
scares means__________________ends resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Zarshad Reply
Why qualify 28 supplies
Milan
what are explicit costs
Nomsa Reply
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
David Reply
economic overview notes
Amahle Reply
identify a demand and a supply curve
Salome Reply
i don't know
Parul
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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